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CRM > English > Activities > Curs 2019-2020 > IRP LOW DIMENSIONAL DYNAMICAL SYSTEMS AND APPLICATIONS: ADVANCED COURSE ON Recent Trends in Nonlinear Science
General information
This Advanced Course is part of the


From February 3rd to 7th, 2020.


Advanced course's information

This Advanced Course, 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.

The school will consist on the following three advanced courses of 9 hours each: 

Course 1: Finiteness theorems for limit cycles 

by Yu. Ilyashenko 

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. 1. reduction from the ODE to complex analysis
  2. 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. 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

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

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.

Poster session 
There will be a poster session addressed to young researchers. In order to apply please submit the following form:

Deadline: December 15th, 2019. (Resolutions will be sent by email​ by December 20th, 2019).


Regular registration fee: 300€
Late registration fee (after December 1st, 2019): 400€

Deadline for registration: January 24th, 2020
Registration fee covers the documentation package, coffee breaks, lunches and the social dinner
Refund policy:
The registration fee depends on the date of the payment.
Cancellations received 1 month before the start of the activity will incur in an administrative fee of 50of the total amount.
Cancellations received less than one month prior to the start of the activity are not refundable.


You will be able to pay by card or bank transfer through the on-line registration system.

If you need an invoice please send an email to​ after you have completed the registration process until the "save data" button. ​

To apply for a grant it is necessary to click first on the 'Register button' (registration section). Only applications from registered (+ 'Save Data') participants will be considered.
In order to increase the number of young researchers participating in this activity, the CRM announces a call for those participants interested in taking part in this activity. This grant includes a reduced registration fee and/or housing in a shared apartment on campus from February 2nd to February 7th, 2020 (5 nights).
Application Grant Form (closed)

Application deadline for grants: October 31st, 2019 (Resolutions will be sent by email by November 8th, 2019).​

Lodging information

For lodging in the area please click here​

For off-campus and family accommodation click here​​​

Organizer committee

​Clara Cufí ​Universitat Autònoma de Barcelona
Antonio Garijo ​Universitat Rovira i Virgili
Xavier Jarque ​Universitat de Barcelona
​Jordi Villadelprat Universitat Rovira i Virgili​

Further information

The EMS offers some travel grants to young mathematicians from less-favoured regions within the geographical area of EMS membership for presenting results at conferences or attending courses, or for research stays in foreign countries, normally up to a maximum of 900 euros in each case or 500 euros for trips within Europe. Eligible researchers should use this online form​ in order to apply for travel grants.

For inquiries about the program please contact the research programme's coordinator Ms. Núria Hernandez at​

Back to IRP​ in Low Dimensional Dynamical Systems and Applications​ web page​​​