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". In the spirit of honors mathematics, we will pay a lot of attention to proofs as a way to get a deeper understanding of the theory.
Differential Equations, Dynamical Systems and an Introduction to Chaos, 2nd edition, by Hirsch, Smale and Devaney. This book is an updated version of a classic advanced undergraduate text.
|Midterm Exams||1/4 each|
|Midterm I||Wednesday, October 22|
|Midterm II||Wednesday, December 10|
|Final Project Due||Monday, December 15|
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.
This is your chance to explore a part of the subject not covered in class. The form and topic are very flexible and creativity is encouraged. For some concrete suggestions for projects follow the link above. You could do one of those or, even better, come up with an idea of your own.