Luigi Chierchia, Roma Tre University, Italy

KAM THEORY FOR SECONDARY TORI

Joint work with **Luca Biasco**

Date: 2019-09-20 (Friday); Time: 14:15-14:55; Location: building B-7, room 1.8.

Abstract

As well known, classical KAM (Kolmogorov, Arnold, Moser) theory deals with the persistence, under small perturbations, of real-analytic (or smooth) Lagrangian tori for nearly-integrable non-degenerate Hamiltonian systems. In this talk I will present a new *uniform* KAM theory apt to deal also with secondary tori, i.e., maximal invariant tori (with different homotopy) "generated" by the perturbation (and that do not exist in the integrable limit). The word "uniform" means that primary and secondary tori are constructed simultaneously; in particular, in the case of Newtonian mechanical systems on \(\textbf{T}^d\), it is proven that, for generic perturbations, the union of primary and secondary tori leave out a region of order \(\varepsilon |\log \varepsilon|^a\), if \(\varepsilon\) is the norm of the perturbation, in agreement (up to the logarithmic correction) with a conjecture by Arnold, Kozlov and Neishtadt.

Some of these results have been announced in the note [1].

References

- L. Biasco, L. Chierchia,
*On the measure of Lagrangian invariant tori in nearly-integrable mechanical systems*, Rend. Lincei Mat. Appl.**26**(2015), 423-432.

Rafael de la Llave, Georgia Institute of Technology, USA

SOME GEOMETRIC MECHANISMS FOR ARNOLD DIFFUSION

Date: 2019-09-19 (Thursday); Time: 15:00-15:40; Location: building B-7, room 1.8.

Abstract

We consider the problem whether small perturbations of integrable mechanical systems can have very large effects. Since the work of Arnold in 1964, it is known that there are situations where the perturbations can accumulate. This can be understood by noting that the small perturbations generate some invariant structures that, with their stable and unstable manifolds can cover a large region in phase space. We will present recent developments in identifying these invariant objects, both in finite and in infinite dimensions.

Desmond Higham, University of Edinburgh, UK

DIFFERENTIAL EQUATIONS FOR NETWORK CENTRALITY

Date: 2019-09-19 (Thursday); Time: 14:15-14:55; Location: building B-7, room 1.8.

Abstract

I will derive and discuss two circumstances where ODEs arise in the study of large, complex networks. In both cases, the overall aim is to identify the most important nodes in a network-this task is useful, for example, in digital marketing, security and epidemiology. In one case, we define our node centrality measure using the concept of nonbacktracking walks. This requires us to derive an expression for an exponential-type generating function associated with the walk counts of different length. Solving the ODE leads to a computationally useful characterisation of the centrality measure. In another case, we are presented with a time-ordered sequence of networks; for example, recording who emailed who over each one-minute time-window. Here, by considering the asymptotic limit as the window size tends to zero, we arrive at a limiting ODE that may be treated with a numerical method. Results for both algorithms will be illustrated on real network examples.

Hinke Osinga, University of Auckland, New Zealand

ROBUST CHAOS: A TALE OF BLENDERS, THEIR COMPUTATION, AND THEIR DESTRUCTION

Joint work with **Stephanie Hittmeyer, Bernd Krauskopf, and Katsutoshi Shinohara**

Date: 2019-09-18 (Wednesday); Time: 10:40-11:20; Location: building B-7, room 1.8.

Abstract

A blender is an intricate geometric structure of a three- or higher-dimensional diffeomorphism [1]. Its characterising feature is that its invariant manifolds behave as geometric objects of a dimension that is larger than expected from the dimensions of the manifolds themselves. We introduce a family of three-dimensional Hénon-like maps and study how it gives rise to an explicit example of a blender [2, 3]. We employ our advanced numerical techniques to present images of blenders and their associated one-dimensional stable manifolds. Moreover, we develop an effective and accurate numerical test to verify what we call the \(\textit{carpet property}\) of a blender. This approach provides strong numerical evidence for the existence of the blender over a large parameter range, as well as its disappearance and geometric properties beyond this range. We conclude with a discussion of the relevance of the carpet property for chaotic attractors.

References

- C. Bonatti, S. Crovisier, L.J. Díaz, A. Wilkinson,
*What is... a blender?*, Not. Am. Math. Soc.**63**(2016), 1175-1178. - L.J. Díaz, S. Kiriki, K. Shinohara,
*Blenders in centre unstable Hénon-like families: with an application to heterodimensional bifurcations*, Nonlinearity**27**(2014), 353-378. - S. Hittmeyer, B. Krauskopf, H.M. Osinga, K. Shinohara,
*Existence of blenders in a Hénon-like family: geometric insights from invariant manifold computations*, Nonlinearity**31**(2018), R239-R267.

Vladimir Protasov, University of L'Aquila, Italy & Lomonosov Moscow State University, Russia

THE JOINT SPECTRAL RADIUS AND FUNCTIONAL EQUATIONS: A RECENT PROGRESS

Date: 2019-09-17 (Tuesday); Time: 14:15-14:55; Location: building B-7, room 1.8.

Abstract

Joint spectral radius of matrices have been used since late eighties as a measure of stability of linear switching dynamical systems. Nearly in the same time it has found important applications in the theory of refinement equations (linear difference equations with a contraction of the argument), which is a key tool in the construction of compactly supported wavelets and of subdivision schemes in approximation theory and design of curves and surfaces. However, the computation or even estimation of the joint spectral radius is a hard problem. It was shown by Blondel and Tsitsiklis that this problem is in general algorithmically undecidable. Nevertheless recent geometrical methods [1,2,3,4] make it possible to efficiently estimate this value or even find it precisely for the vast majority of matrices. We discuss this issue and formulate some open problems.

References

- N. Guglielmi, V.Yu. Protasov,
*Exact computation of joint spectral characteristics of matrices*, Found. Comput. Math**13**(2013), 37-97. - C. Möller, U. Reif,
*A tree-based approach to joint spectral radius determination*, Linear Alg. Appl.**563**(2014), 154-170. - N. Guglielmi, V.Yu. Protasov,
*Invariant polytopes of linear operators with applications to regularity of wavelets and of subdivisions*, SIAM J. Matrix Anal.**37**(2016), 18-52. - T. Mejstrik,
*Improved invariant polytope algorithm and applications*, arXiv:1812.03080.

Emmanuel Trélat, Sorbonne Université, France

OPTIMAL CONTROL AND APPLICATIONS TO AEROSPACE

Date: 2019-09-16 (Monday); Time: 14:15-14:55; Location: building B-7, room 1.8.

Abstract

I will report on nonlinear optimal control theory and show how it can be used to address problems in aerospace, such as orbit transfer. The knowledge resulting from the Pontryagin maximum principle is in general insufficient for solving adequately the problem, in particular due to the difficulty of initializing the shooting method. I will show how the shooting method can be successfully combined with numerical homotopies, which consist of deforming continuously a problem towards a simpler one. In view of designing low-cost interplanetary space missions, optimal control can also be combined with dynamical system theory, using the nice dynamical properties around Lagrange points that are of great interest for mission design.

Warwick Tucker, Uppsala University, Sweden

SMALL DIVISORS AND NORMAL FORMS

Joint work with **Zbigniew Galias**

Date: 2019-09-18 (Wednesday); Time: 11:25-12:05; Location: building B-7, room 1.8.

Abstract

In this talk, we will discuss the computational challenges of computing trajectories of a non-linear ODE in a region close to a saddle-type fixed-point. By introducing a carefully selected close to identity change of variables, we can bring the non-linear ODE into an "almost" linear system. This normal form system has an explicit transfer-map that transports trajectories away from the fixed point in a controlled manner. Determining the domain of existence for such a change of variables poses some interesting computational challenges. The proposed method is quite general, and can be extended to the complex setting with spiral saddles. It is also completely constructive which makes it suitable for practical use. We illustrate the use of the method by a few examples.

Walter Van Assche, KU Leuven, Belgium

ORTHOGONAL POLYNOMIALS AND PAINLEVÉ EQUATIONS

Date: 2019-09-17 (Tuesday); Time: 15:00-15:40; Location: building B-7, room 1.8.

Abstract

Painlevé equations are nonlinear differential equations for which the branch points do not depend on the initialcondition (no movable branch points). There are also discrete Painlevé equations which are non-linearrecurrence relations with enough structure (symmetry and geometry) that make them integrable.Both the discrete and continuous Painlevé equations appear in a natural way in the theory of orthogonalpolynomials. The recurrence coefficients of certain families of orthogonal polynomials often satisfya discrete Painlevé equation. The Toda equations describing the movement of particles with an exponentialinteraction with their neighbors, is equivalent to an exponential modification \(e^{xt}\, d\mu(x)\) of the orthogonality measure \(d\mu \) for a family of orthogonal polynomials, and the corresponding recurrencecoefficients satisfy the Toda equations, which is a system of differential-difference equations. Combining this with the discrete Painlevé equations then gives a Painlevé differential equation.We will illustrate this by a number of examples. The relevant solutions of these Painlevé equationsare usually in terms of known special functions, such as the Bessel functions, the Airy function, parabolic cylinder functions, or (confluent) hypergeometric functions.

References

- W. Van Assche,
*Orthogonal Polynomials and Painlevé Equations*, Australian Mathematical Society Lecture Notes**27**, Cambridge University Press, (2018).

Sjoerd Verduyn Lunel, Utrecht University, Netherlands

DELAY EQUATIONS AND TWIN SEMIGROUPS

Joint work with **Odo Diekmann**

Date: 2019-09-16 (Monday); Time: 15:00-15:40; Location: building B-7, room 1.8.

Abstract

A delay equation is a rule for extending a function of time towards the future on the basis of the (assumed to be) known past. By translation along the extended function (i.e., by updating the history), one defines a dynamical system. If one chooses as state-space the continuous initial functions, the translation semigroup is continuous, but the initial data corresponding to the fundamental solution is not contained in the state space.

In ongoing joint work with Odo Diekmann, we choose as state space the space of bounded Borel functions and thus sacrifice strong continuity in order to gain a simple description of the variation-of-constants formula.

The aim of the lecture is to introduce the perturbation theory framework of twin semigroups on a norming dual pair of spaces, to show how renewal equations fit in this framework and to sketch how neutral equations can be covered. The growth of an age-structured population serves as a pedagogical example.

Jiangong You, Nankai University, China

DYNAMICAL SYSTEM APPROACH TO SPECTRAL THEORY OF QUASI-PERIODIC SCHRÖDINGER OPERATORS

Date: 2019-09-20 (Friday); Time: 15:00-15:40; Location: building B-7, room 1.8.

Abstract

The spectral theory of quasiperiodic operators is a fascinating field which continuously attracts a lot of attentions for its rich background in quantum physics as well as its rich connections with many mathematical theories and methods. In this talk, I will briefly introduce the problems in this field and their connections with dynamical system. I will also talk about some recent results joint with Avila, Ge, Leguil, Zhao and Zhou on both spectrum and spectral measure by reducibility theory in dynamical systems.

References

- A. Avila, J. You and Q. Zhou,
*Sharp phase transitions for the almost Mathieu operator*, Duke Math. J.**166**(2017), 2697-2718. - A. Avila, J. You and Q. Zhou,
*The dry Ten Martini problem in the non-critical case*, Preprint. - M. Leguil, J. You, Z. Zhao, Q. Zhou,
*Asymptotics of spectral gaps of quasi-periodic Schrödinger operators*, arXiv:1712.04700. - L. Ge, J. You and Q. Zhou,
*Exponential dynamical localization: Criterion and applications*, arXiv:1901.04258. - L. Ge, J. You,
*Arithmetic version of Anderson localization via reducibility*, Preprint.

Everaldo Bonotto, University of São Paulo, Brazil

NEW TRENDS ON GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS

Joint work with **Marcia Federson and Rodolfo Collegari**

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-7, room 1.9.

Abstract

The theory of generalized ordinary differential equations lies on the fact that these equations encompass various types of other equations such as ordinary differential equations (ODEs), impulsive differential equations (IDEs), measure differential equations (MDEs), functional differential equations and dynamic equations on time scales. Moreover, the theory of generalized ordinary differential equations deals with problems, especially, when the functions involved have many discontinuities and/or are of unbounded variation.

In this talk, we present the general theory of generalized ordinary differential equations and also the most recent results on this topic. In special, we show that this theory also encompass the stochastic differential equations.

References

- M. Federson, Š. Schwabik,
*Generalized ODE approach to impulsive retarded differential equations*, Differential Integral Equations**11**(2006), 1201-1234. - Š. Schwabik,
*Generalized Ordinary Differential Equations*, World Scientific, Singapore, Series in real Anal., vol. 5, 1992.

Alberto Boscaggin, University of Turin, Italy

GENERALIZED PERIODIC SOLUTIONS TO PERTURBED KEPLER PROBLEMS

Joint work with **Walter Dambrosio, Duccio Papini, Rafael Ortega, and Lei Zhao**

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-7, room 1.9.

Abstract

For the perturbed Kepler problem
\[
\ddot x = - \frac{x}{\vert x \vert^3} + \nabla_x W(t,x), \qquad x \in \mathbb{R}^d,
\]
with \(d =2\) or \(d = 3\), we discuss the existence of periodic solutions, possibly interacting with the singular set (\(x = 0\)).
First, a suitable notion of *generalized* solution is introduced, based on the theory of regularization of collisions in Celestial Mechanics; second, existence and multiplicity results are provided, with suitable assumptions on the perturbation term \(W\), by the use of sympletic and variational methods.

References

- A. Boscaggin, W. Dambrosio, D. Papini,
*Periodic solutions to a forced Kepler problem in the plane*, Proc. Amer. Math. Soc., to appear. - A. Boscaggin, R. Ortega, L. Zhao,
*Periodic solutions and regularization of a Kepler problem with time-dependent perturbation*, Trans. Amer. Math. Soc., to appear.

Teresa Faria, University of Lisbon, Portugal

GLOBAL DYNAMICS FOR NICHOLSON’S BLOWFLIES SYSTEMS

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-7, room 1.9.

Abstract

We study the global asymptotic behaviour of solutions for a Nicholson's blowflies system with patch structure and multiple discrete delays: \[ \begin{split} x_i'(t)=-d_i(t)x_i(t)+\sum_{j=1,j\ne i}^n a_{ij}(t)x_j(t)+\sum_{k=1}^m \beta_{ik}(t) x_i(t-\tau_{ik}(t))e^{-c_{i}(t)x_i(t-\tau_{ik}(t))},\\ \hskip 30mm i=1,\dots,n,\ t\ge 0, \end{split}\tag{1} \] where all the coefficient and delay functions are continuous, nonnegative and bounded, \(d_i(t)>0,c_i(t)\ge c_i>0\) and \(\beta_i(t):=\sum_{k=1}^m \beta_{ik}(t) >0\) for \(t\ge 0,\) \(i,j=1,\dots, n, k=1,\dots, m\). For the autonomous version of (1), an overview of results concerning the total or partial extinction of the populations, uniform persistence, existence and absolute global asymptotic stability of a positive equilibrium is presented, see [3, 4]. A criterion for the global attractivity of the positive equilibrium depending on the size of delays is also given [2], extending results in [1]. Most of these results rely on some properties of the so-called community matrix and on the specific shape of the nonlinearity.

For non-autonomous systems (1), sufficient conditions for both the extinction of the populations in all the patches and the permanence of the system were established in [3]. In this case, (1) is treated as a perturbation of the linear homogeneous cooperative ODE system \(x_i'(t)=-d_i(t)x_i(t)+\sum_{j=1,j\ne i}^n a_{ij}(t)x_j(t)\ (1\le i\le n)\), for which conditions for its asymptotic stability are imposed; although the nonlinearities in (1) are non-monotone, techniques of cooperative DDEs are used.

References

- H.A. El-Morshedy, A. Ruiz-Herrera,
*Geometric methods of global attraction in systems of delay differential equations*, J. Differential Equations**263**(2017), 5968-5986. - D. Caetano, T. Faria,
*Stability and attractivity for Nicholson systems with time-dependent delays*, Electron. J. Qual. Theory Differ. Equ.**2017**(2017), Paper no. 63, 19 pp. - T. Faria, R. Obaya, A.M. Sanz,
*Asymptotic behaviour for a class of non-monotone delay differential systems with applications*, J. Dynam. Differential Equations**30**(2018), 911-935. - T. Faria, G. Röst,
*Persistence, permanence and global stability for an \(n\)-dimensional Nicholson system*, J. Dynam. Differential Equations**26**(2014), 723-744.

Guglielmo Feltrin, University of Udine, Italy

PARABOLIC ARCS FOR TIME-DEPENDENT PERTURBATIONS OF THE KEPLER PROBLEM

Joint work with **Alberto Boscaggin, Walter Dambrosio, and Susanna Terracini**

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-7, room 1.9.

Abstract

We prove the existence of parabolic arcs with prescribed asymptotic direction for the equation \[ \ddot{x} = - \dfrac{x}{\lvert x \rvert^{3}} + \nabla W(t,x), \qquad x \in \mathbb{R}^{d}, \] where \(d \geq 2\) and \(W\) is a (possibly time-dependent) lower order term, for \(\vert x \vert \to +\infty\), with respect to the Kepler potential \(1/\vert x \vert\). The result applies to the elliptic restricted three-body problem and, more generally, to the restricted \((N+1)\)-body problem. The proof relies on a perturbative argument, after an appropriate formulation of the problem in a suitable functional space.

