## Syllabus for Math 5525 --- Spring 2018

### Description:

A dynamical system is a set of "states" together with a deterministic rule for how the states evolve with time. An ordinary differential equation or ODE is a rule describing the rate of change of the state. The laws of motion of many physical systems are expressed as differential equations and one would like to solve these equations to find the corresponding time evolution rules, i.e., to go from the ODE to the dynamical system. Unfortunately it is not usually possible to find explicit formulas for the solutions. A more realistic goal is to deduce the most important qualitative properties of the solutions without actually finding them.

In this course we will cover the basic existence and uniqueness theory, the theory of linear ODE's and develop qualitative methods for understanding nonlinear problems. Many different examples will be covered which exhibit a range of dynamical behaviors from equilibrium to periodicity to quasiperiodicity to "chaos". We will pay a lot of attention to proofs as a way to get a deeper understanding of the theory.

### Text:

Differential Equations, Dynamical Systems and an Introduction to Chaos, 3rd edition, by Hirsch, Smale and Devaney. This book is an updated version of a classic advanced undergraduate text.

### Homework:

Homework will be due about every two or three weeks. Students are encouraged to work together on the assignments but everyone should write up their own version of the solutions. Not all of the assigned problems will be graded. To see the assignments as they become available, click on the link above.

### Computers:

Computer experiments can give a lot of insight about differential equations and dynamical systems. I will present some computer demonstrations in class using the software Mathematica. Mathematica notebooks with some of these demonstrations will be available for download below. You are encouraged to use Mathematica or some other program to do some explorations of your own. You can use Mathematica at the CSE computer labs. To get a CSE computer account go to CSE Accounts. Once you have a CSE lab account, you can also download a copy for your personal computer at: Get Mathematica

I will be writing up some Mathematica notebooks for use in class. These can be found here: Mathematica Notebooks