Instructor: Laurent O. Jay, 225L MLH, ljay@math.uiowa.edu

Classes: at 10:30-11:20 MWF in 218 MLH

Prerequisites: 22M:100 or consent of instructor. A course in differential equations. Some knowledge of computer programming. The language Maple will be introduced.

Course description: This course will cover nonlinear differential equations, one- and two-dimensional flows, stability, phase plane, limit cycles, bifurcations, chaos, fractals, and applications. At the end of the course the student should master essential issues in nonlinear dynamics and chaos.

Textbook (reference): Dynamical Systems with Applications using Maple by Stephen Lynch, Birkhauser, Boston, 2001, 398 pages, ISBN 0-8176-4150-5, Price: $59.95, (ENGINEERING Library QA614.8 .L96 2001).

Course outline:

• Linear and nonlinear differential equations;
• Trajectories, phase space, flows, limit sets;
• Derivatives with respect to initial conditions, the resolvent;
• Critical points, invariant manifolds;
• Stable, unstable, and center manifolds;
• Periodic orbits, limit cycles, Poincare maps, Floquet theory;
• Hamiltonian systems, Liapunov functions and stability;
• Modelling interacting species;
• Bifurcation theory;
• Local and global bifurcations;
• Three dimensional autonomous systems and chaos, Lyapunov exponents;
• Cascades, the Feigenbaum number;
• Homoclinic orbits, heteroclinic orbits;
• Strange attractors;
• The second part of David Hilbert's 16'th problem;
• Limit cycles of Lienard systems;
• Linear and nonlinear discrete dynamical systems;
• Complex iterative maps;
• Fractals;
• Multifractals;
• Controlling chaos;

Grading procedures: One mid-term examination (30%), one final examination (30%), and homework (40%).