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

Classes: courses at 10:30-11:20 MWF in 118 MLH and discussion section at 10:30-11:20 T in 218 MLH

Prerequisites: 22M:100 or consent of instructor. A course in differential equations. Some knowledge of computer programming. The languages Matlab and Maple will be used.

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

Textbook: An Introduction to Dynamical Systems: Discrete and Continuous by R. Clark Robinson, Prentice Hall, 2004, 672 pages, ISBN: 0131431404, list price: $87.00. Table of contents, information and errata on the book by the author. The book on amazon.com. Library reference: ENGINEERING Library QA614.8 .R65 2004.

Course outline: see Table of contents. 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;
• 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;
• Complex iterative maps;
• Fractals;
• Multifractals;
• Controlling chaos;
• etc.

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