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Research seminar "Modern theory of dynamical systems"

2021/2022
Academic Year
ENG
Instruction in English
5
ECTS credits

Instructor

Course Syllabus

Abstract

The course is devoted to the description of the current state of the theory of dynamic systems from the point of view of topological classification of important classes of systems with regular and chaotic dynamics arising in the modeling of various processes of natural science.
Learning Objectives

Learning Objectives

  • To show the current state of research in the field of dynamic systems, the beginning of which dates back to the classical works of A. Poincare, A. M. Lyapunov, A. A. Andronov and his school of nonlinear oscillations, which determined many achievements of specialists in the direction of studying the stability and theory of bifurcations of dynamic systems for many years to come.
  • To explain the phenomenon of "hyperbolic revolution", begun in the works of D. V. Anosov and S. Smale, and the associated cascade of works in the field of topological classification of dynamic systems on manifolds with complex dynamics.
  • To give an idea of the methods of the qualitative theory of dynamic systems, closely related to the study of the topology of the phase spaces on which they are defined.
Expected Learning Outcomes

Expected Learning Outcomes

  • Know calssification theorem for cascades on surfaces
  • understanding of dynamics on the circle
  • Understanding of the interrelation between dynamics and topology of ambient manifold. T
  • Understanding resuluts on topological structure of multidimensional manifolds (dimensions greater than three) that admit Morse-Smale flows and cascades without heteroclinic intersections of saddle periodic points with Morse index 1 or n-1.
Course Contents

Course Contents

  • Theme 1. Dynamics of one-dimensional mappings.
  • Theme 2. Classification of flows, foliations and cascades on surfaces with regular and chaotic dynamics.
  • Theme 3. The topology of three-dimensional manifolds and a description of structurally stable flows and cascades, the dynamic properties of which are most closely related to the topological characteristics of the carrier phase space.
  • Theme 4. Topological structure of multidimensional manifolds (dimensions greater than three) that admit Morse-Smale flows and cascades without heteroclinic intersections of saddle periodic points with Morse index 1 or n-1.
Assessment Elements

Assessment Elements

  • non-blocking реферат (доклад)
  • non-blocking итоговый опрос
Interim Assessment

Interim Assessment

  • 2021/2022 1st module
  • 2021/2022 2nd module
    0.7 * итоговый опрос + 0.3 * реферат (доклад)
  • 2021/2022 3rd module
  • 2021/2022 4th module
    0.7 * итоговый опрос + 0.3 * реферат (доклад)
Bibliography

Bibliography

Recommended Core Bibliography

  • Grines V., Medvedev Timur, Pochinka O. Dynamical Systems on 2- and 3-Manifolds. Switzerland : Springer, 2016.
  • Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L.O. Methods Of Qualitative Theory In Nonlinear Dynamics (Part II). World Sci //Singapore, New Jersey, London, Hong Kong. – 2001.

Recommended Additional Bibliography

  • Dynamical Systems on 2- and 3-Manifolds, XXVI, 295 p., Grines, V. Z., Medvedev, T. V., Pochinka, O. V., 2016