## Dynamical Systems and Ordinary Differential Equations

Dynamical systems problems range from the vibrations of molecules to planetary motion and span the breadth of the physical, biological and social sciences as well as engineering. Research in the subject stretches from investigation of realistic models of complex systems like the brain and the power grid to mathematically rigorous investigations of highly abstract systems such as the iteration of quadratic functions.

In the past few years, CAM faculty and students have used dynamical systems to study the oscillations of bubbles, the flight of mosquitoes, the emergence of cooperation, and rapid evolution in predator -prey systems, among other phenomena.

Cornell has a long history as a center for research in the subject, and CAM has been a focal point for that research. CAM faculty who regularly teach dynamical systems courses and serve as advisors of students doing research in the subject are Steve Ellner, John Guckenheimer, John Hubbard , Richard Rand, James Sethna, John Smillie, Paul Steen, Steve Strogatz, Alex Vladimirsky and Jane Wang. Many students have taken advantage of the interdisciplinary opportunities provided by CAM to engage in research that connects experiment, theory and simulation.

**Dynamical systems theory studies models of how things change in time.**

Even the simplest nonlinear dynamical systems can generate phenomena of bewildering complexity. Because formulas that describe the long time behavior of a system seldom exist, we rely on computer simulation to show how initial conditions evolve for particular systems. Simulations with many different systems display common patterns that have been observed repeatedly. One of the main goals of dynamical systems theory is to discover these patterns and characterize their properties. The theory can then be used to describe and interpret the dynamics of specific systems. It can also be used as the foundation for numerical algorithms that seek to analyze system behavior in ways that go beyond simulation. Throughout the theory, dependence of dynamical behavior upon system parameters has been an important topic. Bifurcation theory is the part of dynamical systems theory that systematically studies how systems change with varying parameters.