Chaotic Dynamical Systems

Entry requirements

120 credit points and Several Variable Calculus, Linear Algebra II, Ordinary Differential Equations I

Learning outcomes

In order to pass the course (grade 3) the student should

• be able to linearize dynamical systems and to determine fix points and their stability properties;
• be able to determine periodic orbits and limit cycles;
• be able to create bifurcation diagrams for families of dynamical systems;
• be able to formulate, prove and apply theorems of existence and uniqueness for the solutions of ordinary differential equations, Poincaré–Bendixson’s theorem and Grönwall’s lemma;
• be able to construct Lyapunov functions and Poincaré maps;
• be able to perform numerical studies of dynamical systems;
• have acquired a good knowledge of hyperbolicy, stable and unstable manifolds, homoclinic phenomena, structural stability, symbolic dynamics, and dependence of parameters and initialvalues;
• be able to describe the construction of some common rare attractors;
• be able to describe some common applications of the theory.

Content

Existence and uniqueness theorems for solutions of ordinary differential equations, numerical methods, flows, parameter and initial value dependence, fix points, periodic orbits, limit cycles, linearisation, stability and Lyapunov functions, phase portraits, Poincaré–Bendixson’s theorem, Grönwall’s lemma, Poincaré maps. Structural stability, symbolic dynamics, conjugation, bifurcation theory, stable and unstable manifolds, homoclinic phenomena, hyperbolicy, chaos and sensitive dependence on initial data, strange attractors. Applications.

Instruction

Lectures and problem solving sessions.

Assessment

Oral examination.