Syllabus Math 551

Math 551. Dynamical Systems II Instructor Syllabus

I. Discrete space/discrete time

  • Arnol’d Cat Map on a prime lattice
  • Bernoulli shift on a prime lattice
  • Cellular automata ­ cells form the space and time is discrete
  • Dynamical systems that restrict to lattices

II. Continuous space/discrete time

  • Logistic map on the interval [0, 1]
  • Arnol’d Cat Map on the 2-torus
  • Bernoulli shift on the interval
  • Mapping of a surface of section into itself in a flow

III. Discrete space/continuous time

  • Finite state system with continuous time such as an on-off switch that controls a circuit
  • Finite state game with continuous time
  • Finite state machine with continuous time

IV. Continuous space/continuous time

  • Ordinary differential equation with time as independent variable
  • Partial differential equation with space and time as independent variables
  • Flow on a manifold generated by a vector field

V. Computer Simulations as Dynamical Systems

  • Discrete event simulation with branching
  • Continuous time simulation

VI. Other topics to select from:

  • Structural stability, KAM theory, Hopf index theory of vector fields
  • Morse theory of gradient vector fields, infinite dimensional dynamical systems, variational methods, Hamiltonian dynamics, area preserving twist maps ­ Poincaré’s conjecture and Birkhoff’s proof, Aubry-Mather theory, hyperbolic dynamics, Lefschetz theory of algebraic differential equations on projective space

Recommended references for Math 551 could be chosen by the instructor from among:

  • V. I. Arnold, Geometric methods in Ordinary Differential Equations
  • V. I. Arnold, Ordinary Differential Equations
  • D.K. Arrowsmith and C.M. Place, An Introduction to Dynamical Systems
  • C. Chicone, Ordinary Differential Equations and Applications
  • Coddington and Levinson, Theory of Ordinary Differential Equations
  • J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Note: this includes a chapter on chaos.)
  • A. Katok and B. Hasselblatt, Introduction to the modern theory of Dynamical Systems

Approved by GAC; syllabus effective Fall 2010 semester.