Alberto BOSCAGGIN
“Generalized periodic solutions to perturbed Kepler problems”
Abstract: For a perturbed Kepler problem (in dimension 2 or 3) we discuss the existence of periodic solutions, possibly interacting with the singular set. 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, by the use of symplectic and variational methods. Joint works with W. Dambrosio (Torino), D. Papini (Udine), R. Ortega (Granada) and L. Zhao (Augsburg).
Cinzia ELIA
“Periodic orbits of planar discontinuous systems under discretization”
Abstract: We consider a model planar system with discontinuous right-hand side possessing an attracting periodic orbit, and we investigate its persistence under numerical discretization. We were motivated by well known results in the case of discretizations of smooth systems. In this case, the discrete system has an invariant curve close to the original periodic orbit. For discontinuous systems instead, there is no such curve in general, however we show that there is an invariant band for the resulting discrete dynamical system and moreover the band-width is proportional to the discretization stepsize.
We further consider an event-driven discretization of the model problem, whereby the solution is forced to step exactly on the discontinuity line. For this discretised system there is a periodic solution near the one of the original problem (for sufficiently small discretization stepsize).
Finally, we consider what happens to the Euler discretization of the scalar regularized system rewritten in polar coordinates, and give numerical evidence that the discrete solution now undergoes a period doubling cascade with respect to the regularization parameter. The interesting phenomenon here is that the period doubling cascade is triggered by the discontinuity.
This is a joint work with Luca Dieci and Timo Eirola.
Roberta FABBRI
“On the solvability of the Yakubovich minimization problem”
Abstract: The Yakubovich Frequency Theorem, in its periodic and in its general nonautonomous extension, establishes conditions which result to be equivalent to the global solvability of a minimization problem of infinite horizon type, given by an integral of a quadratic functional subject to a control system. The solvability of the minimization problem is formulated in terms of the property of a corresponding linear Hamiltonian systems associated to the problem.
In the talk, some conditions under which the problem is partially solvable are presented ; moreover the set of initial data for which the minimum exists is characterized and the values of the minimum as well of the minimizing pair are provided. In the analysis result to be fundamental the occurence of an exponential dichotomy and the null character of the rotation number for the corresponding linear nonautonomous Hamiltonian system. Joint work with Carmen Nunez.
Maurizio GARRIONE
“Periodic solutions and stability for some models involving rectangular plates”
Abstract: We will present some nonlinear evolution models for rectangular plates, related to the description of the dynamics of suspension bridges. The attention will be focused on time-periodic solutions, for which we will discuss the existence and some stability properties, both from a linear and from a nonlinear point of view. Joint work with J. Chu (Shanghai) and F. Gazzola (Milan).
Carmen NUÑEZ
“Li-Yorke chaos in a noautonomous Hopf bifurcation pattern”
Abstract: We analyze the characteristics of the global attractor of a type of dissipative nonautonomous dynamical systems in terms of the Sacker and Sell spectrum of its linear part.
The model gives rise to a pattern of nonautonomous Hopf bifurcation which can be understood as a generalization of the classical autonomous one. We pay special attention to the dynamics at the bifurcation point, showing the possibility of occurrence of Li-Yorke chaos in the corresponding attractor and hence of a high degree of unpredictability.
This a joint work with Rafael Obaya.
Ken PALMER
“On Sil’nikov saddle-focus homoclinic orbits”
Abstract: Sil’nikov showed that that there is chaos in the neighbourhood of a saddle focus homoclinic orbit. This talk reviews joint work of the speaker with Flaviano Battelli, Brian Coomes and Huseyin Kocak on these orbits.
First an example due to Rodrigues of such an orbit in three dimensions is reviewed. This example is found by perturbing a system which is degenerate in some sense. Other examples in three dimensions have been found from numerical simulations. It is shown how rigorous numerics can be used to prove that these examples are valid.
Sil’nikov originally considered these orbits in 3 dimensions but later he extended his theory to higher dimensions. For these orbits it is necessary to verify three further conditions. Originally these conditions were geometric in nature but they are in fact equivalent to conditions on bounded solutions of the variational equation along the homoclinic orbit. These conditions are more easily verifiable and are used to construct of an orbit in 4 dimensions by again perturbing a degenerate system. A rigorous numerical method is also given for verifying that a homoclinic orbit (in higher dimensions) found by numerical simulations does in fact exist and satisfies all of Sil’nikov’s conditions.
Christian PÖTZSCHE
“Fine structure of the dichotomy spectrum”
Abstract: Our focus is on the dichotomy spectrum (also called dynamical or Sacker-Sell spectrum). It is a crucial notion in the theory of dynamical systems, since it contains information on stability, as well as appropriate robustness properties. However, recent applications in nonautonomous bifurcation theory showed that a detailed insight into the fine structure of this spectral notion is necessary. On this basis, we explore a helpful connection between the dichotomy spectrum and operator theory. It relates the asymptotic behavior of linear nonautonomous difference equations to the point, surjectivity and Fredholm spectra of weighted shifts. This link yields several dynamically meaningful subsets of the dichotomy spectrum, which allows to classify and detect nonautonomous bifurcations
Andrea SFECCI
“Periodic motions in presence of an infinite twist: the harmony of compactness”
Abstract: We present some recent results in the study of existence of periodic solutions for dynamical systems. In 2016, Fonda and Urena proved the higher-dimensional Poincaré-Birkhoff theorem. In the last years, several examples of application appeared. A first attempt to deal with infinite-dimensional Hamiltonian systems has been proposed by Fonda, Boscaggin and Garrione. We propose some recents ideas in the study of infinite-dimensional systems by the use of the Poincaré-Birkhoff Theorem.
Luca ZAMPOGNI
“Some results on the scattering theory for the Sturm-Liouville operator and applications”
Abstract: We state some results related to the scattering theory for the Sturm-Liouville operator on the whole line. We introduce a Gel’fand-Levitan-Marchenko theory, discuss some energy oscillation phenomena and provide some physical applications of an appropriate trace formula representation.
Fabio ZANOLIN
“Subharmonic solutions and chaotic dynamics for a class of periodically perturbed Duffing equations: from the Poincaré-Birkhoff twist theorem to Linked Twist Maps”
Abstract: The Poincaré-Birkhoff fixed point theorem is a fixed point theorem for planar area-preserving homeomorphisms defined on an annular region. It was conjectured and proved in some special cases by Poincaré in 1912. In 1913 Birkhoff provided a first proof in the general case. For more than hundred years there have been attempts to improve and generalize the theorem as well as to provide corrected proofs to some delicate and not completely convincing variants of the main result. In the first part of the talk we survey the history of the theorem and we present some main variants and applications for planar Hamiltonian systems with periodic coefficients, in particular to the search subharmonic solutions to non-autonomous Duffing equations. In the second part of the talk, we present some applications of the theory of the Linked Twist Maps, thus proving the presence of chaotic dynamics for certain classes of periodically perturbed Duffing equations. Connections between the two approaches are investigated.