Galina Filipuk, University of Warsaw, Poland

ASPECTS OF SPECIAL FUNCTIONS

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-7, room 1.9.

Abstract

Special functions often solve ordinary differential equations. The well-known hypergeometric and Heun functions solve linear second order differential equations, whereas the Painlevé transcendents solve nonlinear second order differential equations. In this talk I shall overview some aspects of linear and nonlinear special functions and their differential equations. I shall also describe connections of linear equations to Okubo type systems.

References

- G. Filipuk, A. Lastra,
*On the solutions of Okubo-type systems*, Preprint 2019. - G. Filipuk, A. Ishkhanyan, J. Dereziński,
*On the Heun equation (tentative title)*, Preprint 2019.

Marlène Frigon, Université de Montréal, Canada

EXISTENCE AND MULTIPLICITY RESULTS FOR SYSTEMS OF FIRST ORDER DIFFERENTIAL EQUATIONS VIA THE METHOD OF SOLUTION-REGIONS

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-7, room 1.9.

Abstract

We present existence and multiplicity results for systems of first order differential equations of the form: \[ \begin{aligned} &u'(t) = f(t,u(t)) \quad \text{for a.e. } t\in [0,T], \\ &u \in \mathcal{B}; \end{aligned} \tag{1} \] where \(f :[0,T]\times \mathbb{R}^N\to \mathbb{R}^N\) is a Carathéodory function and \(\mathcal{B}\) denotes a boundary value condition. No growth conditions will be imposed on \(f\). Even though this problem was widely treated, few multiplicity results can be found in the literature.

In the case where there is only one equation (\(N=1\)), the method of upper and lower solutions is well known and very useful to obtain existence and multiplicity results. In particular, this was done in the pioneering work of Mawhin [4].

Very few multiplicity results were obtained in the case where the system (1) has more than one equation (\(N > 1\)). In [3], Frigon and Lotfipour introduced the notion of strict solution-tubes on which rely their multiplicity results. This method was used in [1] to obtain multiplicity results for systems of differential equations with a nonlinear differential operator.

We will present the method of solution-regions to establish existence and multiplicity results for the system (1). A solution-region will be a suitable set \(R\) in \([0,T] \times \mathbb{R}^N\) for which we will deduce that it contains the graph of viable solutions. We will show that this method generalizes the methods of upper and lower solutions and of solution-tubes. We will introduce also the notion of strict solution-regions and we will give conditions insuring the existence of at least three viable solutions of (1). Many non trivial examples will be presented throughout this presentation to show that the method of solution-regions is a powerful tool to establish the existence of solutions of systems of differential equations.

References

- N. El Khattabi, M. Frigon, N. Ayyadi,
*Multiple solutions of problems with nonlinear first order differential operators*, J. Fixed Point Theory Appl.**17**(2015), 23-42. - M. Frigon,
*Existence and multiplicity results for systems of first order differential equations via the method of solution-regions*, Adv. Nonlinear Stud.**18**(2018), 469-485. - M. Frigon, M. Lotfipour,
*Multiplicity results for systems of first order differential inclusions*, J. Nonlinear Convex Anal.**16**(2015), 1025-1040. - J. Mawhin,
*First order ordinary differential equations with several periodic solutions*, Z. Angew. Math. Phys.**38**(1987), 257-265. - F. Adrián, F. Tojo,
*A constructive approach towards the method of solution-regions*, J. Math. Anal. Appl.**472**(2019), 1803-1819.

Maurizio Garrione, Polytechnic University of Milan, Italy

PERIODIC SOLUTIONS OF THE BRILLOUIN ELECTRON BEAM FOCUSING EQUATION: SOME RECENT RESULTS

Joint work with **Roberto Castelli and Manuel Zamora**

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-7, room 1.9.

Abstract

We present some recent results dealing with the existence of (positive) \(2\pi\)-periodic solutions of the Brillouin electron beam focusing equation \[ \ddot{x} + b(1+\cos t) x = \frac{1}{x}, \tag{1} \] in dependence on the real parameter \(b\). In literature, such a study was particularly stimulated by a conjecture that arose some decades ago, saying that for every \(b \in (0, 1/4)\) there exists a \(2\pi\)-periodic solution of (1). So far, the conjecture was neither shown to be true nor disproved, but the existence results were improved and refined step by step, reaching existence for \(b \in (0, b_0]\), with \(b_0 \approx 0.1645\), having a strong relation with the first stability interval of the associated Mathieu equation \(\ddot{x} + b(1+\cos t) x=0\). In this talk, we will try to make a little step further in understanding the picture for the \(2\pi\)-periodic solvability of (1). On the one hand, we will see that existence may hold true also for values of \(b\) belonging to stability intervals of the Mathieu equation other than the first, explicitly exhibiting one of such intervals. On the other hand, we will show that there exists \(b^* \approx 0.248\) such that existence holds for \(b \in (0, b^*]\) and for \(b=b^*\) the branch of solutions obtained through symmetry extension of Neumann ones undergoes a fold bifurcation, so that, as a by-product, multiplicity of \(2\pi\)-periodic solutions for \(b\) close to \(b^*\) is proved. This raises further questions about (1) and the validity of the related conjecture. The techniques used rely, respectively, on careful winding number estimates and computer-assisted proofs.

References

- V. Bevc, J.L. Palmer, C. Süsskind,
*On the design of the transition region of axisymmetric, magnetically focused beam valves*, J. British Inst. Radio Engineer.**18**(1958), 696-708. - R. Castelli, M. Garrione,
*Some unexpected results on the Brillouin singular equation: fold bifurcation of periodic solutions*, J. Differential Equations**265**(2018), 2502-2543. - M. Garrione, M. Zamora,
*Periodic solutions of the Brillouin electron beam focusing equation*, Commun. Pure Appl. Anal.**13**(2014), 961-975. - P.J. Torres,
*Mathematical models with singularities. A zoo of singular creatures*, Atlantis Press, Paris, 2015.

Hermen Jan Hupkes, University of Leiden, Netherlands

DYNAMICS ON LATTICES

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-7, room 1.9.

Abstract

We study dynamical systems posed on lattices, with a special focus on the behaviour of basic objects such as travelling corners, expanding spheres and travelling waves. Such systems arise naturally in many applications where the underlying spatial domain has a discrete structure. Think for example of the propagation of electrical signals through nerve fibres, where the the myeline coating has gaps at regular intervals. Or the study of magnetic spins arranged on crystal lattices.

Throughout the talk we will explore the impact that the spatial topology of the lattice has on the dynamical behaviour of solutions. We will discuss lattice impurities, the consequences of anistropy and make connections with the field of crystallography.

Renato Huzak, Hasselt University, Belgium

FINITE CYCLICITY OF THE CONTACT POINT IN SLOW-FAST INTEGRABLE SYSTEMS OF DARBOUX TYPE

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-7, room 1.9.

Abstract

Using singular perturbation theory and family blow-up we prove that nilpotent contact points in deformations of slow-fast Darboux integrable systems have finite cyclicity. The deformations are smooth or analytic depending on the region in the parameter space. This paper is a natural continuation of [M. Bobieński, P. Mardesic and D. Novikov, 2013] and [M. Bobieński and L. Gavrilov, 2016] where one studies limit cycles in polynomial deformations of slow-fast Darboux integrable systems, around the ''integrable'' direction in the parameter space. We extend the existing finite cyclicity result of the contact point to analytic deformations, and under some assumptions we prove that the contact point has finite cyclicity around the ''slow-fast'' direction in the parameter space.

Joanna Janczewska, Gdańsk University of Technology, Poland

BIFURCATION OF EQUILIBRIUM FORMS OF AN ELASTIC ROD ON A TWO-PARAMETER WINKLER FOUNDATION

Joint work with **Marek Izydorek, Nils Waterstraat, and Anita Zgorzelska**

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-7, room 1.9.

Abstract

Bifurcation theory is one of the most powerful tools in studying deformations of elastic rods, plates and shells. Numerous works have been devoted to the study of bifurcation in elasticity theory. We consider two-parameter bifurcation of equilibrium states of an elastic rod on a deformable foundation of Winkler's type. The rod is being compressed by forces at the ends. The left end is free, but we require the shear force to vanish. At the right end, we assume the rod is simply supported. Our main theorem shows that bifurcation occurs if and only if the linearization of our problem has nontrivial solutions. In fact our proof, based on the concept of the Brouwer degree, gives more, namely that from each bifurcation point there branches off a continuum of solutions.

References

- M. Izydorek, J. Janczewska, N. Waterstraat, A. Zgorzelska,
*Bifurcation of equilibrium forms of an elastic rod on a two-parameter Winkler foundation*, Nonlinear Anal. Real World Appl.**39**(2018), 451-463. - J. Janczewska,
*Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point*, Cent. Eur. J. Math.**2**(2004), 561-572. - A. Borisovich, J. Dymkowska,
*Elements of Functional Analysis with Applications in Elastic Mechanics*, Gdańsk University of Technology, Gdańsk, 2003 (in Polish).

Víctor Jiménez López, University of Murcia, Spain

ON THE MARKUS-NEUMANN THEOREM

Joint work with **José Ginés Espín Buendía**

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-7, room 1.9.

Abstract

According to a well-known result by L. Markus [2], extended by D.A. Neumann in [3], two continuous surface flows are equivalent if and only if there is a homeomorphism preserving orbits and time directions of their separatrix configurations. In this talk, based on the paper [1], some examples are shown to illustrate that the Markus-Neumann theorem, as stated in the original papers, needs not work. Also, we show how the (nontrivial) gap of the proof can be amended to get a correct (and somewhat more general) version of the theorem.

References

- J. G. Espín Buendía, V. Jiménez López,
*On the Markus-Neumann theorem*, J. Differential Equations**265**(2018), 6036-6047. - L. Markus,
*Global structure of ordinary differential equations in the plane*, Trans. Amer. Math. Soc.**76**(1954), 127-148. - D.A. Neumann,
*Classification of continuous flows on 2-manifolds*, Proc. Amer. Math. Soc.**48**(1975), 73-81.

Wojciech Kryński, Polish Academy of Sciences, Poland

ODES AND GEOMETRIC STRUCTURES ON SOLUTION SPACES

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-7, room 1.9.

Abstract

We shall consider ordinary differential equations (ODEs) from the geometric vewpoint. Our aim is study geometric structures appearing on the solution spces to ODEs. In particular, for the third order ODEs one can get canonical conformal structures on the solution spaces. Higher order generalizations lead to \(GL(2)\)-geometry and, in general, to the so-called causal or cone geometry. In the talk, we shall also present applications to the twistor theory and to the integrable systems of partial differential equaitons.

Yong-Hoon Lee, Pusan National University, South Korea

BIFURCATION OF A MEAN CURVATURE PROBLEM IN MINKOWSKI SPACE ON AN EXTERIOR DOMAIN

Joint work with **Rui Yang**

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-7, room 1.9.

Abstract

We study the existence of positive radial solutions for a mean curvature problem in Minkowski space on an exterior domain. Based on \(C^1\)-regularity of solutions, which is closely related to the property of nonlinearity \(f\) near 0, we make use of the global bifurcation theory to establish some existence results of positive radial solutions when \(f\) is sublinear at \(\infty\).

Gergely Röst, University of Szeged, Hungary & University of Oxford, UK

BLOWFLY EQUATIONS: HISTORY, CURRENT RESEARCH AND OPEN PROBLEMS

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-7, room 1.9.

Abstract

The nonlinear delay differential equation today known as Nicholson's blowfly equation was introduced in 1980 to offer an explanation for a curious dataset that had been found in experiments with a laboratory insect population. Complex dynamics arises due to the interplay of the time delay and a non-monotone feedback.

In addition to being an elegant biological application, this equation has inspired the development of a large number of analytical and topological tools for infinite dimensional dynamical systems, including local and global Hopf-bifurcation analysis for delay differential equations, asymptotic analysis, stability criteria, invariant manifolds, singular perturbation techniques, invariance principles, order preserving semiflows by non-standard cones in Banach spaces, and the study of slowly and rapidly oscillatory solutions.

In this talk we give an overview of these developments, and discuss three current research directions, namely

(i) a more refined model of age-dependent intraspecific competition in pre-adult life stages and its effects on adult population dynamics;

(ii) the effect of environmental heterogeneity on nonlinear oscillations;

(iii) the evolution of maturation periods.

Inbo Sim, University of Ulsan, South Korea

ON THE STUDY OF POSITIVE SOLUTIONS FOR SEMIPOSITONE \(p\)-LAPLACIAN PROBLEMS

Joint work with **Lee, Shivaji, and Son**

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-7, room 1.9.

Abstract

In this talk, I introduce what semipositone problems are and some issues on it. I focus on constructions of a subsolution to show the existence of positive solutions for sublinear (infinite) semipositone problems on bounded domains. Moreover, I discuss the existence and uniqueness of positive radial solutions for sublinear (infinite) semipositone \(p\)-Laplacian problems on the exterior of a ball with nonlinear boundary conditions. This talk is mainly based on joint works with Lee, Shivaji and Son.

Ewa Stróżyna, Warsaw University of Technology, Poland

ANALYTIC PROPERTIES OF THE COMPLETE FORMAL NORMAL FORM FOR THE BOGDANOV–TAKENS SINGULARITY

Joint work with **Henryk Żołądek**

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-7, room 1.9.

Abstract

In [19] a complete formal normal forms for germs of 2-dimensional holomorphic vector fields with nilpotent singularity was obtained. That classification is quite nontrivial (7 cases), but it can be divided into general types like in the case of the elementary singularities. One could expect that also the analytic properties of the normal forms for the nilpotent singularities are analogous to the case of the elementary singularities. This is really true. In the cases analogous to the focus and the node the normal form is analytic. In the case analogous to the nonresonant saddle the normal form is often nonanalytic due to the small divisors phenomenon. In the cases analogous to the resonant saddles (including saddle-nodes) the normal form is nonanalytic due to bad properties of some homological operators associated with the first nontrivial term in the orbital normal form.

References

- A. Algaba, C. García and M. Reyes,
*Invariant curves and integration of vector fields*, J. Diff. Equat.**266**(2019), 1357–1376. - V. I. Arnold,
*Geometrical Methods in the Theory of Differential Equations*, Springer–Verlag, Berlin–Heidelberg–New York, 1983, [Russian: Nauka, Moscow, 1978]. - V.I. Arnold and Yu. S. Ilyashenko,
*Ordinary differential equations*, in: Ordinary Differential Equations and Smooth Dynamical Systems, Springer, Berlin, (1997), 1–148; (in Russian: Fundamental Directions 1, VINITI, Moskva, 1985, 1–146). - A. Baider and J. Sanders,
*Further reduction of the Bogdanov–Takens normal form*, J. Differential Equations**99**(1992), 205–244. - R.I. Bogdanov,
*Local orbital normal forms of vector fields on a plane*, Trans. Petrovski Sem.**5**(1979), 50–85 [in Russian]. - N.N. Brushlinskaya,
*A finiteness theorem for families of vector fields in the neighborhood of a singular point of a Poincaré type*, Funct. Anal. Applic.**5**(1971), 10–15 [Russian]. - A.D. Bryuno,
*Analytic form of differential equations*, I. Trans. Moscow Math. Soc.**25**(1971), 131– 288; II. Trans. Moscow Math. Soc.**26**(1972), 199–239 [Russian: I. Tr. Mosk. Mat. Obs.**25**(1971), 119–262; II. Tr. Mosk. Mat. Obs.**26**(1972), 199–239]. - P. Bonckaert and F. Verstringe,
*Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part*, Ann. Inst. Fourier**62**(2012), 2211–2225. - M. Canalis-Durand and R. Schäfke,
*Divergence and summability of normal forms of systems of differential equations with nilpotent linear part*, Ann. Fac. Sci. Toulouse Math.**13**(2004), 493–513. - X. Gong,
*Integrable analytic vector field with a nilpotent linear part*, Ann. Inst. Fourier**45**(1995), 1449–1470. - Yu. Ilyashenko,
*Divergence of series that reduce an analytic differential equation to linear normal form at a singular point*, Funkts. Anal. Prilozh.**13(3)**(1979), 87–88 [Russian]. - Yu. Ilyashenko and S. Yakovenko,
*Lectures on Analytic Differential Equations*, Graduate Studies in Math.,**86**, Amer. Math. Soc., Providence, 2008. - E. Lombardi and L. Stolovich,
*Normal forms of analytic perturbations of quasi-homogeneous vector fields: rigidity, invariant analytic sets and exponentially small approximation*, Ann. Sci. Éc. Norm. Sup. \(4^{\mathrm{e}}\) série**41**(2010), 659–718. - F. Loray,
*Preparation theorem for codimension one foliations*, Ann. Math.**163**(2006), 709–722. - R. Pérez-Marco,
*Total convergence or general divergence in small divisors*, Commun. Math. Phys.**223**(2001), 451–464. - L. Stolovich and F. Verstringe,
*Holomorphic normal form of nonlinear perturbations of nilpotent vector fields*, Regular Chaotic Dyn.**21**(2016), 410–436. - E. Stróżyna and H. Żołądek,
*The analytic and formal normal form for the nilpotent singularity*, J. Diff. Equations**179**(2002), 479–537. - E. Stróżyna and H. Żołądek,
*Divergence of the reduction to the multidimensional Takens normal form*, Nonlinearity**24**(2011), 3129–3141. - E. Stróżyna and H. Żołądek,
*The complete normal form for the Bogdanov–Takens singularity*, Moscow Math. J.**15**(2015), 141–178. - E. Stróżyna,
*Normal forms for germs of vector fields with quadratic leading part. The polynomial first integral case*, J. Diff. Equations**259**(2015), 709–722. - E. Stróżyna,
*Normal forms for germs of vector fields with quadratic leading part. The remaining cases*, Studia Math.**239**(2017), 6718–6748. - F. Takens,
*Singularities of vector fields*, Publ. Math. IHES**43**(1974), 47–100. - H. Żołądek,
*The Monodromy Group*, Monografie Matematyczne,**67**, Birkhäuser, Basel, 2006.

