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

Classes: courses at 1:30-2:20 MWF in 213 MLH and discussion section at 1:30-2:20 T in 118 MLH

Prerequisites: 22M:100 or consent of instructor. A course in differential equations. Some knowledge of computer programming. The language Matlab will be used. Programming in Maple or Mathematica will also be accepted.

Course description: This course will cover continuous dynamical systems (linear/nonlinear differential equations), one- and two-dimensional flows, stability, phase plane, limit cycles, bifurcations, chaos, numerical methods for differential equations, and some applications. At the end of the course the student should master essential issues in those topics.

Textbook (complement, recommended): Differential Equations and Dynamical Systems (Hardcover) by Lawrence Perko, Springer, Series: Texts in Applied Mathematics, Vol. 7, 3rd edition, 2001, 568 pages, ISBN: 0-387-95116-4, list price: $84.95. Table of contents. The book on amazon.com. Library reference: MATH QA372 .P47 1991.

Course outline: Examples of topics:

• 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;
• Homoclinic orbits, heteroclinic orbits;
• Strange attractors;
• The second part of David Hilbert's 16'th problem;
• Limit cycles of Lienard systems;
• etc.

Grading procedures: One mid-term examination (30%), one final examination (35%), and homework (35%).