MA-624 Differential Equations and Dynamical Systems

Emnet er tilknyttet følgende studieprogram

• Ph.d.-program i teknologi og realfag

Innhold

This course considers a variety of material in ordinary differential equations, difference equations and dynamical systems. Topics covered will be chosen from the following list. Other related topics may be covered at the instructor¿s discretion.

• Nonlinear ordinary differential equations: local and global existence of solutions, uniqueness, maximal interval of existence, dependence on initial data

• Linear systems, fundamental properties

• Floquet theory for periodic systems

• Algebraic and topological classification of equilibria for linear systems

• Non-simple equilibria

• Phase-plane methods

• Stability by linearization

• Stable and unstable manifolds

• Lyapunov¿s direct method and LaSalle¿s invariance principle

• Center manifolds and normal forms

• Limit sets and attractors

• Periodic orbits

• Poincaré map

• Limit cycles and Poincaré-Bendixon theory

• Sturm theory

• Structural stability

• Elementary bifurcations and Poincaré-Andronov-Hopf bifurcation

• Difference equations (maps), period doubling, bifurcations, chaos

Læringsutbytte

On successful completion of the course, the students should be able to:

• analyze local and global existence and nonexistence of solutions to differential equations and determine maximal intervals of existence

• analyze fundamental properties of solutions to differential equations

• apply basic theory to systems of linear differential equations, including periodic systems

• classify equilibrium points and use phase plane methods

• analyze stability properties of nonlinear di¤erential equations using linearization and Lyapunov¿s direct method

• investigate existence of periodic solutions

• apply Sturm theory to second order differential equations

• classify and analyze bifurcations in nonlinear systems

• analyze discrete dynamical systems

Undervisnings- og læringsformer

Lectures + Examples Classes

Eksamen

Oral Examination. Pass/Fail