Satoshi Tanaka, Okayama University of Science, Japan

ON THE UNIQUENESS OF POSITIVE RADIAL SOLUTIONS OF SUPERLINEAR ELLIPTIC EQUATIONS IN ANNULI

Joint work with **Naoki Shioji and Kohtaro Watanabe**

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-7, room 1.9.

Abstract

The Dirichlet problem \[ \left\{ \begin{array}{cl} \Delta u + f(u) =0 & \mbox{in} \ x \in A, \\[1ex] u=0 & \mbox{on} \ \partial A, \end{array} \right. \] is considered, where \(A:=\{x\in \mathbb{R}^N : a\lt |x| \lt b \}\), \(N \in \mathbb{N}\), \(N \ge 2\), \(0 \lt a \lt b \lt\infty \), \(f \in C^1[0,\infty)\), \(f(u) \gt 0\) and \(uf'(u) \ge f(u)\) for \(u \gt 0\).

The uniqueness of radial positive solutions is studied. Hence the boundary value problem \[ u'' + \dfrac{N-1}{r} u' + f(u) = 0, \quad r \in (a,b), \qquad %\\[1ex] u(a) = u(b) = 0 \] is considered. The uniqueness results of positive solutions are shown.

Pedro J. Torres, University of Granada, Spain

PERIODIC SOLUTIONS OF THE LORENTZ FORCE EQUATION

Joint work with **David Arcoya and Cristian Bereanu**

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-7, room 1.9.

Abstract

We consider the existence of \(T\)-periodic solutions of the Lorentz force equation \begin{eqnarray*} \left(\frac{q'}{\sqrt{1-|q'|^2}}\right)'= E(t,q)+q'\times B(t,q) \end{eqnarray*} where the electric and magnetic fields \(E,B\) are written in terms of the electric and magnetic potentials \(V:[0,T]\times\mathbb R^3\to\mathbb R\) and \(W:[0,T]\times\mathbb R^3\to\mathbb R^3\) as \[ E=-\nabla_q V-\frac{\partial W}{\partial t},\qquad B=\mbox{curl}_q\, W. \] Following for instance [3, 4], this is the relativistic equation of motion for a single charge in the fields generated by \(V\) and \(W.\) The above equation is formally the Euler - Lagrange equation of the relativistic Lagrangian \begin{eqnarray*} \mathcal L(t,q,q') = 1 - \sqrt{1 - |q'|^2} + q'\cdot W(t,q) +V(t,q). \end{eqnarray*} and also it is the Hamilton - Jacobi equation of the relativistic Hamiltonian \begin{eqnarray*} \mathcal H(t,p,q)=\sqrt{1+|p-W(t,q)|^2}-1+V(t,q). \end{eqnarray*} The purpose of the talk is to review some methods recently developed for the existence and multiplicity of \(T\)-periodic solutions, by using a topological degree approach [1] or a novel critical point theory [2].

References

- C. Bereanu, J. Mawhin,
*Boundary value problems for some nonlinear systems with singular \(\phi\)-Laplacian*, J. Fixed Point Theor. Appl.**4**(2008), 57-75. - D. Arcoya, C. Bereanu, P.J. Torres,
*Critical point theory for the Lorentz force equation*, Arch. Rational Mech. Anal.**232**(2019), 1685-1724. - R. Feynman, R. Leighton, M. Sands,
*The Feynman Lectures on Physics. Electrodynamics, vol. 2*, Addison-Wesley, Massachusetts, 1964. - L.D. Landau, E.M. Lifschitz,
*he Classical Theory of Fields, Fourth Edition: Volume 2*, Butterworth-Heinemann, 1980.

Jordi Villadelprat, Rovira i Virgili University, Spain

DULAC TIME FOR FAMILIES OF HYPERBOLIC SADDLE SINGULARITIES

Joint work with **David Marín**

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-7, room 1.9.

Abstract

We consider a smooth unfolding of a saddle point and we fix two transverse sections, the first one at the stable separatrix and the second one at the unstable separatrix. The *Dulac time* is the time that spends the flow for the transition from the first to the second transverse section. We present a structure theorem for the asymptotic expansion of the Dulac time, with the principal part expressed in terms of Roussarie's monomial scale, and the remainder having flat properties that are well preserved through the division-derivation algorithm. We also provide explicit formulae for the coefficients of the first monomials in the principal part by means of a new integral operator that generalizes Mellin transform. We explain its applicability in the study of the bifurcation diagram of the period function of the quadratic reversible centers.

Manuel Zamora, University of Oviedo, Spain

SECOND-ORDER INDEFINITE SINGULAR EQUATIONS. THE PERIODIC CASE

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building B-7, room 1.9.

Abstract

In this talk we will discuss the existence of a \(T-\)periodic solution to the second order differential equation \[ u''=\frac{h(t)}{u^{\lambda}}, \] where \(\lambda>0\) and the weight \(h\in L(\mathbb{R}/T\mathbb{Z})\) is a sign-changing function. When \(\lambda\geq 1\), the obtained results have the form of relation between the multiplicities of the zeroes of the weight function \(h\) and the order of the singularity of the nonlinear term. Nevertheless, when \(\lambda\in (0,1)\), the key ingredient to solve the aforementioned problem is connected more with the oscillation and the symmetry aspects of the weight function \(h\) than with the multiplicity of its zeroes.

References

- J. Godoy, M. Zamora,
*A General Result to the Existence of a Periodic Solution to an Indefinite Equation with a Weak Singularity*, J. Dyn. Diff. Equat.**31**(2019), 451–468. - R. Hakl, M. Zamora,
*Periodic solutions to second-order indefinite singular equations*, J. Diff. Equat.**263**(2017), 451–469.

Henryk Żołądek, University of Warsaw, Poland

INVARIANTS OF GROUP REPRESENTATIONS, DIMENSION/DEGREE DUALITY AND NORMAL FORMS OF VECTOR FIELDS

Joint work with **Ewa Stróżyna**

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-7, room 1.9.

Abstract

We develop a constructive approach to the problem of polynomial first integrals for linear vector fields. As an application we obtain a new proof of the theorem of Wietzenböck about finiteness of the number of generators of the ring of constants of a linear derivation in the polynomial ring. Moreover, we propose an alternative approach to the analyticity property of the normal form reduction of a germ of vector field with nilpotent linear part in a case considered by Stolovich and Verstringe.

Xingfu Zou, University of Western Ontario, Canada

DYNAMICS OF DDE SYSTEM GENERALIZED FROM NICHOLSON’S EQUATION TO A TWO-PATCH ENVIRONMENT WITH DENSITY-DEPENDENT DISPERSALS

Joint work with **Chang-Yuan Cheng and Shyan-Shiou Chen**

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-7, room 1.9.

Abstract

We derive a model system that describes the dynamics of a single species over two patches with local dynamics governed by Nicholson's DDE and coupled by density dependent dispersals. Under a biologically meaningful assumption that the dispersal rate during the immature period depends only on the mature population, we are able to analyze model to some extent: well-posedness is confirmed, criteria for existence of a positive equilibrium are obtained, threshold for extinction/persistence is established. We also identify a positive invariant set and establish global convergence of solutions under certain conditions. We find that although the levels of the density-dependent dispersals play no role in determining extinction/persistence, our numerical results show that they can affect, when the population is persistent, the long term dynamics including the temporal-spatial patterns and the final population sizes.

Federico Zullo, University of Brescia, Italy

ON THE DYNAMICS OF THE ZEROS OF SOLUTIONS OF AIRY EQUATION (AND BEYOND)

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-7, room 1.9.

Abstract

We discuss the dynamics of the zeros of entire functions in the complex plane. In particular we present the dependence of the zeros of solutions of the Airy equation on two parameters introduced in the equation. The parameters characterize the general solution of the equation. A system of infinitely many nonlinear evolution differential equations are obtained, displaying interesting properties. The possibility to extend the approach to other entire functions in the complex plane will be discussed.

Christian Aarset, Alpen-Adria Universität Klagenfurt, Austria

BIFURCATIONS IN PERIODIC IDEs

Joint work with **Christian Pötzsche**

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-7, room 2.2.

Abstract

In theoretical ecology, one often models population growth with the help of discrete-time difference equations. One method to account for the effects of dispersal throughout the habitat is to employ *integrodifference equations* [1], or IDEs, as opposed to employing scalar difference equations. Given a compact habitat \(\Omega\subset\mathbb{R}^d\), usually with \(d=1,2,3\), together with some appropriate parameter space \(\Lambda\), we consider IDEs on the form
\[u_{t+1} := \int_\Omega f(\cdot,y,u_t(y),\alpha)dy \tag{1}
\]
with \(u_t\in C(\Omega)\) for all \(t\in\mathbb{N}\), where \(f: \Omega\times\Omega\times\mathbb{R}\times\Lambda\rightarrow\mathbb{R}\) is some appropriate function; a commonly employed form for such \(f\) is e.g. \(f(x,y,z,\alpha):=k(x,y)g(y,z,\alpha)\), where \(k\) is some dispersal kernel (e.g. Laplace, Gaussian) and \(g\) is some parameter-dependent growth function (e.g. Beverton-Holt, Ricker).

One is frequently interested in the stability behaviour of *fixed points*, solutions \(u^*\) of (1). However, certain IDEs, in particular - but not limited to - those using the Ricker growth function, may feature transfer of stability from a branch of fixed points to a branch of two- or higher- *periodic solutions*, solutions of the *iterated equation*. We explore such *flip bifurcations* in details, and generalize this theory to cover bifurcations of *periodic solutions* of any integer period, with the particular goal of formulating our assumptions so that they can easily be verified numerically.

References

- M. Kot, W.M. Schaffer,
*Discrete-Time Growth-Dispersal Models*, Mathematical Biosciences**80**(1986), 109-136.

Artur Babiarz, Silesian University of Technology, Poland

LYAPUNOV SPECTRUM ASSIGNABILITY PROBLEM OF DYNAMICAL SYSTEMS

Joint work with **Irina Banshchikova, Adam Czornik, Evgenii Makarov, Michał Niezabitowski, and Svetlana Popova**

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-7, room 2.2.

Abstract

For discrete linear time-varying systems with bounded coefficients, the pole assignment problem utilizing linear state feedback is discussed. It is shown that uniform complete controllability is sufficient for the Lyapunov exponents being arbitrarily assignable by choosing a suitable feedback. Our aim is to prove that all the systems from the closure (in the topology of pointwise convergence) of all shifts of the original system have assignable Lyapunov spectrum if and only if the original system is uniformly completely controllable. Using an appropriate time-varying linear feedback we obtain sufficient conditions to place the Lyapunov spectrum of the closed-loop system in an arbitrary position within some neighborhood of the Lyapunov spectrum of the free system. Moreover, we prove that diagonalizability, Lyapunov regularity and stability of the Lyapunov spectrum each separately are the required sufficient conditions provided that the open-loop system is uniformly completely controllable.

References

- A. Babiarz, A. Czornik, E. Makarov, M. Niezabitowski, S. Popova,
*Pole placement theorem for discrete time-varying linear systems*, SIAM Journal on Control and Optimization**55**(2017), 671– 692. - A. Babiarz, I. Banshchikova, A. Czornik, E. Makarov, M. Niezabitowski, S. Popova,
*Necessary and sufficient conditions for assignability of the Lyapunov spectrum of discrete linear time-varying systems*, IEEE Transactions on Automatic Control**63**(2018), 3825–3837. - A. Babiarz, I. Banshchikova, A. Czornik, E. Makarov, M. Niezabitowski, S. Popova,
*Proportional local assignability of Lyapunov spectrum of linear discrete time-varying systems*, SIAM Journal on Control and Optimization**57**(2019), 1355–1377.

Stephen Baigent, University College London, UK

CONCAVE AND CONCAVE CARRYING SIMPLICES

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-7, room 2.2.

Abstract

The carrying simplex is a codimension-one invariant hypersurface that is the common boundary of the basins of repulsion of the origin and infinity of both continuous- and discrete-time competitive Kolmogorov systems [1, 2].

Geometrically carrying simplices are ‘nice’ invariant manifolds. They project radially one-to-one and onto the unit probability simplex and are graphs of locally Lipschitz functions. Moreover, in some cases they may be graphs of convex, concave or saddle-like functions [4, 3, 5, 6].

I will introduce the carrying simplex and discuss how the bending of hyperplanes under the map can be used to determine when the carrying simplex is convex or concave.

References

- M.W. Hirsch,
*Systems of differential equations which are competitive or cooperative: III Competing species*, Nonlinearity**1**(1988), 51–71. - M.W. Hirsch,
*On existence and uniqueness of the carrying simplex for competitive dynamical systems*, Journal of Biological Dynamics,**2(2)**(2008), 169–179. - S. Baigent,
*Geometry of carrying simplices of 3-species competitive Lotka-Volterra systems*, Nonlinearity**26**(2013), 1001–1029. - S. Baigent,
*Convexity-preserving flows of totally competitive planar Lotka-Volterra equations and the geometry of the carrying simplex*, Proceedings of the Edinburgh Mathematical Society,**55**(2012), 53–63. - S. Baigent,
*Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems*, Journal of Difference Equations and Applications,**22(5)**(2016), 1–14. - S. Baigent,.
*Convex geometry of the carrying simplex for the May-Leonard map*, Discrete and Continuous Dynamical Systems Series B,**24(4)**(2018) 1697–1723.

Inese Bula and Diāna Mežecka, University of Latvia, Latvia

ON THE SOLUTIONS OF RICCATI DIFFERENCE EQUATION VIA FIBONACCI NUMBERS

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-7, room 2.2.

Abstract

A difference equation of the form \[x_{n+1}= \frac{\alpha + \beta x_n}{A+ Bx_n}, \quad n=0,1,...,\] where the parameters \(\alpha\), \(\beta\), \(A\), \(B\) and the initial condition \(x_0\) are real numbers is called a Riccati difference equation. This equation has been studied in many articles (for example, see general review in [1]). In [2, 3] authors studied special cases of Riccati difference equation whose solutions can be expressed via Fibonacci numbers.

In our talk we consider a Riccati difference equation in the form \[ x_{n+1}= \frac{F_m + F_{m-1} x_n}{F_{m+1} + F_m x_n}, \quad n=0,1,..., \tag{1} \] where \(F_0=0\), \(F_1=1\), ..., \(F_{m+1}=F_{m}+F_{m-1}\), \(m=1,2,...\), are Fibonacci numbers. We show some properties of equations (1), including following result.

**Theorem.** For every \(m=1,2,...\), and every initial condition \(x_0 \neq - \frac{F_{k+1}}{F_{k}}, \, k=1,2,...\), the solution of equation (1) is in the form
\[x_n=\frac{F_{mn}+F_{mn-1}x_0}{F_{mn+1}+F_{mn} x_0},\quad n=1, 2,... \, .\]

References

- M.R.S. Kulenovic, G. Ladas,
*Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures*, Chapman & Hall/CRCBoca, Raton, Florida, 2002. - D.T. Tollu, Y. Yazlik, N. Taskara,
*On the solutions of two special types of Riccati difference equation via Fibonacci numbers*, Advances in Difference Equations**174**(2013), 1-7. - D.T. Tollu, Y. Yazlik, N. Taskara,
*The Solutions of Four Riccati Difference Equations Associated with Fibonacci Numbers*, Balkan Journal of Mathematics**02**(2014), 163-172.

Álvaro Castañeda, University of Chile, Chile

DICHOTOMY SPECTRUM AND ALMOST TOPOLOGICAL CONJUGACY ON NONAUTONOMOUS UNBOUNDED DIFFERENCE SYSTEM

Joint work with **Gonzalo Robledo**

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-7, room 2.2.

Abstract

We will consider the nonautonomous linear system \[ x(n + 1) = A(n)x(n)\tag{1} \] where \(x(n)\) is a column vector of \(\mathbb{R}^d\) and the matrix function \(n \mapsto A(n) \in \mathbb{R}^{d\times d}\) is non singular. We also assume that (1) has an exponential dichotomy on \(\mathbb{Z}\) with projector \(P = I\) (see [1] for a formal definition). We also consider the perturbed system \[ w(n + 1) = A(n)w(n) + f (n, w(n))\tag{2} \] where \(f : \mathbb{Z} \times \mathbb{R}^d \to \mathbb{R}^d\) is continuous in \(\mathbb{R}^d\) is a Lipschitz function such that \(n \mapsto f(n,0)\) is bounded for any \(n \in \mathbb{Z}.\) We will present a result with sufficient conditions ensuring that (1) and (2) are almost topologically equivalent, namely the existence of a map \(H : \mathbb{Z} \times \mathbb{R}^d \to \mathbb{R}^d\) with the following properties: i) For each fixed \(n \in \mathbb{Z},\) the map \(u \mapsto H(n,u)\) is a bijection. ii) For any fixed \(n \in \mathbb{Z},\) the maps \(u \mapsto H(n,u)\) and \(u \mapsto H^{-1}(n,u) = G(n,u)\) are continuous with the possible exception of a set with Lebesgue measure zero. iii) If \(x(n)\) is a solution of (1), then \(H[n,x(n)]\) is a solution of (2). Similarly, if \(w(n)\) is a solution of (2), then \(G[n, w(n)]\) is a solution of (1). This result can also be seen as a generalization of a continuous result obtained by F. Lin in [2].

