This course will be the first semester of a graduate level introduction to differential equations and dynamical systems with emphasis on qualitative, geometrical methods for nonlinear systems. It will be followed in the spring by Math 8502 -- Dynamical Systems and Differential Equations, which will explore further topics.
Understanding the modern theory of dynamical systems requires a lot of ideas from many parts of mathematics. I hope to cover both the differential equations theory itself and these background ideas in a way which can be understood not only by math graduate students but by any mathematically inclined student with a solid knowledge of linear algebra, advanced calculus and elementary differential equations.
Here are some of the topics I would like to cover in the first semester: basic existence and uniqueness theory, flows and flow boxes, invariant sets, flows on manifolds, alpha and omega limit sets, flows in the plane, Poincare-Bendixson theory, index theory, Poincare maps, rotation numbers, variational equations, theory of linear systems, dynamics near equilibria including Hartman's theorem and the stable manifold theorem. In addition a good deal of time will be spent on interesting examples, mostly from classical mechanics, to illustrate and apply the theory.
Although no book is required, here are some recommended references.
|"Ordinary Differential Equations", by V.I. Arnold|
|"Geometrical Methods in the Theory of Ordinary Differential Equations", by V.I. Arnold|
|"Ordinary Differential Equations with Applications", by C. Chicone|
|"Dynamical Systems: Stability, Symbolic Dynamics and Chaos", by Clark Robinson|
|"Ordinary Differential Equations", by Jack Hale|
|"Ordinary Differential Equations", by Philip Hartman|
|"Differential Equations, Geometric Theory", by S. Lefschetz|
|"Theory of Ordinary Differential Equations", by E. Coddington and N. Levinson|
Based on several homework assignments throughout the semester. No exams. The link leads PDF versions of the homework assignments so far.