Theory of Dynamical Systems

Annotation

This course is an introduction to the modern theory of dynamical systems.  In the course we present basic concepts, ideas and theorems illustrating them with a wide range of examples.  Emphasis is put on the  topics having fundamental applications in  mathematics and physics. In particular,  they include entropy, chaos, and strange attractors, which are crucial for understanding many physical processes and phenomena.

Prerequisites: Calculus, Linear Algebra, Ordinary Differential Equations, Differential Geometry

Course plan

1. First definitions and examples.  Continuous and discrete dynamical systems, some fundamental examples, the simplest asymptotic properties.
2. Structural stability and typical systems.  Hyperbolic systems, the Smale horseshoe, Arnold’s cat map.  The Andronov–Pontryagin theorem.
3. Topological entropy.  Non-wandering set.
4. Recurrences and attractors.
5. Ergodicity.  Invariant measures.  The Poincare recurrence theorem.  The Birkhoff ergodic theorem.
6. Metric entropy. The Perron measure.  Symbolic dynamics.
7. Local stability. Hyperbolic fixed points.  The Hadamard–Perron theorem. The Hartman–Grobman theorem.
8. Local normal forms.  Resonances.  The Hopf–Andronov bifurcation.
9. Chaos. Strange attractors.  The Feigenbaum universality.

Literature

1. A. Katok, B. Hasselblatt. Introduction to the Modern Theory of Dynamical systems. Cambridge University Press, 2012.
2. V.I.Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, 1988.