References

- B. Aulbach, S. Siegmund,
*The dichotomy spectrum for noninvertible systems of linear difference equations*, J. Diff. Eqs. Appl.**7**(2001), 895–913. - F. Lin,
*Hartman’s linearization on nonautonomous unbounded system*, Nonlinear Analysis**66**(2007), 38–50.

Adam Czornik, Silesian University of Technology, Poland

DISCRETE TIME-VARYING FRACTIONAL LINEAR EQUATIONS AS VOLTERRA CONVOLUTION EQUATIONS

Joint work with **Pham The Anh, Artur Babiarz, Michał Niezabitowski, and Stefan Siegmund**

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-7, room 2.2.

Abstract

We study the discrete-time fractional linear systems. We show how the different type (Caputo, Riemann-Liouville, forward and backward) of fractional linear difference equation may be converted to Volterra convolution equation. Using this representation we obtain some results about rate of convergency and divergency of solutions and variation of constant formulae. Moreover we show that the norm of difference between two different solution can not tends to infinity faster than a polynomial which degree depends of the fractional order of difference.

**Acknowledgements**

The research was funded by the National Science Centre in Poland granted according to decisions DEC-2015/19/D/ST7/03679 (A.B.) and DEC-2017/25/B/ST7/02888 (A.C.), respectively. The research was supported by the Polish National Agency for Academic Exchange according to the decision PPN/BEK/2018/1/00312/DEC/1 (M.N.). The research was partially supported by an Alexander von Humboldt Polish Honorary Research Fellowship (S.S.).

References

- P. Anh, A. Babiarz, A. Czornik, M. Niezabitowski, S. Siegmund,
*Variation of constant formulas for fractional difference equations*, Archives of Control Sciences**28**(2018), 617-633. - P. Anh, A. Babiarz, A. Czornik, M. Niezabitowski, S. Siegmund,
*Asymptotic properties of discrete linear fractional equations*, Bulletin of the Polish Academy of Sciences: Technical Sciences, accepted for publication.

Zuzana Došlá, Masaryk University, Czech Republic

DISCRETE BOUNDARY VALUE PROBLEMS ON UNBOUNDED DOMAINS

Joint work with **Mauro Marini and Serena Matucci**

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-7, room 2.2.

Abstract

We study the boundary value problem \[ \begin{cases} \Delta(a_{n}\Phi(\Delta x_{n}))=b_nF(x_{n+1}), \ \ n\in\mathbb{N}& \\ x_{1}=c,\quad x_{n}>0,\quad \lim_{n\to\infty} x_n=d,\ & \end{cases} \tag{1}\] where \(\Delta\) is the forward difference operator \(\Delta x_{n}=x_{n+1}-x_{n}\), \(\Phi\) is an increasing odd homeomorphisms, \(\Phi :(-\rho,\rho)\rightarrow(-\sigma,\sigma)\) such that \(\Phi(u)u>0\) for \(u\neq0,\) and \(\rho,\sigma\leq\infty.\) We assume that the sequences \((a_n)\), \((b_n)\) are positive, and boundary conditions satisfy \(c>0\) and \(d\geq 0.\) Solutions of (1) with the terminal condition \(\lim_{n\to\infty} x_n=0\) are usually called decaying solution.

Problem (1) appears in the discretization process for searching spherically symmetric solutions of certain nonlinear elliptic differential equations with generalized phi-Laplacian. The case of noncompact domains seems to be of particular interest in view of applications to radially symmetric solutions to PDEs on the exterior of a ball.

Prototypes of \(\Phi\) are the classical \(\Phi\)-Laplacian, \[ \Phi_{p}(u)=|u|^{p-2}u, \quad p\geq1\, ; \] and when \(\sigma <\infty\) and \(\rho<\infty\) operators \[ \Phi_{C}(u)=\frac{u}{\sqrt{1+|u|^{2}}} \quad \text {and}\quad \Phi_{R}(u)=\frac{u}{\sqrt{1-|u|^{2}}}\, \] arising in studying radial symmetric solutions of partial differential equations with the mean curvature and the relativity operator, respectively.

If \(\Phi\) is the classical \(\Phi\)-Laplacian, the solvability of (1) has been investigated in [2], using properties of the recessive solution to suitable half-linear difference equations, a half-linearization technique and a fixed point theorem in Frechét spaces (see also [3]). Problem (1) is also motivated by [1] where general \(\Phi\) has been considered and extremal solutions have been investigated in case that \((b_n)\) is negative.

References

- M. Cecchi, Z. Došlá, M. Marini,
*Regular and extremal solutions for difference equations with generalized phi-Laplacian*, J. Difference Equ. Appl.**18**(2012), 815-831. - Z. Došlá, M. Marini, S. Matucci,
*Decaying solutions for discrete boundary value problems on the half line*, J. Difference Equ. Appl.**22**(2016), 1244-1260. - M. Marini, S. Matucci, P. Řehák,
*Boundary value problems for functional difference equations on infinite intervals*, Adv. Difference Equ.**2006**Article 31283 (2006), 14 pp.

Ábel Garab, Alpen-Adria Universität Klagenfurt, Austria

DELAY DIFFERENCE EQUATIONS: PERMANENCE AND THE STRUCTURE OF THE GLOBAL ATTRACTOR

Joint work with **Christian Pötzsche**

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-7, room 2.2.

Abstract

In the first part of the talk we give sufficient conditions on the uniform boundedness and permanence of non-autonomous multiple delay difference equations of the form \[x_{k+1}=x_k f_k(x_{k-d},\dots,x_{k-1},x_k),\] where \(f_k\colon D \subseteq (0,\infty)^{d+1}\to (0,\infty)\). This also implies the existence of the global (pullback) attractor, provided the right-hand side is continuous.

In the second part, under some feedback conditions the right-hand side, we give a so-called Morse decomposition of the global attractor for equations of the form \(x_{k+1}=g(x_{k-d}, x_{k}).\) The decomposition is based on an integer valued Lyapunov functional introduced by J. Mallet-Paret and G. Sell.

Both results are applicable for a wide range of single species discrete time population dynamical models, such as models by Ricker, Pielou, Mackey-Glass, Wazewska-Lasota, and Clark.

Ewa Girejko, Bialystok University of Technology, Poland

ON CONSENSUS UNDER DoS ATTACK IN THE MULTIAGENT SYSTEMS

Joint work with **Agnieszka B. Malinowska**

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-7, room 2.2.

Abstract

In the paper multiagent systems under Denial-of-Service (DoS) attack are considered. We provide convergence results to ensure the consensus in the system under the attack. Since DoS attack is usually unpredictable with respect to duration of time and lasts one second or more, we examine the problem on various time domains.

Agnieszka Malinowska, Bialystok University of Technology, Poland

OPTIMAL CONTROL OF FRACTIONAL MULTI-AGENT SYSTEMS

Joint work with **Tatiana Odzijewicz**

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-7, room 2.2.

Abstract

We deal with control strategies for discrete-time fractional multi-agent systems. By using the discrete fractional order operator we introduce memory effects to the considered problem. Necessary optimality conditions for discrete-time fractional optimal control problems with single- and double-summator dynamics are proved. We demonstrate the validity of the proposed control strategy by numerical examples.

A.B. Malinowska is supported by the Bialystok University of Technology grant S/WI/1/2016 and funded by the resources for research by Ministry of Science and Higher Education.

Serena Matucci, University of Florence, Italy

ASYMPTOTIC PROBLEMS FOR SECOND ORDER NONLINEAR DIFFERENCE EQUATIONS WITH DEVIATING ARGUMENT

Joint work with **Zuzana Došlá
**

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-7, room 2.2.

Abstract

A fixed point approach, based on Schauder linearization device in the Frechét space of all the sequences, is presented and compared with corresponding results in the space of continuous functions, extending results in [3]. As an applications, the problem of the existence of the so- called intermediate solutions is analyzed for the half-linear and sublinear Emden-Fowler type equations with deviating argument \[ \Delta(a_n |\Delta x_n|^\alpha \, \mathrm{ sgn } \, \Delta x_n) + b_n |x_{n+q}|^\beta \, \mathrm{ sgn } \, x_{n+q}=0,\tag{1} \] where \(\Delta\) is the forward difference operator, \(a=\{a_n\}, \, b=\{b_n\}\) are positive real sequences, \(0< \beta \le \alpha\) and \(q \in \mathbb Z\). In particular, we analyze the effect of the deviating argument on the existence of unbounded nonoscillatory solutions for (1), by means of a comparison with the equation \[ \Delta(a_n |\Delta y_n|^\alpha \, \mathrm{ sgn } \, \Delta y_n) + b_n |y_{n+1}|^\beta \, \mathrm{ sgn } \, y_{n+1}=0.\tag{2} \] As a consequence, necessary and sufficient conditions for the existence of intermediate solutions for (1) (that is, eventually positive solutions \(x\) s.t. \(\lim_n x_n=+\infty\), \(\lim_n a_n |\Delta x_n|^\alpha=0\)) are given. The results presented generalize some in [1] in case \(\alpha=\beta\), and in [2] when \(\alpha>\beta\).

References

- M. Cecchi, Z. Došlá, M. Marini,
*On the growth of nonoscillatory solutions for difference equations with deviating argument*, Adv. Difference Equ. (2008), Art. ID 505324, 15 pp. - M. Cecchi, Z. Došlá, M. Marini,
*Intermediate solutions for nonlinear difference equations with p- Laplacian*, Adv. Stud. Pure Math.,**53**(2009), 33—40. - M. Marini, S. Matucci, P. Řehàk,
*Boundary value problems for functional difference equations on infinite intervals*, Adv. Difference Equ. (2006), Art. 31283, 14 pp.

Małgorzata Migda, Poznań University of Technology, Poland

ASYMPTOTIC PROPERTIES OF SOLUTIONS TO SUM-DIFFERENCE EQUATIONS OF VOLTERRA TYPE

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-7, room 2.2.

Abstract

Volterra difference equations appeared as a discretization of Volterra integral and integro-differential equations. They also often arise during the mathematical modeling of some real life situations where the current state is determined by the whole previous history. In this talk we consider some difference equations of Volterra type. In particular we discuss the equations of the form \[ \Delta(r_n\Delta x_n)=b_n+\sum_{k=1}^{n}K(n,k)f(x_k). \] We give sufficient conditions for the existence of a solution \(x\) of the above equation with the property \[ x_n=y_n+{\mathrm{o}} (n^s), \] where \(y\) is a given solution of the equation \(\Delta(r_n\Delta y_n)=b_n\) and \(s\in(-\infty,0]\). We show also applications of the obtained results to a linear Volterra equation. Sufficient conditions for the existence of asymptotically periodic solutions will be discussed as well.

References

- J. Migda, M. Migda,
*Asymptotic behavior of solutions of discrete Volterra equations*, Opuscula Math.**36**(2016), 265-278. - J. Migda, M. Migda, M. Nockowska-Rosiak,
*Asymptotic properties of solutions to second-order difference equations of Volterra type*, Discrete Dynamics in Nature and Society (2018), Article ID 2368694, 10 pp. - J. Migda, M. Migda, Z. Zbąszyniak,
*Asymptotically periodic solutions of second order difference equations*, Appl. Math. Comput.**350**(2019), 181-189.

Tatiana Odzijewicz, SGH Warsaw School of Economics, Poland

OPTIMAL LEADER-FOLLOWER CONTROL FOR THE FRACTIONAL OPINION FORMATION MODEL

Joint work with **Ricardo Almeida and Agnieszka B. Malinowska**

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-7, room 2.2.

Abstract

This work deals with an opinion formation model, that obeys a nonlinear system of fractional-order differential equations. We introduce a virtual leader in order to attain a consensus. Sufficient conditions are established to ensure that the opinions of all agents globally asymptotically approach the opinion of the leader. We also address the problem of designing optimal control strategies for the leader so that the followers tend to consensus in the most efficient way. A variational integrator scheme is applied to solve the leader-follower optimal control problem. Finally, in order to verify the theoretical analysis, several particular examples are presented.

References

- R. Almeida, A. B. Malinowska, T. Odzijewicz,
*Optimal leader–follower control for the fractional opinion formation model*, J. Optimiz. Theory App.**182**(2019), 1171–1185.

Kenneth James Palmer, National Taiwan University, Taiwan

EXPONENTIAL DICHOTOMY AND SEPARATION IN LINEAR DIFFERENCE EQUATIONS

Joint work with **Flaviano Battelli**

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-7, room 2.2.

Abstract

We consider linear difference equations \(x(n+1)=A(n)x(n)\), in which \(A(n)\) may not be invertible or bounded. The main issues considered here are robustness (or roughness) and the relation between a triangular system and its corresponding diagonal system. In general, exponential separation is weaker than exponential dichotomy but, for certain systems, it turns out that in some sense exponential separation implies exponential dichotomy. Differences between the differential equations case and the difference equations case are highlighted.

Mihály Pituk, University of Pannonia, Hungary

ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF LINEAR DIFFERENCE EQUATIONS

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-7, room 2.2.

Abstract

In this talk, we will summarize some results on the asymptotic behavior of the positive solutions of linear difference equations. Under appropriate assumptions, we will study the growth rates and the existence of wieghted limits of the positive solutions.

References

- R. Chieocan, M. Pituk,
*Weighted limits for Poincaré difference equations*, Applied Mathematics Letters**49**(2015), 51-57. - R. Obaya, M. Pituk,
*A variant of the Krein-Rutman theorem for Poincaré difference equations*, Journal of Difference Equations and Applications**18**(2012), 1751-1762. - M. Pituk, C. Pötzsche,
*Ergodicity beyond asymptotically autonomous linear difference equations*, Applied Mathematics Letters**86**(2018), 149-156.

Christian Pötzsche, Alpen-Adria Universität Klagenfurt, Austria

GLOBAL ATTRACTIVITY AND DISCRETIZATION IN INTEGRODIFFERENCE EQUATIONS

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-7, room 2.2.

Abstract

Integrodifference equations are popular models in theoretical ecology to describe the temporal evolution and spatial dispersal of populations having nonoverlapping generations. As a contribution to the numerical dynamics of such infinite-dimensional dynamical systems, we establish that global attractivity of periodic solutions is robust under a wide class of spatial discretizations. Beyond robustness also the convergence order of the numerical schemes is preserved.

References

- M. Kot, W. Schaffer,
*Discrete-time growth-dispersal models*, Math. Biosci.**80**(1986), 109-136. - C. Pötzsche,
*Numerical dynamics of integrodifference equations: Basics and discretization errors in a \(C^0\)-setting*, Appl. Math. Comput.**354**(2019), 422-443. - C. Pötzsche,
*Numerical dynamics of integrodifference equations: Global attractivity in a \(C^0\)-setting*, submitted (2019).

Pavel Řehák, Brno University of Technology, Czech Republic

REFINED DISCRETE REGULAR VARIATION AND ITS APPLICATIONS IN DIFFERENCE EQUATIONS

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-7, room 2.2.

Abstract

We introduce a new class of the so-called regularly varying sequences with respect to an auxiliary sequence \(\tau\), and state its properties. This class, on one hand, generalizes regularly varying sequences. On the other hand, it refines them and makes it possible to do a more sophisticated analysis in applications. We show a close connection with regular variation on time scales; thanks to this relation, we can use the existing theory on time scales to develop discrete regular variation with respect to \(\tau\). We reveal also a connection with generalized regularly varying functions. As an application, we study asymptotic behavior of solutions to linear difference equations; we obtain generalization and extension of known results. The theory also yields, as a by-product, a knew view on the Kummer type test for convergence of series, which generalizes, among others, Raabe's test and Bertrand's test.

References

- P. Řehák,
*Refined discrete regular variation and its applications*, to appear in Math. Meth. Appl. Sci., DOI: 10.1002/mma.5670.

Ewa Schmeidel, University of Bialystok, Poland

CONSENSUS OF MULTI-AGENTS SYSTEMS ON ARBITRARY TIME SCALE

Joint work with **Urszula Ostaszewska and Małgorzata Zdanowicz**

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-7, room 2.2.

Abstract

In my talk an emergence of leader-following model based on graph theory on the arbitrary time scales is investigated. It means that the step size is not necessarily constant but it is a function of time. We propose and prove conditions ensuring a leader-following consensus for any time scales using Grönwall inequality. The presented results are illustrated by examples.

References

- U. Ostaszewska, E. Schmeidel, M. Zdanowicz,
*Exponentially stable solution of mathematical model based on graph theory of agents dynamics on time scales*, Adv. Difference Equ., to appear. - U. Ostaszewska, E. Schmeidel, M. Zdanowicz,
*Emergence of consensus of multi-agents systems on time scales*, Miskolc Math. Notes, (to appear, article code: MMN-2704).

Peter Šepitka, Masaryk University, Czech Republic

SINGULAR STURMIAN THEORY FOR WEAKLY DISCONJUGATE LINEAR HAMILTONIAN DIFFERENTIAL SYSTEMS

Joint work with **Roman Šimon Hilscher**

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-7, room 2.2.

