# 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.