S. Marò (University of Oviedo, Spain), Dynamics of a breathing circle billiardAbstract. We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map admits invariant probability measures with positive metric entropy. The proof relies on variational techniques based on Aubry-Mather theory. Joint work with Claudio Bonanno.
December 5, 2022, 14:00 (UTC+1)
M. Sabatini (University of Trento, Italy), Local vs global properties of planar vector fieldsAbstract. It is well-known that a locally invertible map need not be globally invertible. In this talk we review some results in which the global invertibility of locally invertible maps is proved under suitable additional assumptions. In particular we consider results related to the celebrated, yet unsolved Jacobian Conjecture. We also deal with similar problems arising in stability theory, and show the connections among the two fields.
November 10, 2022, 15:00 (UTC+1)
E. Liz (University of Vigo, Spain), A dynamical model of happinessAbstract. It is now recognized that the personal well-being of an individual can be evaluated numerically. The related hedonic utility (happiness) profile would give at each instant t the degree u(t) of happiness. The moment-based approach to the evaluation of happiness introduced by the Nobel laureate Daniel Kahneman establishes that the experienced utility of an episode can be derived from real-time measures of the pleasure and pain that the subject experienced during that episode. Since these evaluations consist of two types of utility concepts: instant utility and remembered utility, a dynamical model of happiness based on this approach must be defined by a delay differential equation. Furthermore, the application of the peak-end rule leads to a class of delay-differential equations called differential equations with maxima. We propose a dynamical model for happiness based on differential equations with maxima and provide rigorous mathematical results which support some experimental observations such as the U-shape of happiness over the life cycle and the unpredictability of happiness. The talk is based on joint work with Elena Trofimchuk and Sergei Trofimchuk.
May 26, 2022, 14:30 (UTC+2)
A. Ruiz-Herrera (University of Oviedo, Spain), Topology of attractors and periodic pointsAbstract. The dynamics of a dissipative and area contracting planar homeomorphism is described in terms of the attractor. This is a subset of the plane defined as the maximal compact invariant set. We prove that the coexistence of two fixed points and an N-cycle produces some topological complexity: the attractor cannot be arcwise connected. The proofs are based on the theory of prime ends. We discuss several applications in periodic systems of differential equations. This is a joint work with Rafael Ortega.
May 12, 2022, 14:30 (UTC+2)
O. Makarenkov (UT Dallas), The occurrence of stable limit cycles in the model of a planar passive biped walking down a slopeAbstract. We consider the simplest model of a passive biped walking down a slope given by the equations of switched coupled pendula. Following the fundamental work by Garcia et al. [J. Biomech. Eng. 120 (1998)], we view the slope of the ground as a small parameter γ≥0. When γ=0, the system can be solved in closed form and the existence of a family of cycles (i.e. potential walking cycles) can be computed in closed form. As observed in the paper by Garcia et al., the family of cycles disappears when γ increases and only isolated asymptotically stable cycles (walking cycles) persist. The talk presents a proof of this statement using a suitable perturbation theorem for maps. I will also note that the above-mentioned occurrence of limit cycles observed by Garcia et al. is a so-called border-collision bifurcation in the modern language of nonsmooth dynamical systems.
April 28, 2022, 16:30 (UTC+2)
A. Fonda (University of Trieste), The Poincaré-Birkhoff theorem: coupling twist with lower and upper solutionsAbstract. In 1983, Conley and Zehnder proved a remarkable theorem on the periodic problem associated with a general Hamiltonian system, giving a partial answer to a conjecture by V.I. Arnold. In the same paper they also mentioned a possible relation of their result with the Poincaré-Birkhoff Theorem, which was first conjectured by Poincaré in 1912, shortly before his death, and then proved by Birkhoff some years later. The pioneering paper by Conley and Zehnder has then been extended in different directions by several authors.
More recently, in 2017, a deeper relation between these results and the Poincaré-Birkhoff Theorem has been established by A.J. Urena jointly with myself. Our theorem has found several applications and has been further extended in two papers written jointly with P. Gidoni. It is the aim of this talk to propose a further extension of this fertile theory to Hamiltonian systems which, besides the periodicity-twist conditions always required in the Poincaré-Birkhoff Theorem, also present a pair of well-ordered lower and upper solutions.
April 7, 2022, 14:30 (UTC+2)
L. Zhao (University of Augsburg), Conformal Transformations and Integrable Mechanical BilliardsAbstract. The models we shall discuss are motions of a particle in the plane moving under the influence of a conservative force field which in addition reflect elastically against certain smooth reflection "wall". The dynamics of such a system depends on the force field and the shape of the reflection wall. While one could believe that the dynamics should generally be complicated, some of these systems are actually integrable and thus carry dynamics with order. In this talk we shall explain how conformal correspondence of natural mechanical systems extends to correspondence between integrable mechanical billiards. This provides a link between some apparently different integrable mechanical billiards, and also allows us to identify certain new integrable mechanical billiards defined with the Kepler and the two-center problems. The talk is based on joint work with Airi Takeuchi from Karlsruhe Institute of Technology.