Abstract

In this talk we introduce several new results in the Sturmian theory of weakly disconjugate (or equivalently, eventually controllable) linear Hamiltonian systems. We present singular comparison theorems on unbounded intervals for two nonoscillatory systems satisfying the Sturmian majorant condition and the Legendre condition. In particular, we show exact formulas and optimal estimates for the numbers of proper focal points of conjoined bases of these systems. This topic was infrequently studied in the literature and the validity of singular comparison theorems on unbounded intervals for general uncontrollable setting is an open problem so far. The presented results complete and generalize the previously obtained (i) singular Sturmian comparison/separation theorems on unbounded intervals by O. Došlý and W. Kratz in [1], and by the author jointly with R. Šimon Hilscher in [3], (ii) as well as the Sturmian comparison theorems on compact intervals by R. Šimon Hilscher in [4] and by J. Elyseeva in [2].

References

- O. Došlý, W. Kratz,
*Singular Sturmian theory for linear Hamiltonian differential systems*, Appl. Math. Lett.**26**(2013), 1187–1191. - J.V. Elyseeva,
*Comparison theorems for conjoined bases of linear Hamiltonian differential systems and the comparative index*, J. Math. Anal. Appl.**444**(2016), 1260–1273. - P. Šepitka, R. Šimon Hilscher,
*Singular Sturmian separation theorems on unbounded intervals for linear Hamiltonian systems*, J. Differential Equations**266**(2018), 7481–7524. - R. Šimon Hilscher,
*Sturmian theory for linear Hamiltonian systems without controllability*, Math. Nachr.**284**(2011), 831–843.

Stefan Siegmund, Technische Universität Dresden, Germany

A HILBERT SPACE APPROACH TO FRACTIONAL DIFFERENCE EQUATIONS

Joint work with **Pham The Anh, Artur Babiarz, Adam Czornik, Konrad Kitzing, Michał Niezabitowski, Sascha Trostorff, and Hoang The Tuan**

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-7, room 2.2.

Abstract

We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractional powers of the operator \(1 - \tau^{-1}\) with the right shift \(\tau^{-1}\) on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems

References

- Pham The Anh, A. Babiarz, A. Czornik, K. Kitzing, M. Niezabitowski, S. Siegmund, S. Trostorff, Hoang The Tuan,
*A Hilbert space approach to fractional difference equations*, submitted.

Roman Šimon Hilscher, Masaryk University, Czech Republic

THE STORY OF FOCAL POINT IN DISCRETE STURMIAN THEORY

Joint work with **Peter Šepitka**

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-7, room 2.2.

Abstract

We will discuss the development of the concepts of generalized zeros and focal points for second order difference equations and symplectic difference systems in the relation with the validity of the Sturmian separation and comparison theorems. Our aim is to present recent progress in this area by discussing singular Sturmian theory for possibly uncontrollable symplectic difference systems on unbounded intervals. We will also present a simple application of the new concept in disconjugacy criteria for the second order Sturm-Liouville difference equations on unbounded intervals.

References

- C. D. Ahlbrandt, A. C. Peterson,
*Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, and Riccati Equations*, Kluwer Texts in the Mathematical Sciences**16**Kluwer Academic Publishers Group, Dordrecht, 1996. - M. Bohner,
*Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions*, J. Math. Anal. Appl.**199(3)**(1996), 804–826. - M. Bohner, O. Došlý, W. Kratz,
*Sturmian and spectral theory for discrete symplectic systems*, Trans. Amer. Math. Soc.**361(6)**(2009), 3109–3123. - O. Došlý, J. V. Elyseeva, R. Šimon
*Hilscher, Symplectic Difference Systems: Oscillation and Spectral Theory*, Birkhäuser, Basel, 2019 (to appear). - J. V. Elyseeva,
*Comparative index for solutions of symplectic difference systems*, Differential Equations,**45(3)**(2009), 445–459. - J. V. Elyseeva,
*Comparison theorems for symplectic systems of difference equations*, Differential Equations,**46(9)**(2010), 1339–1352. - P. Hartman,
*Difference equations: disconjugacy, principal solutions, Green’s function, complete monotonicity*, Trans. Amer. Math. Soc.**246**(1978), 1–30. - W. Kratz,
*Discrete oscillation*, J. Difference Equ. Appl.**9(1)**(2003), 135–147. - P. Šepitka, R. Šimon Hilscher,
*Recessive solutions for nonoscillatory discrete symplectic systems*, Linear Algebra Appl.**469**(2015), 243–275. - P. Šepitka, R. Šimon Hilscher,
*Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems*, J. Difference Equ. Appl.**23(4)**(2017), 657–698. - P. Šepitka, R. Šimon Hilscher,
*Singular Sturmian separation theorems for nonoscillatory symplectic difference systems*, J. Difference Equ. Appl.**24(12)**(2018), 1894–1934.

Robert Skiba, Nicolaus Copernicus University in Toruń, Poland

A CONTINUATION PRINCIPLE FOR FREDHOLM MAPS AND ITS APPLICATION TO DIFFERENTIAL EQUATIONS

Joint work with **Christian Pötzsche**

Date: 2019-09-20 (Friday); Time: 12:05-12:25; Location: building B-7, room 2.2.

Abstract

In this talk we are going to present an abstract and flexible continuation theorem for zeros of parametrized Fredholm maps between Banach spaces. It guarantees not only the existence of zeros to corresponding equations but also that they form a continuum for parameters from a connected manifold. Our basic tools will be transfer maps and a specific topological degree. Next, we will explain how using an abstract and flexible continuation theorem to find global branches of homoclinic solutions for parametrized nonautonomous ordinary differential equations. Our approach will be based on degree-theoretical arguments. In particular, Landesman-Lazer conditions will be proposed to obtain the existence of homoclinic solutions by means of a nonzero degree.

References

- C. Pötzsche, R. Skiba,
*Global Continuation of Homoclinic Solutions*, Zeitschrift fur Analysis und ihre Anwendungen**37(2)**(2018), 159-187. - C. Pötzsche, R. Skiba,
*A Continuation Principle for Fredholm maps I: Theory and Basics*, submitted. - C. Pötzsche, R. Skiba,
*A Continuation Principle for Fredholm maps II: Application to homoclinic solutions*, submitted.

Zhi-Tao Wen, Shantou University, China

INTEGRABILITY OF DIFFERENCE EQUATIONS WITH BINOMIAL SERIES

Joint work with **Katsuya Ishizaki**

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-7, room 2.2.

Abstract

We consider binomial series \(\sum_{n=0}^\infty a_n z^{\underline{n}}\), where \(z^{\underline{n}}=z(z-1)\cdots(z-n+1)\). Integrability by binomial series is concerned for difference equations. In this talk, we consider a formal solution of a difference equation written by binomial series. Further, we discuss conditions of convergence of these formal solutions to find a sufficient condition for meromorphic solutions, and investigate the order of growth of them. As an application, we construct a difference Riccati equation possessing a transcendental meromorphic solution of order \(1/2\).

Pál Burai, University of Debrecen, Hungary

MEAN LIKE MAPS GENERATED BY PATHOLOGICAL FUNCTIONS

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-7, room 1.8.

Abstract

The goal of this talk is the investigation of quasi-arithmetic expressions (close relatives of quasi-arithmetic means) generated by invertible (not necessary continuous) functions.

The resulted class can contain maps, which are not means, and which are not regular. However, it contains the whole class of quasi-arithmetic means.

References

- S. Balcerzyk,
*Wstęp do algebry homologicznej*, Państwowe Wydawnictwo Naukowe, Warsaw, 1970, Biblioteka Matematyczna, Tom 34. - K. Baron,
*On additive involutions and Hamel bases*, Aequationes Math.,**87(1-2)**(2014), 159-163. - P. Burai,
*An extension theorem for conditionally additive functions and its application for the equality problem of Quasi-arithmetic expressions*, submitted. - J.G. Dhombres and R. Ger,
*Conditional Cauchy equations*, Glas. Mat. Ser. III**13(33)(1)**(1978), 39-62. - Z. Daróczy and L. Losonczi,
*Über die Erweiterung der auf einer Punktmenge additiven Funktionen*, Publ. Math. Debrecen**14**(1967), 239-245. - W. Jabłoński,
*Additive involutions and Hamel bases*, Aequationes Math.**89(3)**(2015), 575-582. - M. Kuczma,
*An introduction to the theory of functional equations and inequalities*, Birkhäuser Verlag, Basel, second edition, 2009. Cauchy’s equation and Jensen’s inequality, Edited and with a preface by Attila Gilányi. - I. Makai,
*Über invertierbare Lösungen der additiven Cauchy-Funktionalgleichung*, Publ. Math. Debrecen**16**(1969), 239-243.

Liviu Cădariu-Brăiloiu, Politehnica University of Timisoara, Romania

APPLICATIONS OF FIXED POINT RESULTS TO THE GENERALIZED HYERS-ULAM STABILITY OF A FUNCTIONAL EQUATION

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-7, room 1.8.

Abstract

In the last time there are emphasised several methods which allows to obtain Hyers-Ulam stability results for large classes of functional, differential and integral equations, in various spaces. For example, some fixed points theorems for operators (not necessarily linear) satisfying suitable very general properties have been proved recently. These results were used to obtain properties of generalized Hyers-Ulam stability, hyperstability, superstability, best constant, for different classes of functional equations.

The aim of this talk is to present an application of such a fixed point theorem for proving generalized Hyers-Ulam stability properties of a functional equation.

References

- J. Brzdęk, J. Chudziak, Z. Páles,
*A fixed point approach to stability of functional equations*, Nonlinear Analysis - TMA**74**(2011), 6728–6732. - J. Brzdęk, L. Cădariu, K. Ciepliński,
*Fixed point theory and the Ulam stability*, J. Function Spaces**2014**(2014), Article ID 829419, 16 pp. - J. Brzdęk, L. Cădariu,
*Stability for a family of equations generalizing the equation of p-Wright affine functions*, Appl. Math. Comput.**276**(2016), 158–171. - K. Ciepliński,
*Applications of fixed point theorems to the Hyers-Ulam stability of functional equations – a survey*, Ann. Funct. Anal.**3(1)**(2012), 151–164.

Jacek Chudziak, University of Rzeszów, Poland

ON SOME APPLICATIONS OF QUASIDEVIATION MEANS

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-7, room 1.8.

Abstract

We show that willingness to accept (WTA) and willingness to pay (WTP) are particular cases of quasideviation means, introduced in [2]. Using this fact and applying some results in [3], we investigate the properties of WTA and WTP related to the experimentally observed disparity between them [1, 4].

References

- J.L. Knetsch, J.A. Sinden,
*Willingness to pay and compensation demanded: experimental evidence of an unexpected disparity in measures of value*, The Quarterly Journal of Economics**99**(1984), 507-521. - Zs. Páles,
*Characterization of quasideviation means*, Acta. Math. Sci. Hungar.**40**(1982), 243-260. - Zs. Páles,
*General inequalities for quasideviation means*, Aequationes Math.**36**(1988), 32-56. - R. Thaler,
*Toward a positive theory of consumer choice*, Journal of Economic Behavior and Organization**1**(1980), 39-60.

Gregory Derfel, Ben-Gurion University of the Negev, Israel

ON THE ASYMPTOTIC BEHAVIOUR OF THE ZEROS OF THE SOLUTIONS OF THE PANTOGRAPH EQUATION

Joint work with **Peter Grabner and Robert Tichy**

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-7, room 1.8.

Abstract

We study asymptotic behaviour of the solutions of the pantograph equation. From this we derive asymptotic formula for the zeros of these solutions.

References

- C. Zhang,
*An asymptotic formula for the zeros of the deformed exponential function*, J. Math. Anal. Appl.**441(2)**(2016), 565-573. - L. Wang, C. Zhang,
*Zeros of the deformed exponential function*, Advances in Mathematics**332**(2018), 311-348. - G. Derfel, P. Grabner and R. Tichy,
*On the asymptotic behaviour of the zeros of the solutions of a functional-differential equation with rescaling*, Operator Theory: Advances and Applications**263**(2018), 281-295.

Davor Dragičević, University of Rijeka, Croatia

NEW CHARACTERIZATIONS OF HYPERBOLICITY FOR LINEAR COCYCLES

Joint work with **Adina Luminita Sasu and Bogdan Sasu**

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-7, room 1.8.

Abstract

We will describe some new characterizations of stability, expansivity and hyperbolicity of linear cocycles developed in [1] which are based on the ideas from subadditive ergodic theory.

References

- D. Dragičević, A.L. Sasu, B. Sasu,
*On the asymptotic behavior of discrete dynamical systems - an ergodic theory approach*, submitted.

Włodzimierz Fechner, Lodz University of Technology, Poland

NEW INEQUALITIES FOR PROBABILITY FUNCTIONS IN THE TWO-PERSON RED-AND-BLACK GAME

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-7, room 1.8.

Abstract

We discuss a model of a two-person, non-cooperative stochastic game, inspired by the discrete version of the red-and-black gambling problem presented by Dubins and Savage [3]. Assume that two players hold certain amounts of money. At each stage of the game they simultaneously bid some part of their current fortune and the probability of winning or loosing depends on their bids. In many models of the red-and-black game it is assumed that the win probability is a function of the quotient of the bid of the first player and the sum of both bids. In the literature some additional properties, like concavity or super-multiplicativity, are assumed in order to ensure that bold and timid strategy is the Nash equilibrium (e.g. in works of Chen and Hsiau [1, 2]). In the talk we propose a generalization in which the probability of winning is a two-variable function which depends on both bids. We introduce two new functional inequalities whose solutions lead to win probability functions for which a Nash equilibrium is realized by the bold-timid strategy.

References

- M.R. Chen, S.R. Hsiau,
*Two-person red-and-black games with bet-dependent win probability functions*, J. Appl. Probab.**43(4)**(2006), 905-915. - M.R. Chen, S.R. Hsiau,
*wo new models for the two-person red-and-black game*, J. Appl. Probab.**47(1)**(2010), 97-108. - L.E. Dubins, L.J. Savage,
*How to gamble if you must. Inequalities for stochastic processes*, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. - W. Fechner,
*New inequalities for probability functions in the two-person red-and-black game*, arXiv:1811.00359 [math.PR].

Żywilla Fechner, Lodz University of Technology, Poland

BASIC FUNCTIONS ON HYPERGROUP-TYPE STRUCTURES

Joint work with **László Székelyhidi**

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-7, room 1.8.

Abstract

The aim of the talk is to present a characterization of functions like exponential monomials, polynomials and moment functions. We are interested in functions defined on some special type of hypergroups like affine groups, double coset hypergroups and hypergroup joins. We also discuss a connection of these functions with spectral synthesis problems.

References

- L. Székelyhidi,
*Functional equations on hypergroups*, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. - Ż. Fechner, L. Székelyhidi,
*Spherical and moment functions on the affine group of \(SU(n)\)*, Acta Mathematica Hungarica**157(1)**(2019), 10–26. - Ż. Fechner, L. Székelyhidi,
*Functional equations on double coset hypergroups*, Annals of Functional Analysis,**8(3)**(2017), 411–423, - Ż. Fechner, L. Székelyhidi,
*Sine functions on hypergroups*, Archiv der Mathematik,**106(4)**(2016), 371–382.

Dorota Głazowska, University of Zielona Góra, Poland

EMBEDDABILITY OF PAIRS OF WEIGHTED QUASI-ARITHMETIC MEANS INTO A SEMIFLOW

Joint work with **Justyna Jarczyk and Witold Jarczyk**

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-7, room 1.8.

Abstract

Let \(I \subset \mathbb{R}\) be an interval. Given any continuous strictly monotonic function
\( f: I \rightarrow \mathbb{R}\) and \(p \in \left( 0,1\right)\) the formula
\[
A^f_p(x,y)=f^{-1}\left(pf(x)+(1-p)f(y)\right),
\]
defines a mean on \(I\) called the *quasi-arithmetic mean generated by* \(f\) *and weighted by* \(p\).

We determine the form of all semiflows of pairs of weighted quasi-arithmetic means, those over positive dyadic numbers as well as those continuous ones. Then the iterability of such pairs is characterized: necessary and sufficient conditions for a given pair of weighted quasi-arithmetic means to be embeddable into a continuous semiflow are given. In particular, it turns out that surprisingly the existence of a square iterative root in the class of such pairs implies the embeddability.

Justyna Jarczyk, University of Zielona Góra, Poland

GAUSSIAN ALGORITHM FOR MAPPINGS BUILT OF PARAMETRIZED MEANS

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-7, room 1.8.

Abstract

After recalling the notion of iterate of a function depending on a parameter, introduced by K. Baron and M. Kuczma in [1], we present a counterpart of Gaussian algorithm for mappings built of parametrized means. We consider also a special case of the so-called random means and describe some specific properties of the limit of their Gauss iterates.

References

- K. Baron, M. Kuczma,
*Iteration of random-valued functions on the unit interval*, Colloq. Math.**37**(1977), 263-269.

Witold Jarczyk, John Paul II Catholic University of Lublin, Poland

GENERALIZED GAUSSIAN ALGORITHM

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-7, room 1.8.

