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:
- Review of linear systems and eigenvalues
- Dynamical systems and nonlinear ODE's
- Existence, uniqueness, continuous dependence on initial conditions
- locally Lipschitz vector fields
- Gronwall's inequality
- Picard iteration
- the flow of a differential equation
- Equilibrium points, stability and asymptotic stability
- Phase plane analysis for 2-dimensional systems
- Lyapunov functions and global stability
- Periodic solutions and limit cycles
- local sections and flow boxes
- Poincare-Bendixson theorem
- Gradient systems
- Applications
- Van der Pol oscillator
- Lotka-Volterra equations
- Invadability criteria for coexistence
- 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
- L. Perko, Differential Equations and Dynamical Systems, 3rd. edition,
Springer (2002).
- M. Hirsch and S. Smale, Differential Equations,
Dynamical Systems, and Linear Algebra,
Academic Press (1974).
- P. Waltman, A Second Course in Elementary Differential Equations,
Academic Press.
- J. Cronin, Differential Equations: Introduction and Qualitative
Theory, Dekker.
- L. Edelstein-Keshet, Mathematical Models in Biology,
McGraw-Hill.