Dynamical Systems

GENERAL DESCRIPTION

 

This thematic session features talks by several leading experts working across different areas of modern dynamical systems. Topics will range from holomorphic and transcendental dynamics in one and several complex variables to bifurcation theory and piecewise-smooth systems, rigorous computer-assisted dynamics, Hamiltonian and symplectic dynamics, celestial mechanics, and the topology of flows and foliations in dimension three. The talks will highlight connections between complex and smooth dynamics, local bifurcations and global behaviour, Hamiltonian instability and invariant manifolds, and analytic, geometric, topological, and computational methods. The session promises to be accessible not only to the specialists working in dynamical systems, but also to a broad mathematical audience.

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SCHEDULE

 

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ABSTRACTS

 

Universality in trascendental dynamics
by Anna Miriam Benini (University of Parma)

Abstract
The Mandelbrot set is a fractal object encoding the dynamical behaviour of the family of quadratic polynomials z²+c, where c is a parameter varying over the complex plane. It surprisingly appears also in the parameter spaces of all (reasonable) rational maps and in such sense, it also encodes the dynamical behaviour of this much larger class. The explanation is intricate and relies on the concept of renormalization: essentially, renormalization isolates and extrapolates the behaviour of a rational function near its critical values, and brings it back to analogous behaviours for quadratic polynomials. In this work we present an analogous object for transcendental maps, which arises from a model family and yet encodes the dynamical behaviour of all (reasonable) families of transcendental meromorphic maps.
This is joint work with M. Astorg and N. Fagella.

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Melnikov analysis  for piecewise linear vector fields separated by a cubic
by Claudio Buzzi (São Paulo State University)

Abstract
We develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function. We apply this result showing that 7 is a lower bound for the Hilbert number of the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. 

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Non-local ergodic measures in Hamiltonian dynamics
by Jaime Paradela (ICMAT)

Abstract
We consider the direct product of two Hamiltonian diffeomorphisms, one of which exhibits a basic set and the other one a non-degenerate elliptic equilibrium. Under a domination condition we show that a typical real-analytic perturbation of this system displays a large normally hyperbolic set on which the natural invariant measure is ergodic.
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Birkhoff sections
by Ana Rechtman (Université Grenoble-Alpes)

Abstract
In the work of Poincaré on the restricted 3-body problem in celestial mechanics, he stablishes for some energy levels, the existence of an annular section bounded by two periodic orbits of the flow (a section is a surface tranverse to the orbits of a flow that intersects all the orbits). He also found, in another energy level, a hyperbolic periodic orbit with a homoclinic intersection (a horseshoe). These results gave rise to the study of homeomorphisms of the annulus and to the quest for sections of flows on 3-manifolds in one hand, and in the other hand the study of horseshoes. 

In the talk I will review results on the existence of Birkhoff sections (or sections bounded by periodic orbits), starting with the work of Birkhoff and ending with recent developments for Reeb vector fields (associated to a contact form). I will also explain how the existence of a Birkhoff section allows to find many homoclinic intersections for generic Reeb flows.

These results were obtained in collaboration with V. Colin, P. Dehornoy and U. Hryniewicz.

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Lyapunov exponents for a family of tangent-like maps
by Anna Jové (Imperial College)

Abstract 
In a joint work with Matthieu Astorg, we study Lyapunov exponents for a family of meromorphic dynamical systems that arise as natural extensions of the tangent map. We investigate the Lyapunov exponent with respect to the natural invariant measure on the boundary of the attracting region, focusing on its integrability, positivity, and dependence on the parameter. Some of the techniques developed in this setting extend to a broader class of meromorphic dynamical systems with simply connected attracting basins.

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Arnold Diffusion and Blenders in the Three-Body Problem
by Maciej Capiński (University of Kraków)

Abstract 
We present a computer-assisted methodology based on the propagation of topological discs and its use in analysis of global dynamics. The method combines topological techniques with validated numerics and can be applied to detect both diffusion as well as persistence of symbolic dynamics. We discuss its applications to Arnold diffusion and blenders in celestial mechanics.