Abstract

The classical Gaussian algorithm runs as follows: taking any \(x,y \in (0, +\infty)\) put \(x_1:=x\), \(y_1:=y\) and \[ x_{n+1}:=A\left(x_n,y_n\right), \quad y_{n+1}:=G\left(x_n,y_n\right), \qquad n \in {\mathbb N}, \] where \(A\) and \(G\) state for the arithmetic and geometric mean, respectively. Gauss proved that both the sequences converge to a common limit, say \(A\otimes \hspace{-0,1cm}G(x,y)\). The function \(A\otimes G\) is a mean on \((0, +\infty)\), i.e. it satisfies \[ \min \{x,y\} \leq A\otimes G(x,y) \leq \max \{x,y\}, \qquad x,y \in (0, +\infty), \] and has nice properties. Iterating the map \((A,G): (0, +\infty)^2\rightarrow (0, +\infty)^2\) one can write down the convergence of Gaussian iterates to \(A\otimes G\) as \[ (A,G)^i\rightarrow \left(A\otimes G,A\otimes G \right). \] The Gauss procedure has been fairly extended to a pretty large class of pairs \((M,N)\) of means on an arbitrary interval \(I\). The talk is a survey of results concerning the convergence of iterates \((M,N)^i\) and properties of the mean \[ M\otimes N:= \lim_{i\rightarrow \infty} (M,N)^i. \] Starting with ideas and results aggregated by J.M. Borwein and P.B. Borwein from different papers more than 30 years ago we come to those proved by J. Matkowski.

Gergely Kiss, Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Hungary

FUNCTIONAL EQUATIONS, FIELD HOMOMORPHISMS AND DERIVATIONS IN THE LIGHT OF SPECTRAL THEORY

Joint work with **Eszter Gselmann and Csaba Vincze**

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-7, room 1.8.

Abstract

First, in my talk I discuss the solutions of linear functional equations on fields of the form \[ \sum_{i=1}^n f_i(b_ix+c_iy)=0 ~~~~~\forall x,y\in K, \] where \(b_i,c_i\) are given constants, \(K\) is the field and \(f_i\) are unknown functions. I present that typically the set of solutions is a linear space containing field homomorphisms and higher order derivations. This result is based on spectral synthesis. Here I recall the theoretic background and discuss the main tools that we of use. In the second part of my presentation I study functional equations \[ \sum_{i=1}^n f_i^{p_i}(x^{q_i})=0 ~~~~~\forall x\in K, \] and \[ \sum_{i=1}^n x^{p_i}f_i(x^{q_i})=0 ~~~~~\forall x\in K, \] where \(p_i, q_i\) are positive integers and \(f_i\) are additive functions, that characterize field homomorphisms and higher order derivations, respectively. Among other techniques these results deliberately use spectral theory. Finally, I mention some further directions of research in this area. My talk is based on the [1, 2, 3].

References

- G. Kiss, M. Laczkovich,
*Linear functional equations, differential operators and spectral synthesis*, Aequationes Mathematicae**89(2)**(2015), 301-328. - E. Gselmann, G. Kiss, Cs. Vincze,
*On functional equations characterizing derivations: methods and examples*, Results in Mathematics**74**(2018), 27 pp. - E. Gselmann, G. Kiss, Cs. Vincze,
*Characterization of field homomorphisms through Pexiderized functional equations*, Journal of difference equations and applications**25**(2019), 26 pp.

Zbigniew Leśniak, Pedagogical University of Cracow, Poland

ON FRACTIONAL ITERATES OF A BROUWER HOMEOMORPHISM

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-7, room 1.8.

Abstract

We present a method for finding continuous (and consequently homeomorphic) orientation preserving iterative roots of a Brouwer homeomorphism for which there exists a family of pairwise disjoint invariant lines covering the plane.

To obtain the roots we use properties of the equivalence classes of the codivergency relation. In particular, the key role plays the fact that each of the invariant lines of the considered family is contained either in the set of regular points or in the set of irregular points of the given Brouwer homeomorphism.

References

- Z. Leśniak,
*On fractional iterates of a free mapping embeddable in a flow*, J. Math. Anal. Appl.**366**(2010), 310-318. - Z. Leśniak,
*On properties of the set of invariant lines of a Brouwer homeomorphism*, J. Difference Equ. Appl.**24**(2018), 746-752.

Janusz Morawiec, University of Silesia in Katowice, Poland

ON BETWEENNESS-PRESERVING MAPPINGS

Joint work with **Wiesław Kubiś and Thomas Zürcher**

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-7, room 1.8.

Abstract

We are interested in a Euclidean version of betweenness. We say that a point \(z\) is between two points \(x\) and \(y\) if and only if \(z\) is in the convex hull of \(x\) and \(y\). In this setting, we call a betweenness-preserving map monotone. The aim of this talk is to present regularity results for monotone mappings in the plane.

Kazuki Okamura, Shinshu University, Japan

SOME RESULTS FOR CONJUGATE EQUATIONS

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-7, room 1.8.

Abstract

I will talk about conjugate maps between two iterated function systems driven by several weak contractions, depending on [3] and [4]. Specifically, a special case of our framework is as follows: Let \(X\) and \(Y\) be compact metric spaces. Let \(I\) be a finite set. Assume that for each \(i \in I\), weak contractions \(f_i : X \to X\) and \(g_i : Y \to Y\) are given. Consider the solution \(\varphi : X \to Y\) satisfying that \[\varphi (f_i (x)) = g_i (\varphi(x)), \ i \in I, x \in X.\tag{1}\]

Conjugate equations of this kind are a certain generalization of de Rham’s functional equations [5]. They are considered by Zdun [8], Girgensohn-Kairies-Zhang [2], Shi-Yilei [7], Serpa- Buescu [6] and Bárány-Kiss-Kolossváry [1].

I will mainly talk about regularity of a unique solution \(\varphi\) of (1). Then, I will give examples to which our results are applicable. If time is permitted, I will also discuss existence and uniqueness of a more general class of conjugate equations than above.

References

- B. Bárány, G. Kiss, and I. Kolossváry,
*Pointwise regularity of parameterized affine zipper fractal curves*, Nonlinearity,**31**(2018) 1705-1733. - R. Girgensohn, H.-H. Kairies, W. Zhang,
*Regular and irregular solutions of a system of functional equations*, Aequationes Math.**72**(2006), 27-40. - K. Okamura,
*Some results for conjugate equations*, to appear in Aequationes Math. - K. Okamura,
*Hausdorff dimensions for graph-directed measures driven by infinite rooted trees*, preprint. - G. de Rham,
*Sur quelques courbes définies par des équations fonctionalles*, Univ. E Politec. Horino. Rend. Sem. Mat.**16**(1957), 101-113. - C. Serpa, J. Buescu,
*Constructive solutions for systems of iterative functional equations*, Constr. Approx.**45**(2017), 273-299. - Y.-G. Shi, T. Yilei,
*On conjugacies between asymmetric Bernoulli shifts*, J. Math. Anal. Appl.**434**(2016) 209-221. - M. C. Zdun,
*On conjugacy of some systems of functions*, Aequationes Math.**61**(2001) 239-254.

Fedor Pakovich, Ben-Gurion University of the Negev, Israel

COMMUTING RATIONAL FUNCTIONS REVISITED

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-7, room 1.8.

Abstract

Let \(A\) and \(B\) be rational functions on the Riemann sphere. The classical Ritt theorem states that if
\(A\) and \(B\) commute and do not have an iterate in common, then up to a conjugacy they are either powers,
or Chebyshev polynomials, or Lattès maps. This result however provides no information about commuting rational functions which *do* have a common iterate. On the other hand, non-trivial examples of such functions exist and were constructed already by Ritt. In the talk we present new results concerning
this class of commuting rational functions. In particular, we describe a method which permits to describe all rational functions commuting with a given rational function.

Zsolt Páles, University of Debrecen, Hungary

ON DERIVATIONS WITH ADDITIONAL PROPERTIES

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-7, room 1.8.

Abstract

A function \(d:\mathbb{R}\to \mathbb{R}\) is called a *derivation* if, for all \(x,y\in \mathbb{R} \),
\[
d(x+y)=d(x)+d(y) \qquad\mbox{and}\qquad d(xy)=yd(x)+xd(y).
\]
It is a nontrivial fact that, for any non-algebraic number \(t\in \mathbb{R}\), there exists a derivation which does not vanish at
\(t\). Nonzero derivations have many striking applications in the theory of functional equations and functional
inequalities. Derivations derivate many of the elementary functions. For instance, if \(f:I\to \mathbb{R}\) is the ratio of two
polynomials with algebraic coefficients, then, for every \(x\in I\),
\[
d(f(x))=f'(x)d(x).
\]
It has been an old problem of the theory of functional equations whether there exists a nonzero derivation which
derivates the exponential function or any of the trigonometric functions in the above sense. Our main result shows that
the answer to this problem is affirmative.

Paweł Pasteczka, Pedagogical University of Cracow, Poland

QUASI-ARITHMETIC GAUSS-TYPE ITERATION

Date: 2019-09-16 (Monday); Time: 12:05-12:25; Location: building B-7, room 1.8.

Abstract

For a sequence of continuous, monotone functions \(f_1,\dots,f_n \colon I \to \mathbb{R}\) (\(I\) is an interval) we define the mapping \(M \colon I^n \to I^n\) as a Cartesian product of quasi-arithmetic means generated by \(f_j\)-s, that is functions \(A^{[f_j]}(v_1,\dots,v_n):=f_j^{-1}\big(\tfrac1n(f_j(v_1)+\cdots+f_j(v_n))\big)\). It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of \(I^n\).

We prove that whenever all \(f_j\)-s are \(\mathcal{C}^2\) with nowhere vanishing first derivative, then this convergence is quadratic. We present both qualitative- and quantitative-type results concerning this iteration. In particular, we deliver an effective upper estimation of the value \(\text{Var}\, M^{k}(v)\) and calculate the limit \(\frac{\text{Var}\, M^{k+1}(v)}{(\text{Var}\, M^{k}(v))^2}\) in a nondegenerated case.

References

- P. Pasteczka,
*Iterated quasi-arithmetic mean-type mappings*, Colloq. Math.**144**(2016), 215–228. - P. Pasteczka,
*On the quasi-arithmetic Gauss-type iteration*, Aequationes Math.**92**(2018), 1119– 1128.

Maciej Sablik, University of Silesia in Katowice, Poland

ON FUNCTIONAL EQUATIONS CHARACTERIZING GENERALIZED POLYNOMIALS

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-7, room 1.8.

Abstract

We present some results on solving functional equations that characterize generalized polynomials. The results are essentially coming from our earlier works but we are going to investigate some new problems. We will give a description of solutions defined on Abelian groups and ask about the possible solutions in the case where semigroups (Abelian) are considered.

References

- M. Sablik, T. Riede,
*Characterizing polynomial functions by a mean value property*, Publ. Math. Debrecen**52**(1998), 597-609. - M. Sablik,
*Taylor's theorem and functional equations*, Aequationes Math.**60**(2000), 258-267. - M. Sablik,
*An elementary method of solving functional equations*, Annales Univ. Sci. Budapest., Sect. Comp.**48**(2018), 181-188. - L. Székelyhidi,
*Convolution type functional equations on topological commutative groups*, World Scientific Publishing Co. Inc., Teaneck, NJ, 1991. - T. Szostok,
*Functional equations stemming from numerical analysis*, Dissertationes Math. (Rozprawy Mat.)**508**(2015), 57 pp.

László Székelyhidi, University of Debrecen, Hungary

FUNCTIONAL EQUATIONS VIA SPECTRAL SYNTHESIS

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-7, room 1.8.

Abstract

Spectral synthesis studies the structure of translation invariant spaces of continuous functions over topological groups. The masterpiece is Laurent Schwartz's theorem stating that on the real line every translation invariant linear space of continuous complex valued functions which is closed under compact convergence is the closure of all exponential polynomials included in the space. As the solution space of a great variety of systems of convolution type functional equations satisfies these conditions spectral synthesis can be applied to describe the solutions. These ideas are worth for generalizations to obtain extensions of classical results of abstract harmonic analysis for functions without growth conditions (boundedness, integrability, etc.) Recently extensions of Schwartz's theorem have been proved over discrete groups using ring-theoretical methods, and spherical versions of the theorem have been obtained on spaces of functions invariant under various subgroups of the general linear group. Also extensions of the basic results to more general situations have been proved by relaxing the group-structure. In this survey talk we present the fundamental methods, ideas and results together with relevant applications to functional equations.

Bettina Wilkens, University of Namibia, Namibia

A RING-THEORETIC APPROACH TO DISCRETE SPECTRAL SYNTHESIS

Joint work with **László Székelyhidi**

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-7, room 1.8.

Abstract

Let \(G\) be an Abelian group and let \(\mathcal C(G)\) be the vector space of complex-valued functions on \(G\).
With the topology of pointwise convergence, \(\mathcal C(G)\) is a locally convex space. The group \(G\) acts on \({\mathcal C}G\) by translations. Closed submodules of \({\mathcal C}G\) are called *varieties*.
Consider the bilinear product \(\mathcal C(G) \times \mathbb{C} G \rightarrow \mathbb{C}\) given by
\[ \langle \sum\limits_{x \in G} a_x x, \, f \rangle =\sum\limits_{x \in G} a_x f(x).\]
Assigning the function \(f \mapsto \langle a, \, f \rangle\) to \(a\) in \(\mathbb{C} G\) yields an identification of \(\mathbb{C}G G\) with the space of linear functionals on \({\mathcal C}G \) that are continuous with respect to the topology of pointwise convergence. Assigning the map \(a \mapsto \langle a, \, f \rangle\) to \(f\) provides an identification of \(\mathcal C(G)\) with the algebraic dual \(\mathbb{C}G^{\ast}.\) Defining orthogonal complements in the usual way, the Hahn-Banach theorem yields that the map \(V \mapsto V^{\perp}\) is a one-to-one correspondence between varieties and ideals of \(\mathbb{C}G\).

We exploit this to investigate to characterise the dual \(\mathbb{C}G/V^{\perp}\) when \(V\) is a variety with *spectral analysis* - each subvariety of \(V\) contains a one-dimensional module - that is *syntheziable* - topologically generated by its finite-dimensional subvarieties- or possesses *spectral synthesis* - only has synthesizable subvarieties.
We shall see that these properties correspond to well-known and well-researched properties of commutative rings. If \(V\) has spectral synthesis- the strongest of the listed properties, then \(\mathbb{C}G/V^{\perp}\) emerges as "almost" Noetherian. Finally, we discuss how the ring theoretic results may be used to provide a description of the module structure of \(V\).

References

- R. Gilmer, W. Heinzer,
*Ideals contracted from a Noetherian extension ring*, J. Pure Appl. Algebra**24(2)**(1980), 123-144. - L. Székelyhidi,
*Harmonic and Spectral Analysis*, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. - L. Székelyhidi, B. Wilkens,
*Spectral synthesis and residually finite-dimensional algebras*, J. Algebra Appl.**16**(2017), 10 pp.

Weinian Zhang, Sichuan University, China

INVARIANT MANIFOLDS WITH/WITHOUT SPECTRAL GAP

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-7, room 1.8.

Abstract

In this talk we discuss invariant manifolds obtained with or without a spectral gap condition, showing approximation to weak hyperbolic manifolds (with gap condition) and giving the existence and smoothness for invariant submanifolds on a center manifold (without gap condition).

Wenmeng Zhang, Chongqing Normal University, China

SMOOTH LINEARIZATION WITH A NONUNIFORM DICHOTOMY

Joint work with **Davor Dragičević and Weinian Zhang**

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-7, room 1.8.

Abstract

In this talk, we give two smooth linearization theorems for \(C^{1,1}\) nonautonomous systems with a nonuniform strong exponential dichotomy. The first theorem concerns \(C^1\) linearization with a gap condition, while the second one concerns simultaneously differentiable and Hölder continuous linearization without any gap conditions. Restricted in the autonomous case, the second result gives the simultaneously differentiable and Hölder linearization of \(C^{1,1}\) hyperbolic systems without any non-resonant conditions.

Linfeng Zhou, Sichuan University, China

NONUNIFORM EXPONENTIAL DICHOTOMY AND ADMISSIBILITY

Joint work with **Kening Lu and Weinian Zhang**

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-7, room 1.8.

Abstract

Nonuniform exponential dichotomy describes nonuniform hyperbolicity for linear nonautonomous dynamical systems. In this talk, we present results on the relationships between nonuniform exponential dichotomies and admissible pairs for classes of weighted bounded functions, and the equivalent relationships between nonuniform exponential dichotomy and admissible pairs of classes of Lyapunov bounded functions.

References

- L. Zhou, K. Lu, W. Zhang,
*Equivalences between nonuniform exponential dichotomy and admissibility*, J. Differential Equations**262**(2017), 682-747. - L. Zhou, W. Zhang,
*Admissibility and roughness of nonuniform exponential dichotomies for difference equations*, J. Functional Analysis**271**(2016), 1087-1129.

Martin Berz, Michigan State University, USA

RIGOROUS INTEGRATION OF FLOWS OF ODES USING TAYLOR MODELS

Joint work with **Kyoko Makino**

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-7, room 2.4.

Abstract

Taylor models combine the advantages of numerical methods and their efficiency under tightly controlled computational resources even for complex problems, with the advantages of symbolic approaches and their ability to be rigorous and to allow the treatment of functional dependencies instead of merely points. The result is a local representation of a function by its Taylor expansion and a mathematically rigorous bound for the approximation error.

