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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

Fundamental concepts of nonlinear oscillations, structural stability, local and global bifurcations in the context of ordinary and partial differential equations; dynamic systems, structural stability and Lyapunov-Schmidt Reduction, bifurcations of equilibrium points, limit cycles and tori, the center manifold and Poincare normal forms, co-dimension two and higher order bifurcations, bifurcation theory of maps, the Birkhoff-Smale homoclinic theorem and horseshoes, Melnikov's method and Silnikov phenomena, period doubling, and other routes to chaos are covered in this course. Applications to engineering problems, such as aircraft at high angles of attack, pipes conveying fluid, and panel flutter are covered as well.

4 hours

BS degree in engineering or science from an accredited college in the United States or an approved institution of higher learning abroad and a course in Aerospace Dynamical Systems (AE 352) or Intermediate Dynamics (TAM 412).