MATH 7170 - Applied Dynamical Systems

John Guckenheimer, spring 2014.

This year Guckenheimer and Holmes were awarded the Steele Prize for Mathematical Exposition by the American Mathematical Society for their book Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. This book, pubished thirty years ago, was based upon early versions of this course. Although the course has evolved, in tribute to the Steele Prize, I will teach the "classic" version of the course in 2014 by following the text closely. In many places, significant developments of the topics beyond the material in the text will be discussed. Students taking the course will work exercises from the text and complete a more substantial project of their choice related to the course material. Prerequisites are ONE of the following: MATH 6170, TAM 5780 (now MAE 5790), MATH 4200, or a comparable undergraduate course on differential equations and dynamical systems.

The catalog description is: "Topics include review of planar (single-degree-of-freedom) systems; local and global analysis; structural stability and bifurcations in planar systems; center manifolds and normal forms; the averaging theorem and perturbation methods; Melnikov’s method; discrete dynamical systems, maps and difference equations, homoclinic and heteroclinic motions, the Smale Horseshoe and other complex invariant sets; global bifurcations, strange attractors, and chaos in free and forced oscillator equations; and applications to problems in solid and fluid mechanics."