# Recent Trends in Nonlinear Science 2020

- Dates:
- 03.02.20 - 07.02.20
- Place:
- Bellaterra, Spain
- Organizers:
- C. Cufí, A. Garijo, X. Jarque, and J. Villadelprat
- Web Site:
- http://www.crm.cat/en/Activities...

The Advanced Course ** Recent Trends on Dynamical Systems**, apart from being the starting point of the IRP on

*Low dimensional dynamical systems and applications*, will also be the 17th international school of the series

*Recent Trends in Nonlinear Science*by the DANCE (Dinámica, Atractores y Nolinealidad: Caos y Estabilidad) Spanish network.

Course 1: **Finiteness theorems for limit **cycles

by Yu. Ilyashenko (Cornell University / Steklov Math. Institute of Russian Acad. of Sciences)

Abstract: A polynomial vector field in the plain has but a finite number of limit cycles. Limit cycles of an analytic vector field cannot accumulate to a polycycle of this field. The first theorem is an immediate corollary of the second one. My current proof consists of the three parts:

1. reduction from the ODE to complex analysis

2. proofs of the theorems above for vector fields with the so called ``alternant'' polycycles (those in which the maps TO and FROM central manifolds of the saddlenodes alternate)

3. proof for the general case.

There are some gaps in the third part that I am filling now. The lectures will give a brief, yet comprehensive presentation of the first two parts, and a survey of the current state of matters for the third part.

Course 2: **Continuity of Lyapunov exponents**

by Marcelo Viana (IMPA)

Abstract: I will introduce the notions of linear cocycle and Lyapunov exponent, and will sketch of the proof of the fact that the Lyapunov exponents of random products of matrices vary continuously with the data.

This is based on joint work with C. Bocker, A. Avila and A. Eskin.

Course 3: **Dynamical aspects of skew products over minimal dynamics **

by Mario Ponce (Pontificia Universidad Católica de Chile)

Abstract: We will introduce the skew-product dynamics, especially those whose transformation in the fibre corresponds to an isometry. We will show how these simple dynamics give rise to several phenomena that are governed both by the arithmetic properties of the rotation numbers, as well as by geometric properties that force some rigid dynamic behaviours. In particular we will study the dynamics associated to invariant sections, to bounded orbits, among other phenomena. These skew-products constitute a geometric model for the resolution of linearised equations of more complex dynamics. We will investigate about the geometric and dynamic methods that allow us to solve such equations.

This Advanced Courese is part of the Intensive Research Program LOW DIMENSIONAL DYNAMICAL SYSTEMS AND APPLICATIONS at Centre de Recerca Matemàtica from February to April 2020.