## Dynamical Systems

(mastermath)

• Aim:
The aim of this course is to introduce the student to concepts, examples, results and techniques for studying smooth dynamical systems generated by ordinary differential equations or maps. The student learns to apply techniques from topology and analysis to study properties of dynamical systems.

• Description:
We provide a broad introduction to the subject of dynamical systems. In particular we develop theory of topological dynamics, symbolic dynamics and hyperbolic dynamics. Several examples are used to illustrate the theory and clarify the development of the theory.

An aim of dynamical systems theory is to describe asymptotic properties of orbits for typical initial points. The strength and beauty of the theory lies herein that techniques to do so work not only for special examples but for large classes of dynamical systems. The focus of the course will always be on learning techniques to analyse dynamical systems without relying on explicit formulas for the dynamical system.

As an example, the hyperbolic torus automorphism \$(x,y) \mapsto (2 1 // 1 1) (x,y) \mod 1\$ on the torus \$R^2/Z^2\$ is a topologically transitive dynamical system for which most orbits lie dense in the torus. What makes the example relevant is that small perturbations of it share its relevant properties. The automorphism is for instance \$C^1\$-structurally stable, so that a \$C^1\$ small perturbation is also topologically transitive. To see this requires much more advanced techniques than needed to study the linear automorphism. These techniques rely on the construction of stable and unstable manifolds.

The stable manifold theorem is among the highlights of the course. Another central result we cover is the structural stability theorem for hyperbolic sets.

A topical description of contents:
-- Topological dynamics. Notions to describe attractors, limit sets and chaotic dynamics such as recurrence, topological transitivity, topological mixing.
-- Symbolic dynamics and their use to study chaotic dynamics. Full shift. Subshift of finite type. Topological Markov chain.
-- Aspects of bifurcation theory
-- Examples of chaotic dynamical systems such as hyperbolic torus automorphisms, the Smale horseshoe map and the solenoid.
-- Hyperbolic dynamics. Stable manifolds. Shadowing (finding real orbits near approximate orbits).
-- Structural stability and its relation with hyperbolicity. Shadowing as a technique to study structural stability.

• Organisation:
2x45 min lectures + 45 min exercise session per week.

• Examination:
Two larger sets of homework exercises will be given. The end grade is determined from these homework sets and an individual written exam, both counting for half the grade.

• Literature:
Michael Brin and Garrett Stuck
Introduction to Dynamical Systems
Cambridge University Press

• Prerequisites:
Prerequisite is material covered in a standard bachelor program in mathematics, containing in particular a bachelor course on ordinary differential equations and topology. In dynamical systems theory, results for dynamical systems generated by maps or differential equations are developed in parallel. Our focus will be on dynamical systems generated by maps. A bachelor course on differential equations treats how a differential equation gives rise to a flow, i.e. a dynamical system, and starts a study of its qualitative properties. Notions and techniques from topological dynamical systems are used throughout the course and require knowledge of topology as taught in a bachelor programme.

Contents

• September 8-October 20:
lectured by Bob Rink.
• October 27: Section 5.1 (Expanding endomorphisms revisited), Section 5.2 (Hyperbolic sets): Prop. 5.2.1
Suggested exercises: 5.2.1-5.2.7
• November 3: Section 5.2 (Hyperbolic sets): Prop. 5.2.2, Section 5.4 (Invariant cones), Section 5.5 (Stability of hyperbolic sets): Prop. 5.5.1, introduction to Section 5.6 (Stable and unstable manifolds)
Suggested exercises: 5.2.3, 5.4.4
• November 10: Section 5.6 (Stable and unstable manifolds), Section 5.9 (Local product structure and locally maximal hyperbolic sets)
Suggested exercises: 5.7.3, 5.9.1, 5.9.2
• November 17: Section 5.3 (Epsilon-orbits), Section 5.5 (Stability of hyperbolic sets)
Suggested exercises: 5.3.2, 5.3.5, 5.5.1
• November 24: Section 5.10 (Anosov diffeomorphisms), Section 5.11 (Axiom A and structural stability)
Suggested exercises: 5.11.1, 5.11.3
• December 1: Section 5.12 (Markov partitions), additional material is in Appendix 2 from Palis, Takens "Hyperbolicity & sensitive chaotic dynamics at homoclinic bifurcations"
Suggested exercises: 5.12.4
• December 8: Section 7.2 (Circle diffeomorphisms), Section 7.4 (Combinatorial theory of piecewise-monotone mappings)
Suggested exercises: 7.4.3
• December 15: Section 7.5 (The Schwarzian derivative): Theorem 7.5.3, outlook at advanced topics and research (Nonuniformly hyperbolic systems, partially hyperbolic systems, Palis programme on classifying typical dynamics)
Suggested exercises: 7.5.2, 7.5.3