March 24, 2022, 14:30 (UTC+1)
C. Soresina (Universität Graz), Multistability and time-periodic spatial patterns in the cross-diffusion SKT modelAbstract. The Shigesada-Kawasaki-Teramoto model (SKT) was proposed to account for stable inhomogeneous steady states exhibiting spatial segregation, which describes a situation of coexistence of two competing species. Even though the reaction part does not present the activator-inhibitor structure, the cross-diffusion terms are the key ingredient for the appearance of spatial patterns. We provide a deeper understanding of the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearised and weakly non-linear analysis with advanced numerical bifurcation methods via the continuation software pde2path. We study the role of the additional cross-diffusion term in pattern formation, focusing on multistability regions and on the presence of time-periodic spatial patterns appearing via Hopf bifurcation points.
March 10, 2022, 14:30 (UTC+1)
A. Slavík (Charles University, Czech Republic), Reaction-diffusion equations on graphs: stationary states and Lyapunov functionsAbstract. We focus on reaction-diffusion systems on discrete spatial domains represented by finite graphs (networks). In some situations, such systems are more natural than their continuous-space counterparts, and their qualitative behavior might be different. For example, unlike the continuous-space model, the discrete-space Lotka-Volterra competition model has stable spatially heterogeneous stationary states. For a fairly general class of reaction-diffusion systems, the existence of spatially heterogeneous stationary states is guaranteed by the implicit function theorem, provided that the diffusion is sufficiently weak. In some applications, the only relevant stationary states are those with nonnegative components. We present a criterion for determining which states obtained from the implicit function theorem are nonnegative. Finally, we consider the problem of constructing Lyapunov functions for reaction-diffusion equations on graphs. The results will be illustrated on examples from mathematical biology.
February 25, 2022, 16:00 (UTC+1)
P. Amster (University of Buenos Aires), On a theorem by Browder and its application to nonlinear boundary value problemsAbstract. In a paper from 1960, Felix Browder established a result concerning the continuation of the fixed points of a family of continuous functions of a compact convex subset of the n-dimensional space depending continuously on a real parameter. In this talk, we shall present a simple extension of Browder's theorem to Banach spaces and a more general family of parameters and show applications to some nonlinear boundary problems.
December 10, 2021, 15:00 (UTC+1)
J. A. Cid (University of Vigo), Brouwer fixed point theorem and periodic solutions of ODE'sAbstract. In this talk we will present the equivalence between the Brouwer fixed point theorem and the existence of periodic solutions for some second order differential equations defined in convex sets and that satisfy a suitable tangency condition at the boundary. Our approach involves some basic concepts of convex analysis and Stampacchia's method and as an application we will obtain also a very short proof, using the Brouwer fixed point theorem, of a result by Fonda and Gidoni on the existence of zeros of some mappings in convex sets. This is a joint work with J. Mawhin.
November 10, 2021, 17:00 (UTC+1)
E. Affili (University of Deusto), A mathematical model for civil wars: a new Lotka-Volterra competitive systemAbstract. Imagine two populations sharing the same environmental resources in a situation of open hostility. The interactions among these populations are governed not by random encounters but via the strategic decisions of one population that can attack the other according to different levels of aggressiveness. This leads to a non-variational model for the two populations at war, taking into account structural ecological parameters. The analysis of the dynamical properties of the system reveals several equilibria and bifurcation phenomena. Moreover, we present the strategies that may lead to the victory of the aggressive population, i.e., the choices of the aggressiveness parameter, in dependence of the structural constants of the system and possibly varying in time in order to optimize the efficacy of the attacks, which take to the extinction in finite time of the defensive population. The model that we present is flexible enough to include also technological competition models of aggressive companies releasing computer viruses to set a rival companies out of the market. This is a joint work with S. Dipierro, L. Rossi and E. Valdinoci.
October 27, 2021, 17:00 (UTC+2)
P. Gidoni (Czech Academy of Sciences), Existence of a periodic solution for superlinear second order ODEsAbstract. We prove a necessary and sufficient condition for the existence of a T-periodic solution for the time-periodic second order differential equation x''+f(t,x)+p(t,x,x')=0, where f grows superlinearly in x uniformly in time, while p is bounded. Our method is based on a fixed-point theorem which uses the rotational properties of the dynamics.
September 29, 2021, 17:00 (UTC+2)