The resulting differential algebraic calculus comprising an algebra with differentiation and integration is particularly amenable to the study of ODEs and PDEs based on fixed point problems from functional analysis. We describe the development of rigorous tools to determine enclosures of flows of general nonlinear differential equations based on Picard iterations.

The methods can be used for the computation of enclosures of flows over large domains to prescribed accuracy via domain decomposition methods. We study the behavior of the methods for several dynamical systems, and in particular analyze suitable parameter settings to efficiently balance local domain size versus local order. Comparisons to other recently proposed computational approaches are given, showing the advantages of the Taylor model methods for large domains.

Renato Calleja, National Autonomous University of Mexico, Mexico

TORUS KNOT CHOREOGRAPHIES IN THE \(N\)-BODY PROBLEM

Joint work with **Eusebius Doedel, Carlos García Azpeitia, Jason Mireles-James, and Jean-Philippe Lessard**

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-7, room 2.4.

Abstract

\(N\)-body choreographies are periodic solutions to the \(N\)-body equations in which \(N\) equal masses chase each other around a fixed closed curve. In this talk I will present a systematic approach for proving the existence of spatial choreographies in the gravitational \(N\) body problem with the help of the digital computer. These arise from the polygonal system of \(N\) bodies in a rotating frame of reference. In rotating coordinates, after exploiting the symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, correspond to choreographies.

References

- R. Calleja, E. Doedel, and C. García-Azpeitia,
*Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the \(n\)-body problem*, Celestial Mech. Dynam. Astronom.**130**(2018), 130:48. - R. Calleja, C. García-Azpeitia, J.P. Lessard, and J.D. Mireles-James,
*Torus knot choreographies in the \(n\)-body problem*,Preprint, available at http://cosweb1.fau.edu/~jmirelesjames/torusKnotChoreographies.html

Maciej Capiński, AGH University of Science and Technology, Poland

ARNOLD DIFFUSION IN THE ELLIPTIC RESTRICTED THREE-BODY PROBLEM

Joint work with **Marian Gidea**

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-7, room 2.4.

Abstract

We present a topological mechanism for diffusion in Hamiltonian systems and apply it to the Planar Elliptic Restricted Three-Body Problem. We treat the elliptic problem as a perturbation of the circular problem, where the perturbation parameter \(\varepsilon\) is the eccentricity of the orbits of the primaries. We measure the energy as the Hamiltonian \(H\) of the circular problem. Our objective is to prove that for any \(\varepsilon>0\) there exist orbits which start with some value \(H=h\) and finish with \(H=h+c\), where \(c>0\) is independent from \(\varepsilon\). Our method is based on topological shadowing of trajectories along homoclinic intersections of invariant manifolds. We perform a geometric construction, which allows us to obtain orbits for which we can control the increase in energy. The method is suitable for computer assisted proofs and can be used to obtain explicit bounds on the energy changes, for explicit ranges of the perturbation parameter. We apply it to the setting of the Neptune-Tryton system.

The construction also leads to symbolic dynamics in energy and to stochastic properties of the diffusing orbits. This will be the subject of the talk by Marian Gidea in the same session.

Hayato Chiba, Tohoku University, Japan

A BIFURCATION OF THE KURAMOTO MODEL ON NETWORKS

Joint work with **Georgi Medvedev**

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-7, room 2.4.

Abstract

For the mean-field limit of a system of globally coupled phase oscillators defined on networks, a bifurcation from the incoherent state to the partially locked state at the critical coupling strength is investigated based on the generalized spectral theory. This reveals that a network topology affects the dynamics through the eigenvalue problem of a certain Fredholm integral operator which defines the structure of a network.

References

- H. Chiba, G.S. Medvedev,
*The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas*, Discret. Contin. Dyn. S.-A**39**(2019), 131-155. - H. Chiba, G.S. Medvedev,
*The mean field analysis ofthe Kuramoto model on graphs II. Asymptotic stability of the incoherentstate, center manifold reduction, and bifurcations*, Series, Discret. Contin. Dyn. S.-A, (2019). - H. Chiba, G.S. Medvedev, M.S. Muzuhara,
*Bifurcations in the Kuramoto model on graphs*, Chaos**28**(2019), 073109.

Jacek Cyranka, University of California, San Diego, USA & University of Warsaw, Poland

CONTRACTIBILITY OF A PERSISTENCE MAP PREIMAGE

Joint work with **Konstantin Mischaikow**

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-7, room 2.4.

Abstract

This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of solutions snapshots, what conclusions can be drawn about solutions of the original dynamical system? In this paper we provide a definition of a persistence diagram for a point in \(\mathbb{R}^N\) modeled on piecewise monotone functions. We then provide conditions under which time series of persistence diagrams can be used to guarantee the existence of a fixed point of the flow on \(\mathbb{R}^N\) that generates the time series. To obtain this result requires an understanding of the preimage of the persistence map. The main theorem of this paper gives conditions under which these preimages are contractible simplicial complexes.

References

- J. Cyranka and K. Mischaikow,
*Contractibility of a persistence map preimage*, arXiv e-prints 2018, arXiv:1810.12447.

Zbigniew Galias, AGH University of Science and Technology, Poland

ENCLOSURE OF THE DOUBLE SCROLL ATTRACTOR FOR THE CHUA’S CIRCUIT WITH A CUBIC NONLINEARITY

Joint work with **Warwick Tucker**

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-7, room 2.4.

Abstract

We consider the Chua's circuit with a cubic nonlinearity. The dynamics of the circuit is defined by \[ C_1\dot{x}_1 = (x_2-x_1)/R-g(x_1), \quad C_2\dot{x}_2 = (x_1-x_2)/R+x_3, \quad L \dot{x}_3 = -x_2-R_0 x_3, \] where \(g(x_1)=g_1x_1+g_2x_1^3\). The system is considered with the following parameter values \(C_1=0.7\), \(C_2=7.8\), \(L=1.891\), \(R_0=0.01499\), \(g_1=-0.59\), \(g_2=0.02\), and \(R=2\), for which in simulations one observes the double scroll attractor.

Let us define \(\Sigma=\Sigma_1\cup\Sigma_2\), where \(\Sigma_1=\{x\colon x_1=2.1647\}\) and \(\Sigma_2=\{x\colon x_1=-2.1647\}\). The *return map* \(P:\Sigma\mapsto\Sigma\) is defined as
\(P(x)=\varphi(\tau(x),x)\), where \(\varphi(t,x)\) is the trajectory based at \(x\), and \(\tau(x)\) is the *return time* after which the trajectory \(\varphi(t,x)\) returns to \(\Sigma\).

A candidate \(T\subset \Sigma\) for a trapping region enclosing the numerically observed attractor of the return map \(P\) is constructed. The return map \(P\) is not defined on the whole set \(T\). This is a consequence of the fact that the double scroll attractor contains the origin-an unstable equilibrium. For some initial points in \(x\in T\) the corresponding trajectories converge to the origin, i.e. \(\varphi(t,x)\to(0,0,0)\) for \(t\to\infty\). It follows that standard rigorous integration procedures cannot be used to study the dynamics of the system over the whole set \(T\). A method to handle trajectories passing arbitrarily close to an equilibrium is needed. Such trajectories may have arbitrarily large return times. The Jacobian matrix \(J\) at the origin has one real positive eigenvalue \(\lambda\approx0.2066\) and a pair of complex eigenvalues with negative real parts \(\alpha\pm\beta i\approx-0.075\pm 0.1966\). Normal form theory is used to develop a method to find enclosures of trajectories in a neighborhood of an unstable fixed point of a spiral type.

We prove the existence of a trapping region enclosing the double scroll attractor for the Chua's circuit with a cubic nonlinearity. More precisely, we prove the following theorem: for each \(x\in T\) either \(P(x)\in T\) or the trajectory \(\varphi(t,x)\) converges to the origin without intersecting \(\Sigma\), i.e., \(\varphi(t,x)\to(0,0,0)\) for \(t\to\infty\) and \(\{\varphi(t,x)\colon t>0\}\cap \Sigma=\emptyset\).

In the computer assisted proof, to handle trajectories passing close to the origin, we define the cylinder \(C\) centered at the origin. We also define the entry set, which is a part of the cylinder side and the exit set consisting of two parts each enclosed in one of the cylinder bases.

The proof of the main results is composed of three parts. In the first part the set \(T\) is covered by boxes. For each box we prove that either the image of this box under \(P\) is enclosed in \(T\) or all trajectories based in this box enter the cylinder \(C\) through the entry set. In the second part, we show that all trajectories based at the exit set reach \(T\). In the third part of the proof, we show that trajectories based at the entry set either converge to the origin or exit the cylinder through the exit set.

The first two parts of the proof are carried out using the CAPD library for the computation of trajectories and the evaluation of the return map \(P\). The third part of the proof is carried out using the normal form theory.

Marian Gidea, Yeshiva University, USA

SYMBOLIC DYNAMICS AND STOCHASTIC BEHAVIOR IN THE ELLIPTIC RESTRICTED THREE-BODY PROBLEM

Joint work with **Maciej Capiński**

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-7, room 2.4.

Abstract

We study Hamiltonian instability in the elliptic restricted three-body problem, in the context of a concrete model, on the motion of a small body (e.g., asteroid or spaceship) relative the Neptune-Triton system.

The elliptic restricted three-body problem can be regarded as a perturbation of the circular restricted three-body problem, with the perturbation parameter \(\varepsilon\) being the eccentricity of the orbits of the primaries. When the perturbation parameter is set to zero, the total energy \(H_\varepsilon\) of the system is preserved. When the perturbation parameter is non-zero, the total energy may vary. We provide two global instability results concerning the variation of energy along trajectories.

First, we show that for every suitably small, non-zero perturbation parameter, there exist trajectories along which the energy makes chaotic jumps. That is, given a sequence of energy level sets \((h^\sigma)_{\sigma\geq 0}\), with \(\|h^{\sigma+1}-h^\sigma\|>2\eta\), for some suitable \(\eta>0\), there exists a trajectory with \(\|H_\varepsilon (t^\sigma)-h^\sigma\|<\eta\), for some times \(t^\sigma>0\) and all \(\sigma\geq 0\).

Second, we show that the distributions of energies along orbits starting from some sets of initial conditions converge to a Brownian motion with drift as the perturbation parameter tends to zero. Moreover, we can obtain any desired values of the drift and of the variance for the limiting Brownian motion, for appropriate sets of initial conditions. That is, if we consider the stochastic process \(X_{t}^{\varepsilon }(z)\) representing the evolution of the energy along a trajectory starting from some point \(z\), with appropriately rescaled time \(t\), then, for every \(\mu,\sigma\in \mathbb{R}\) there exists a set \(\Omega_\varepsilon\) of initial points \(z\) for which \(X_{t}^{\varepsilon}-X_{0}^{\varepsilon }\) converges in distribution to \(\mu t+\sigma W_{t}\) as \(\varepsilon\to 0\), where \(W_{t}\) is the standard Wiener process.

In both cases we obtain an explicit range of the perturbation parameter \(\varepsilon\) for which the above phenomena occur. The proof of the results is based on topological methods and validated numerics.

Our results address conjectures made by Arnold [1] and Chirikov [2].

References

- V.I. Arnold,
*Instability of dynamical systems with several degrees of freedom*, Sov. Math. Doklady**5**(1964), 581-565. - B. Chirikov,
*A universal instability of many-dimensional oscillator systems*, Phys. Rep.**52**(1979), 264-379.

Marcel Guardia, Polytechnic University of Catalonia, Spain

DIFFUSIVE BEHAVIOR ALONG MEAN MOTION RESONANCES IN THE RESTRICTED 3 BODY PROBLEM

Joint work with **Vadim Kaloshin, Pau Martín, and Pau Roldán**

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-7, room 2.4.

Abstract

Consider the Restricted Planar Elliptic Three Body Problem. This problem models the Sun-Jupiter-Asteroid dynamics. For eccentricity of Jupiter \(e_0\) small enough we show that there exists a family of probability measures \(\nu_{e_0}\) supported at the \(3 : 1\) mean motion resonance such that the pushforward under the associated Hamiltonian flow has the following property. At the time scale \(te_0^{-2}\), the distribution of the eccentricity of the Asteroid weakly converges to an (Ito stochastic) diffusion process on the line as \(e_0\to 0\). This resonance corresponds to the biggest of the Kirkwood gap on the Asteroid belt in the Solar System.

Àlex Haro, University of Barcelona, Spain

SINGULARITY THEORY FOR KAM TORI: FROM SYMPLECTIC GEOMETRY TO APPLICATIONS THROUGH ANALYSIS

Joint work with **Rafael de la Llave and Alejandra González**

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-7, room 2.4.

Abstract

We present a method to find nontwist KAM tori. These are tori for which the twist condition fails. Our method also leads to a natural classification of KAM tori which is based on Singularity Theory. This talk aims to illustrate the main ideas of our approach, going from rigorous results to numerical computations up to the verge of breakdown.

Bernd Krauskopf, University of Auckland, New Zealand

A HETERODIMENSIONAL CYCLE IN A 4D FLOW

Joint work with **Andy Hammerlindl, Gemma Mason and Hinke M. Osinga**

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-7, room 2.4.

Abstract

A heterodimensional cycle consists of a pair of heteroclinic connections between two saddle periodic orbits with unstable manifolds of different dimensions. Recent theoretical work for diffeomorphisms of dimension at least three has shown that the existence of heterodimensional cycles may be a \(C^1\)-robust property. We study a concrete example of a heterodimensional cycle in a flow, specifically in a four-dimensional Atri model of intracellular calcium dynamics. For suitable parameter values, this model has two saddle periodic orbits of different index. We employ a boundary-value problem setup to compute their global invariant manifolds to show that and how they intersect in a connecting orbit of codimension one and an entire cylinder of connecting orbits. We present the different invariant objects in different projections of the four-dimensional phase space, as well as in intersection with a three-dimensional Poincaré section. In this way, we examine how this heterodimensional cycle arises and organises the nearby dynamics.

Hieronim Kubica, AGH University of Science and Technology, Poland

PERSISTENCE OF NORMALLY HYPERBOLIC INVARIANT MANIFOLDS IN THE ABSENCE OF RATE CONDITIONS

Joint work with **Maciej Capiński**

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-7, room 2.4.

Abstract

Normally hyperbolic invariant manifolds (NHIMs) [8] are persistent [2, 3, 4, 5]. The converse is also true - persistent manifolds are precisely the normally hyperbolic ones [9]. These results take advantage of analytical assumptions. Features like rate conditions associated with the map driving the dynamics, smooth structure on the invariant set, or \(C^1\) topology on the space of admissible perturbations play a key role. In a recent result, we prove that a weaker form of persistence is viable also in a setting which relies only on topological, qualitative assumptions about the dynamics [1].

We consider perturbations of normally hyperbolic invariant manifolds, under which they can loose their hyperbolic properties. We show that if the perturbed map which drives the dynamical system preserves the properties of topological expansion and contraction, described in terms of covering relations [6, 7], then the manifold is perturbed to an invariant set. The main feature is that our results do not require the rate conditions to hold after the perturbation. In this case the manifold can be perturbed to an invariant set, which is not a topological manifold. The method used to show this is not itself perturbative. It can be applied to establish the existence of invariant sets within a prescribed neighborhood also in the absence of a normally hyperbolic invariant manifold prior to perturbation. The dynamics is assumed to be given by a continuous map, without the assumption of invertibility.

References

- M. Capiński, H. Kubica,
*Persistence of normally hyperbolic invariant manifolds in the absence of rate conditions*, presented for publication, 2018 - N. Fenichel,
*Persistence and smoothness of invariant manifolds for flows*, Indiana Univ. Math. J.**21**(1971/1972), 193–226. - N. Fenichel,
*Asymptotic stability with rate conditions*, Indiana Univ. Math. J.**23**(1973/74), 1109– 1137. - N. Fenichel,
*Asymptotic stability with rate conditions for dynamical systems*, Bull. Amer. Math. Soc.**80**(1974), 346–349. - N. Fenichel,
*Asymptotic stability with rate conditions II*, Indiana Univ. Math. J.**26**(1977), 81–93. - M. Gidea, P. Zgliczyński,
*Covering relations for multidimensional dynamical systems*, J. Differential Equations**202**(2004), 32–58. - M. Gidea, P. Zgliczyński,
*Covering relations for multidimensional dynamical systems II*, J. Differential Equations**202**(2004), 59–80. - M. Hirsch, C. Pugh, M. Shub
*Invariant Manifolds*, Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 2006. - R. Mañé,
*Persistent manifolds are normally hyperbolic*, Trans. Amer. Math. Soc.**246**(1978), 261– 283.

Ezequiel Maderna, University of the Republic, Uruguay

HYPERBOLIC EXPANSIONS WITH ARBITRARY LIMIT SHAPE

Joint work with **Andrea Venturelli**

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-7, room 2.4.

Abstract

A well-known fact in the classical \(N\)-body problem is that if we normalize by the size of the configuration a completely parabolic motion, then the normalized configuration converge to the set of central configurations.

We will show that there is no such restriction for motions with positive energy. Moreover, we will show the existence of hyperbolic motions with arbitrarily chosen limit shape, and this for any given initial configuration of the bodies. The energy level \(h>0\) of the motion can also be chosen arbitrarily. The proof uses variational methods and represents a new application of Marchal's theorem, whose main use in recent literature has been to prove the existence of periodic orbits.

