Syllabus for
Math 550. Dynamical Systems I
The comprehensive exam material is Sections 1,2 and 3. Sections 4-8 are to be covered to the extent possible and as time permits. 5 hours for midterm/s and leeway.
1. (6 hours) Introduction to Continuous Dynamics
- Phase space, vector fields, flows
- Cauchy-Peano existence theorem, uniqueness theorem
- Dependence on initial conditions and parameters
- Compact differentiable manifolds where local flows are global flows
2. (3 hours) Introduction to Discrete Dynamics
- Iteration of maps, fixed points and stability
- Discrete flows on manifolds sphere, torus, surfaces of genus g
- Chaotic behavior Bernoulli shift, Cat Map
3. (3 hours) Linear Differential Equations
- Real and Complex Jordan normal forms
- Stability of linear systems with constant coefficients
- Variation of parameters
- Linearization about a solution and the equation of variation
4. (5 hours) Geometric Methods for Nonlinear Equations
- Limit sets and asymptotic behavior
- Real Canonical Forms in 2 and 3 dimensions
- Phase portrait and topological methods in 2 and 3 dimensions
- Periodic orbits and the Poincare-Bendixson theorem
- Floquet Theory
5. (6 hours) Nonlinear Systems near Equilibrium
- Linearization, Hartman-Grobman Theorem, Stable manifold Theorem
- Almost Periodic Systems
6. (3 hours) Structural Stability
- Smale's horseshoe
- Hyperbolic systems
7. (6 hours) Hamiltonian Systems
- First integrals
- Symplectic matrices
- Poincare maps
8. (6 hours) Bifurcation Theory
- Center manifold Theorem
- Saddle-node bifurcation
- Pitchfork bifurcation
- Hopf bifurcation
Recommended references for Math 550 could be chosen by the instructor from among:
- V. I. Arnold, Geometric methods in Ordinary Differential Equations
- V. I. Arnold, Ordinary Differential Equations
- D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems
- C. Chicone, Ordinary Differential Equations and Applications
- E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations
- J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Note: this text includes a chapter on chaos.)
- A. Katok and B. Hasselblatt, Introduction to the Modern theory of Dynamical Systems
Approved by GAC; syllabus effective Fall 2010 semester.