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  • Summary of the course

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    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)


  • Chapter 1 : Global properties of nonlinear systems

  • Chapter 2: Local Bifurcations of Co-dimension 1

  • Chapter 3: Chaotic systems

  • Examen+corrigé_type+évaluation

    Marqué

  • Bibliographie

    1)    H.Dang-Vu, C. Delcarte ‘Bifurcations et chaos : Introduction à la dynamique contemporaine avec des programmes en Pascal, Fortran et Mathématica ‘Ellipses  p 420 (2000 )

    2)    L. Perko ‘Differential Equations and Dynamical Systems’ Springer  (2006)

    3)    L. Afifi, A. ElJai, E. Zerrik ‘Systèmes dynamiques’ Presses universitaires de Perpignan p408 (2009)

    4)    J.P. Demailly ‘Analyse numérique et équations différentielles’ EDP science, France (2006)

    5)    W.E. Boyce, R.C. Diprima ‘Elementary differential equations and boundary value problems’  New York  (1996)

    6)    C. Goumez  ‘Systèmes dynamiques’  (2007)

    7)    E. R. Scheinerman ’Invitation to Dynamical Systems’ (2000)

    8)    J.M. Ginoux ‘Stabilités des systèmes dynamiques chaotiques et variétés singulières’ Thesis Corpus  (2006)


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  • Section 8