Summary of the course
Master's degree title: Applied Mathematics
Semester: S3
EU Title: Fundamental EU UEF3.1
Subject title: Bifurcation theory and chaos
Credits: 6
Coefficients: 3
Teaching objectives
This course aims to study dynamic systems qualitatively, Determine the bifurcation points and classify them, know how to characterise a chaotic dynamic system and calculate the fractal dimension of a strange attractor using different approaches
Recommended prior knowledge
Concepts of dynamic systems (critical points, limit cycles, stability, invariant manifolds)
Content of the subject:
Chapter 1 : Global properties of nonlinear systems
- Limit sets - Periodic orbits
- Poincaré map (application) - Stable manifold, Unstable manifold, Central manifold,
-
Poincaré-Bendixson
theorem in
Chapter 2 : Local bifurcations of codimension 1 .
- Structural stability and normal forms - Fork bifurcation
- Transcritical Bifurcation (Stability Exchange) - Col-Node Bifurcation
- Hopf bifurcation
Chapter 3 : Chaotic Systems
- Properties of Chaos - Transition Scenarios to Chaos,
- Lyapunov Exponents - Fractal Dimensions of Strange Attractors.
Evaluation method: Continuous evaluation ( 33%) and final exam ( 67%).
References
1) H.Dang-Vu , C. Delcarte 'Bifurcations and chaos: Introduction to contemporary dynamics with programs in Pascal, Fortran and Mathematica' Ellipses p 420 (2000)
2) L. Perko 'Differential Equations and Dynamical Systems' Springer (2006)
3) L. Afifi , A. ElJai , E. Zerrik 'Dynamic systems' Perpignan University Press p408 (2009)
4) JP Demailly 'Numerical analysis and differential equations' EDP science, France (2006)
5) WE Boyce, RC Diprima 'Elementary differential equations and boundary value problems' New York(1996)
6) C. Goumez 'Dynamic systems' (2007)
7) ER Scheinerman 'Invitation to Dynamical Systems' ( 2000)