C2 F UNDAMENTAL THEORY OF DYNAMICAL SYSTEMS
COURSE OUTLINE
1. WHAT IS A DYNAMICAL SYSTEM?
1.1
1.2
1.3
1.4
Maps (discrete time dynamical systems)
Flows (continuous time dynamical systems)
Solutions and orbits
Poincaré and time T maps (relationship between maps and flows)
2. TYPES OF BEHAVIOUR
2.1
2.2
2.3
2.4
Fixed points of maps and flows
Periodic orbits of maps and flows
Invariant sets
Recurrence and transitivity
3. STABILITY AND ATTRACTING BEHAVIOUR
3.1
3.2
3.3
Stability of fixed points and periodic orbits
Stability of invariant sets, attractors
Determining stability
4. CO-ORDINATE CHANGES
4.1
4.2
4.3
4.4
4.5
Making the system look as simple as possible
Linear co-ordinate changes, revision of eigenvalues and eigenvectors
Non-linear co-ordinate changes for maps
Non-linear co-ordinate changes for flows
Symbolic coding
5. LINEARIZATION
5.1
5.2
Hartman-Großmann theorem
Implications for stability
6. BEHAVIOUR OF LINEAR SYSTEMS
6.1
6.2
6.3
Linear co-ordinate changes revisited: Jordan Normal Form
Explicit solution of linear systems
Linear stability
7. P ERSISTENCE AND STRUCTURAL STABILITY
7.1
7.2
7.3
7.4
Persistence of fixed points; continuation
Implicit function theorem
Structural stability
Bifurcations
8. STABLE AND UNSTABLE MANIFOLDS (TIME PERMITTING)
8.1
8.2
8.3
Local definition
Decomposition of dynamics
Examples
Course materials will be posted at http://www.ucl.ac.uk/cnda/coursework#c2