Fiche matière Introduction aux systèmes dynamiques
Summary of the course
Master's degree title: Applied Mathematics
Semester: S1
EU Title: Fundamental EU UEF3.1
Subject title: Introduction To Dynamics Systems
Credits: 6
Coefficients: 3
Evaluation method: Continuous evaluation ( 33%) and final exam ( 67%).
by Rouibah KhaoulaTeaching objectives
Presentation of basic notions of the theory of dynamic systems. The qualitative study of dynamic systems, links between the stability of the nonlinear system and that of the linearized, Lyapunov function. Applications in biology, mechanics, etc.
Recommended prior knowledge
General notions of differential equations
Linear algebra (vector space, matrices, eigenvalues, etc.)
Content of the subject:
chapter: Ordinary differential equations
- Cauchy-Lipschitz theorem (Existence and local and global uniqueness, dependence on initial data and parameters)
- Linear differential systems (global existence, resolvent, Duhamel formula)
Chapter 2: Notions of dynamical systems
- Definitions (Vector fields, flow, orbit, phase portrait)
- Some general properties of the flow
- Examples of dynamical systems
Chapter 3: Stability theory
- Stability of linear systems (under stable, unstable and central space)
- Stability of nonlinear systems (linearization, Lyapunov function)
- Hyperbolic fixed points (theorem of Hartman-Grobman, theorem of the stable, unstable and central manifold)
- Invariant manifolds
- P. Auger, C. Lett et J. C. Poggiale. Modélisation mathématique en écologie : Cours et exercices corrigés. Dunod, Paris, 2010.
- G. C. Layek. An introduction to dynamical systems and chaos. Springer, India, 2015.
- J. L. Pac. Systèmes dynamiques : Cours et exercices corrigés. Dunod, Paris, 2016.
- H. O. Peitgen, H. J\"{u}rgens et D. Saupe. Chaos and Fractals. Springer-Verlag, New York,Inc , 2004.
- L. Perko. Differential Equations and Dynamical Systems. Springer, 2006.