Math 539 (Ordinary Differential Equations). Fall 2009

Instructor: Steve Krone

Office: 421 Brink Hall. Phone: 885-6317
Office Hours: any time you can find me

Class Time: MWF 2:30 - 3:20
Place: FRC 201

Prerequisite: Advanced Calculus

Text:J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press (1998).

This is a graduate course on ordinary differential equations and dynamical systems. The main focus of the course will be on the geometric (or qualitative) theory of nonlinear differential equations, and applications of these ideas to problems in mathematical biology. There will be a bit of overlap with Math 437 (mathematical biology), but the mathematical content will be significantly higher. We will study things like existence and uniqueness of solutions, continuous dependence on initial conditions, eigenvalues, stability of equilibria, flows, limit cycles, bifurcation, Lyapunov functions and global stability, attractors, invadability conditions, and Nash equilibria. Despite the strange title, our text is a beautifully-written book which is full of nice mathematics and good ideas. It is one of the most cited references in current research on differential equations and biological applications.

Grades will be based on homework (30%), a midterm exam (30%), and a final exam (40%).

Course Outline:

  1. Review of linear systems and eigenvalues
  2. Dynamical systems and nonlinear ODE's
  3. Existence, uniqueness, continuous dependence on initial conditions
    • locally Lipschitz vector fields
    • Gronwall's inequality
    • Picard iteration
    • the flow of a differential equation
  4. Equilibrium points, stability and asymptotic stability
  5. Phase plane analysis for 2-dimensional systems
  6. Lyapunov functions and global stability
  7. Periodic solutions and limit cycles
    • local sections and flow boxes
    • Poincare-Bendixson theorem
    • Gradient systems
  8. Applications
    • Van der Pol oscillator
    • Lotka-Volterra equations
  9. Invadability criteria for coexistence
  10. Nash equilibria and evolutionarily stable strategies
    • Applications to biological models like Lotka-Volterra and hawk-dove systems
    • Replicator equations and relation to Lotka-Volterra systems

Additional References