## 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.