Syllabus Math 550

Math 550. Dynamical Systems I Instructor Syllabus

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.