Math 518X

Spring 2020

References:

Some primary references are listed below

• Differential Equations and Dynamical Systems 3rd ed. by Lawrence Perko
• Dynamical System Models in the Life Sciences by Frederic Y. M. Wan
• Ordinary Differential Equations and Dynamical Systems by Gerald Teschl
• Ordinary Differential Equations with Applications by Carmen Chicone

   

Course Topics:

This course covers some of the basic theory of differential equations and concepts of dynamical systems. Topics include existence and uniqueness theorems, parameter dependence, well-posedness, solutions of linear systems, linearization of nonlinear systems, stability analysis and bifurcation theory. In addition, there will be an emphasis on modeling aspects of ODE and PDE models arising in biology and physics. Techniques such as parameter estimation, sensitivity analysis, parameter optimization will be applied to a variety of mathematical models.

I plan to cover basic theory in a standard "theorem-proof" format, but many specific topics may be presented conceptually using of examples or computer projects.

A list of topics I hope to cover are the following:

• Linear differential equations; existence and uniqueness, solution methods, well-posedness, stability
• Application to population models; least-squares data fitting
• Nonlinear Systems; local existence and uniqueness, well-posedness, linearization, stability theorems
• Ideas from dynamical systems; planar systems, Hamiltonian systems, Poincare-Bendixson Theorem, Index Theory
• Bifurcation Theory; Structural stability, Hopf Bifurcation, physical applications
• Variational methods; Hamilton's Principle, Optimization, Sensitivity analysis

   

Course Policy, Grading

The semester grade is based on about 6 homework assignments and a takehome final, which counts as a double homework. Some assignments and certainly the final will involve presentation to the class of an assigned project. Assignments will be posted on this web page.