Dynamical systems : deterministic dynamics and fluctuations

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Stages L3
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Actualités : Séminaire de Recherche ICFP
du 15 au 19 novembre 2021 :

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Contact - Secrétariat de l’enseignement :
Tél : 01 44 32 35 60
enseignement@phys.ens.fr

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Faculty : Stephan Fauve
Tutors : Adrian Van Kan, Lionnel Djadaojee
ECTS credits : 6
Language of instruction : English
Key words : Dynamical systems, instabilities, fluctuations, stochastic processes
Exam : written
Covid-19 alternative examination : Oral examination

Description

The first part of the lectures concerns deterministic dynamical systems, i.e. systems governed by coupled differential equations. Most dynamical problems in physics and elsewhere can be modeled that way. Interesting dynamics occur when the governing equations are nonlinear which is the usual case. In contrast to linear systems, nonlinear ones generally have multiple solutions of different qualitative nature, for instance different symmetries. The stability of these solutions is modified at bifurcation points when a control parameter of the system is changed. We will show that universal behaviors are observed in the vicinity of these bifurcations mostly dependent on symmetries. We will then show how a sequence of successive bifurcations can lead to chaos, i.e. a random dynamical behavior although the system is governed by deterministic rules.

In the second part of the lectures, randomness is modeled by assuming that the system is submitted to fluctuations with simple statistical characteristics. After studying the Langevin equation and fluctuation-dissipation relations, we will show in which limits a system submitted to random noise can be modeled using the Fokker-Planck or the master equations. We will then consider some effects of external noise on instability phenomena, such as the amplification of fluctuations in the vicinity of a bifurcation, squeezing of noise or on-off intermittency.

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Prochain Séminaire de la FIP :
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Retrouvez toutes les informations pour vos stages :
Stages L3
Stages M1 ICFP

Actualités : Séminaire de Recherche ICFP
du 15 au 19 novembre 2021 :

Retrouvez le programme complet

Contact - Secrétariat de l’enseignement :
Tél : 01 44 32 35 60
enseignement@phys.ens.fr

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