University of Illinois Urbana-Champaign 4 credits Aerospace Engineering
This course is structured to introduce the graduate students into advanced concepts of the geometric theory of nonlinear dynamics. Topics to be discussed include vector fields and maps, conjugacies, structural stability and Peixoto’s theorem, dynamical systems on two-manifolds; center manifold theory and normal forms for vector fields and maps; local bifurcations of vector fields and maps, co-dimension 1 and 2 bifurcations; global bifurcations, the Smale horseshoe map and invariant Cantor sets, the shift map and symbolic dynamics, chaos in the horseshoe, Conley – Moser conditions for chaos, hyperbolic invariant sets, Moser’s theorem and Smale-Birkhoff homoclinic theorem, homoclinic bifurcations and Newhouse sinks; homoclinic and subharmonic Melnikov theories, conditions for homoclinic chaos, chaos in perturbed Hamiltonian systems; applications to mechanics. This course will demonstrate how these advanced concepts can be applied to the study of response, stability and bifurcation behavior of engineering systems. Same as TAM 516. Prerequisite: TAM 416 and either ME 340, TAM 412 or AE 352.
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