Difference between revisions of "Math 635: Dynamical Systems"
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Latest revision as of 15:44, 3 April 2013
Contents
Catalog Information
Title
Dynamical Systems.
Credit Hours
3
Prerequisite
Description
(A rigorous treatment of the theory of dynamical systems.?)
Desired Learning Outcomes
This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include topological, symbolic, and hyperbolic dynamical systems.
Prerequisites
Students are expected to have completed Math 634. This will provide the students with an understanding on the theory of ordinary differential equations.
Minimal learning outcomes
Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve nontrivial problems related to these concepts, and prove theorems in analogy to proofs given by the instructor.
 Topological dynamical systems
 Nonwandering set, chain recurrence
 Topological mixing and transitivity
 Expansive systems
 Topological entropy
 Topological and smooth conjugacy
 Symbolic dynamical systems
 Subshifts of finite type
 PerronFrobenius theorem
 Topological entropy for subshifts of finite type
 Hyperbolic dynamical systems
 HartmanGrobman theorem
 Stable manifold theorem
 Hyperbolic sets
 Anosov diffeomorphisms
 Smale Horseshoe and transverse homoclinic points
 Shadowing
 Axiom A dynamical systems and spectral decomposition
 Low dimensional dynamical systems
 Circle homeomorphisms, circle diffeomorphisms, and rotation number
 Real quadratic maps
 Expanding endomorphisms
Textbooks
Possible textbooks for this course include (but are not limited to):
Additional topics
These are at the instructor's discretion as time allows examples are: Ergodic theory, Markov partitions, Sharkovsky theorem, Hamiltonian dynamics, and measure theoretic entropy.
Recommended texts
Dynamical Systems, stability, symbolic dynamics, and chaos, by Clark Robinson; Introduction to Dynamical Systems, by Michael Brin and Garrett Stuck
Courses for which this course is prerequisite
None.