| M540 Dynamical Systems |
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Credits -- 3 (3-0-0)
Term Offered -- Fall
Prerequisite -- M318; M369 or consent of instructor
- Description
--- This course deals with the qualitative analysis of nonlinear dynamical systems, a central topic in applied mathematics. The geometrical techniques we consider originated with the work of Henri Poincaré in his study the stability of the solar system and emphasize the use of a phase portrait and the characterization of a dynamical system as a field of vectors in phase space. This view permits an elegant description of the long term behavior of a system and provides information that a direct numerical simulation cannot.
In addition, we discuss the bifurcation of solutions of dynamical systems, i.e., radical changes in the geometry of phase space as we perturb the coefficients of the equations.
This course is recommended for students who have had undergraduate courses in ordinary differential equations and linear algebra. It forms the basis for more advanced study in dynamical systems theory and related areas such as modeling, control theory and fluid dynamics.
- Content
--- We will begin with a review of the linear theory of ODEs from a geometrical viewpoint followed by iterated maps, asymptotic behavior, local bifurcations, topological equivalence, structural stability, center manifold theory, normal forms, unfoldings and higher codimension bifurcations, chaotic dynamical systems and strange attractors. The course will be taught from a qualitative perspective, no computer coding will be required.