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The aim of the lecture course is to show the diversity of problems and
methods of dynamical systems. The lecture course consists of several
independent topics. We consider finite and infinite-dimensional dynamical
systems and discuss the classical results as well as the results obtained in the
last twenty years.
- Dynamical systems with discrete time: the Poincare mapping at an energy
level in a Hamiltonian system; other examples. Invariant measure, the ergodic
theorems of Yosida, von Neumann, and Birkhoff. The Poincare recurrence
theorem. Introduction to ergodic systems: basic theorems, the Koopman
operator. Ergodicity of the tent map, ergodicity of the shift in the torus.
- Other causes of the dynamical chaos’ appearance: separatrix splitting. The
Poincare-Melnikov integral.
- Discrete Lagrangian systems: the anti-integrable limit.
- Time averaging in another context: an infinite-dimensional version of the
Massera theorem and its application to elasticity theory.
- Bounded solutions to the systems of the second order ODE.
- A phase flow of an ODE with a smooth right-hand side: w—limit set and its
properties, attractors.
- ODE with discontinuous right-hand side, differential inclusions Filippov’s
regularization, periodic solutions to systems with dry friction.
References
[1] V. Arnold, Ordinary Differential Equations. Springer, 1992.
[2] R. Temam, Infinite-dimensional dynamical systems. Springer, 1993.
[3] D. Treschev, O. Zubelevich, Introduction to the Perturbation Theory of Hamiltonian
Systems. Springer, 2010.
[4] A.F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Transaction of
A.M.S., 42 (1964), pp. 199-231.
 RSS: Forthcoming seminars
 Lecturer
Zubelevich Oleg Èduardovich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