Kyoko Makino, Michigan State University, USA

RIGOROUS GLOBAL SEARCH, DETERMINATION OF MANIFOLDS AND THEIR HOMOCLINIC POINTS, AND ENTROPY ESTIMATES

Joint work with **Martin Berz**

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-7, room 2.4.

Abstract

Taylor models provide enclosures of functions over a domain within a relaxation band of their Taylor expansion around a point inside the domain. They are obtained automatically by evaluating the code list of the underlying function in Taylor model arithmetic, and under minimal requirements on the underlying floating point arithmetic, the enclosures are mathematically rigorous. The widths of the resulting band scales with a high order of the width of the domain, and so in practice enclosures are obtained that are usually much sharper than those from conventional rigorous methods like intervals, centered forms, and related linearizations for all but the simplest cases. In general, the complexity and nonlinearity of the underlying function dictates optimum order and domain widths to achieve a desired accuracy. The resulting rigorous relaxations can be used for the local description of objective functions and constraints in rigorous global search. Furthermore, the resulting representations can be used efficiently for higher order domain reduction based on conditions on the objective function and the constraints.

The methods can be applied to various problems in dynamical systems. First, Taylor models allow for the computation of tight enclosures of manifolds of dynamical systems. Once these enclosures are given, they can be used with the Taylor model-based rigorous global optimizer to find and isolate all homoclinic points. From these it is possible to determine so-called homoclinic tangles, which contain information on lower bounds of topological entropy of the underlying systems. Various examples of the practical use of the methods are given.

Jason D. Mireles James, Florida Atlantic University, USA

VALIDATED NUMERICS FOR STABLE/UNSTABLE MANIFOLDS OF DELAY DIFFERENTIAL EQUATIONS

Joint work with **Jean-Philippe Lessard and Olivier Hénot**

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-7, room 2.4.

Abstract

The parameterization method is a general functional analytic framework for studying invariant manifolds. The idea is to formulate a chart or covering map for the manifold as the solution of an appropriate invariance equation. Studying the invariance equation leads to both numerical schemes for approximating the invariant manifold and to a posteriori methods for quantifying discretization and truncation errors. This talk considers the parameterization method for unstable manifolds of delay differential equations (DDEs), focusing on the numerical implementation as well as the derivation of mathematically rigorous computer assisted error bounds. One challenge is the fact that the solution of a DDE depends on both present and past states, so that a DDE generates an infinite dimensional dynamical system. The invariant manifolds studied here play an important role in describing the global dynamics of this system.

Markus Neher, KIT, Germany

INTERVAL AND TAYLOR MODEL METHODS FOR ODES

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-7, room 2.4.

Abstract

Verified integration methods for ODEs are methods that compute rigorous bounds for some specific solution or for the flow of some initial set of a given ODE. Interval arithmetic has been used for calculating such bounds for solutions of initial value problems. The origin of these methods dates back to Ramon Moore [2].

Unfortunately, interval methods sometimes suffer from overestimation. This can be caused by the \(\textit{dependency problem}\), that is the lack of interval arithmetic to identify different occurrences of the same variable, and by the \(\textit{wrapping effect}\), which occurs when intermediate results of a calculation are enclosed into intervals. In verified integration this happens when enclosures of the flow at intermediate time steps of the interval of integration are computed. Overestimation may then degrade the computed enclosure of the flow, enforce miniscule step sizes, or provoke premature abortion of the integration.

Taylor models, developed by Martin Berz in the 1990s, combine interval arithmetic with symbolic computations [1]. A Taylor model consists of a multivariate polynomial and a remainder interval. In all computations, the polynomial part is handled by symbolic calculations, which are essentially unaffected by the dependency problem or the wrapping effect. Only the interval remainder term and polynomial terms of high order, which are usually small, are bounded using interval arithmetic. Taylor models also benefit from their capability of representing non-convex sets. For nonlinear ODEs, this increased flexibility in admissible boundary curves for the flow is an intrinsic advantage over traditional interval methods.

In our talk, we analyze Taylor model mehods for the verified integration of ODEs and compare these methods with interval methods.

References

- M. Berz,
*From Taylor series to Taylor models*, AIP Conference Proceedings**405**(1997), 1-23. - R.E. Moore,
*Interval Analysis*, Prentice Hall, Englewood Cliffs, N.J., 1966.

Gabriella Pinzari, University of Padua, Italy

RENORMALIZABLE INTEGRABILITY OF THE PARTIALLY AVERAGED NEWTONIAN POTENTIAL

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-7, room 2.4.

Abstract

**Definition**
Let \(h\), \(g\) be two (commuting) functions
of the form
\[ h(p, q, y, x)=\widehat h({\rm I}(p,q), y, x)\ ,\qquad g(p, q, y, x)=\widehat g({\rm I}(p,q), y, x)\]
where
\[(p, q, y, x)\in {\cal D}:={\cal B}\times U\]
with \(U\subset {\mathbb R}^2\), \({\cal B}\subset{\mathbb R}^{2n}\) open and connected, \((p,q)=(p_1, \cdots, p_n, q_1, \cdots, q_n)\) conjugate coordinates with respect to the two-form \(=dy\wedge dx+\sum_{i=1}^{n}dp_i\wedge dq_i\) and \({\rm I}(p,q)=({\rm I}_1(p,q), \cdots, {\rm I}_n(p,q))\), with
\[{\rm I}_i:\ {\cal B}\to {\mathbb R}\ ,\qquad i=1,\cdots, n\]
pairwise Poisson commuting:
\[\{{\rm I}_i, {\rm I}_j\}=0 \qquad \forall 1\le i \lt j\le n \qquad i+1, \cdots, n.\]
We say that \(h\) is *renormalizably integrable via* \(g\) if there exists a function \[\widetilde h:\qquad {\rm I}({\cal B})\times g(U)\to {\mathbb R}\ , \] such that
\[h(p,q,y,x)=\widetilde h({\rm I}(p,q), \widehat g({\rm I}(p,q),y,x))\]
for all \((p, q, y, x)\in {\cal D}\).

It is proved that the partial average i.e., the Lagrange average with respect to *just one* of the two mean anomalies,
of the Newtonian part of the perturbing function in the three-body problem Hamiltonian is renormalizably integrable. Consequences on the dynamics of the three-body problem are briefly discussed. The talk is based on [1], ArXiv: arXiv:1808.07633 and work in progress.

References

- G. Pinzari,
*A first integral to the partially averaged Newtonian potential of the three-body problem*, Celest Mech. Dyn. Astr.**131(22)**(2019).

Pablo Roldán, Yeshiva University, USA

CONTINUATION OF PERIODIC SOLUTIONS FROM THE CLASSICAL TO THE CURVED THREE-BODY PROBLEM

Joint work with **Abimael Bengochea and Ernesto Perez-Chavela**

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-7, room 2.4.

Abstract

In the classical 3-body problem it is known that any three masses lying on the vertices of an equilateral triangle generate a relative equilibria, which is a periodic solution. We will discuss the possible continuation of this periodic solution to the curved 3-body problem. (The curved problem is the extension of the classical one to a manifold of constant curvature.)

Marco Sansottera, University of Milan, Italy

ANALYTIC STUDY OF THE SECULAR DYNAMICS OF EXOPLANETARY SYSTEMS

Joint work with **Anne-Sophie Libert, Ugo Locatelli, and Antonio Giorgilli**

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-7, room 2.4.

Abstract

The search for exoplanets around nearby stars has produced a massive amount of observational data, pointing out the peculiar character of the Solar system. To date, more than 600 multiple planet systems have been found and the number of discovered exoplanets with unexpected orbital properties (such as highly eccentric orbits, mutually inclined planetary orbits, hot Jupiters, compact multiple systems) constantly increases.

The Laplace-Lagrange secular theory uses the circular approximation as a reference, thus its applicability to extrasolar systems can be doubtful. In this talk we aim to show that perturbation theory reveals very efficient for describing the long-term evolution of extrasolar systems.

First we study the long-term evolution of coplanar extrasolar systems
with two planets by extending the Laplace-Lagrange theory
(see [1, 2]). We identify three categories of
systems: (i) *secular systems*, whose long-term evolution is
accurately described using high order expansions in the
eccentricities; (ii) *near a mean-motion resonance systems*,
for which an approximation at order two in the masses is required;
(iii) *really close to or in a mean-motion resonance systems*,
for which a resonant model has to be used.

Then, being the inclinations of exoplanets detected via radial
velocity method essentially unknown, we show how perturbation theory
can be used in order to provide estimations of the ranges of mutual
inclinations that are compatible with the long-term stability of the
system. We propose a novel procedure (see [3]): a
*reverse KAM approach* by using normal forms depending on a
free parameter related to the unknown mutual inclinations of the
exoplanets. Our approach can interestingly complement the concept of
*AMD-stability* (see [4, 5])
to analyze the dynamics of the multiple-planet extrasolar systems.

References

- A.-S. Libert, M. Sansottera,
*On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems*, Celest. Mech. Dyn. Astr.**117**(2013), 149–168. - M. Sansottera, A.-S. Libert,
*Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance*, Celest. Mech. Dyn. Astr., to appear (2019). - M. Volpi, U. Locatelli, M. Sansottera,
*A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems*, Celest. Mech. Dyn. Astr.**130(36)**(2018). - J. Laskar, A.C. Petit,
*AMD-stability and the classification of planetary systems*, Astron. & Astroph.**605**(2017), A72. - A.C. Petit, J. Laskar, G. Boué,
*AMD-stability in the presence of first-order mean motion resonances*, Astron. & Astroph.,**607**(2017), A35.

Tere M. Seara, Polytechnic University of Catalonia, Spain

ON THE BREAKDOWN OF SMALL AMPLITUDE BREATHERS FOR THE REVERSIBLE KLEIN-GORDON EQUATION

Joint work with **Marcel Guardia and Otavio Gomide**

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-7, room 2.4.

Abstract

Breathers are periodic in time spatially localized solutions of evolutionary PDEs. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations. Exchanging the roles of time and position, breathers can be interpreted as homoclinic solutions to a steady solution. In this talk, I will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solution when the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds. Their distance is exponentially small with respect to the amplitude of the breather and therefore classical perturbative techniques cannot be applied.

Bruno Vergara, ICMAT, Spain

CONVEXITY OF WHITHAM’S HIGHEST WAVE

Joint work with **Alberto Enciso and Javier Gómez-Serrano**

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-7, room 2.4.

Abstract

In this talk I will discuss a conjecture of Ehrnström and Wahlén [1] on the profile of solutions of extreme form to Whitham’s model of shallow water waves. This is a non-local dispersive equation featuring travelling waves and singularities. Analogously to Stokes waves for Euler, we will see that there exists a highest, cusped and periodic solution to this model which is convex between consecutive crests [2].

References

- M. Ehrnström, E. Wahlén,
*On Whitham's conjecture of a highest cusped wave for a nonlocal shallow water wave equation*, Ann. Inst. H. Poincaré Anal. Non. Linéaire, in press (2019), arXiv:1602.05384. - A. Enciso, J. Gómez-Serrano, B. Vergara,
*Convexity of Whitham's highest cusped wave*, (2018), arXiv:1810.10935.

Adrian Weisskopf, Michigan State University, USA

NORMAL FORM METHODS AND RIGOROUS GLOBAL OPTIMIZATION FOR THE ASTRODYNAMICAL BOUNDED MOTION PROBLEM

Joint work with **Martin Berz and Roberto Armellin**

Date: 2019-09-19 (Thursday); Time: 12:05-12:25; Location: building B-7, room 2.4.

Abstract

Due to their common origin and mathematical underpinnings, it is sometimes possible to transfer specific advanced methods to analyze dynamical systems from one field of applications to another. In this work, we illustrate the transfer of differential algebra (DA) based normal form methods and rigorous global optimization for Nekhoroshev-type stability estimates, which were first developed in the field of particle beam physics and accelerator physics, to the field of astrodynamics to design bounded motion in the Earth's zonal problem.

The DA framework [1] and in particular the DA normal
form algorithm [4], and their associated techniques
are hybrid methods of numerical and analytic calculations and have
been established by Berz *et al*.
The methods can be turned mathematically rigorous by fully accounting
for expansion errors and floating point inaccuracies, as is done in
the Taylor model methods discussed in this session.

Many of the DA tools have been applied in the field of accelerator physics, where they reveal details of those dynamical systems that are otherwise very difficult to obtain by conventional methods. More recently, researchers have begun on the fruitful transfer of those DA methods to the astrodynamics community [3, 6, 5, 2]. An advancement of this transfer to normal form methods provides new possibilities, as it will be demonstrated in this work, including the capability of determining entire sets of bounded motion orbits in the full zonal problem.

Given an origin preserving Poincaré return map of a repetitive Hamiltonian system expanded in its phase space coordinates and system parameters, the DA normal form algorithm provides a nonlinear change of variables by an order-by-order transformation to normal form coordinates in which the motion is rotationally invariant. This circular phase space behavior allows for a straightforward extraction of the phase space rotation frequency and an action-angle parameterization of the normal form motion.

The normal form parameterization is used to find orbit sets satisfying the bounded motion condition, i.e. same average nodal period and drift in the ascending node of the bounded orbits. For the right Poincaré surface of the Poincaré return map, the inverse normal form transformation is used to parameterize the map by the action-angle parameterization of the corresponding normal form motion. Averaging the map over a full phase space revolution by a path integral along the angle-parameterization yields the averaged nodal period and drift in the ascending node for which the bounded motion conditions are straightforwardly imposed. Sets of highly accurate bounded orbits are obtained, extending over several thousand kilometers and valid for more than ten years.

References

- M. Berz,
*Differential algebraic description of beam dynamics to very high orders*, Part. Accel.**24**(1988), 109-124. - A. Wittig, R. Armellin,
*High order transfer maps for perturbed Keplerian motion*, Celestial Mechanics and Dynamical Astronomy**122**(2015), 333-358. - P. Di Lizia, R. Armellin, M. Lavagna,
*Application of high order expansions of two-point boundary value problems to astrodynamics*, Celestial Mechanics and Dynamical Astronomy**102**(2008), 355-375. - M. Berz,
*Modern Map Methods in Particle Beam Physics*, Academic Press, 1999. - R. Armellin, P. Di Lizia, F. Topputo, M. Lavagna, F. Bernelli-Zazzera, M. Berz,
*Gravity assist space pruning based on differential algebra*, Celestial Mechanics and Dynamical Astronomy**106**(2010), 1-24. - R. Armellin, P. Di Lizia, M. Berz, K. Makino,
*Computing the critical points of the distance function between two Keplerian orbits via rigorous global optimization*, Celestial Mechanics and Dynamical Astronomy**107**(2010), 377-395.

Daniel Wilczak, Jagiellonian University in Kraków, Poland

CONTINUATION AND BIFURCATIONS OF HALO ORBITS IN THE CIRCULAR RESTRICTED THREE BODY PROBLEM

Joint work with **Irmina Walawska**

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-7, room 2.4.

Abstract

We propose a general framework for computer-assisted verifcation of isochoronus, period- tupling or touch-and-go bifurcations of symmetric periodic orbits for reversible maps. The framework is then adopted to Poincaré maps in reversible autonomous Hamiltonian systems.

In order to justify the applicability of the method, we study bifurcations of halo orbits in the Circular Restricted Three Body Problem. We give a computer-assisted proof [1] of the existence of wide branches of halo orbits bifurcating from \(L_{1,2,3}\)-Lyapunov families and for wide range of mass parameter. For two physically relevant mass parameters (Sun-Jupiter and Earth-Moon systems) we prove, that \(L_{1,2}\) branches of halo orbits undergo multiple period doubling, quadrupling and third-order touch-and-go bifurcations.

The computer-assisted proof uses rigorous ODE solvers and algorithms for computation of Poincare maps and their derivatives from the CAPD library [2].

References

- I. Walawska, D. Wilczak,
*Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems*, Commun. Non. Sci. Num. Simul.**74C**(2019), 0-54. - CAPD library: C++ package for validated numerics for discrete and continuous dynamical systems, http://capd.ii.uj.edu.pl.

Piotr Zgliczyński, Jagiellonian University in Kraków, Poland

CENTRAL CONFIGURATIONS IN PLANAR \(n\)-BODY PROBLEM FOR \(n = 5,6,7\) WITH EQUAL MASSES

Joint work with **Małgorzata Moczurad**

Date: 2019-09-16 (Monday); Time: 12:05-12:25; Location: building B-7, room 2.4.

Abstract

We give a computer assisted proof of the full listing of central configuration for \(n\)-body problem for Newtonian potential on the plane for \(n=5,6,7\) with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For \(n=8,9,10\) we establish the existence of central configurations without any reflectional symmetry.

References

- M. Moczurad, P. Zgliczyński,
*Central configurations in planar \(n\)-body problem for \(n=5,6,7\) with equal masses*, arXiv:1812.07279.

Ke Zhang, University of Toronto, Canada

DIFFUSION LIMIT FOR THE SLOW-FAST STANDARD MAP

Joint work with **Alex Blumenthal and Jacopo De Simoi**

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building B-7, room 2.4.

Abstract

We discuss a simple two-dimensional slow-fast system, which is conjugate to the Chirikov standard map with a large parameter. Consider a random initial condition and view the \(n\)th iterate of the slow variable as a sequence of random variables, we prove a central limit theorem for this sequence, under suitable parameter values and time horizon. Our main motivation for studying this model is a phenomenon called "scattering by resonance" in physical systems.