Complicated dynamics from simple deterministic equations: Theory and Computer Simulation

Dr. R. Clark Robinson

Department of Mathematics

Northwestern University

There has been a fruitful interaction between computer simulation and mathematical theory in the development of our understanding of dynamical systems.

Edward Lorenz found that a small perturbations of initial conditions for a simple model for weather could lead to large effect at a later time: he interpreted this to mean that a butterfly flapping its wings on one place could have large effects at another place a month later. Such sensitive dependence on initial conditions is now understood to be a central feature of may deterministic equations. The term chaos has been introduced to identify many of complicated nature of the dynamics.

Another feature that arises for many dynamical systems is the fractal nature of the attractor (the set of points a single trajectory approaches asymptotically.) These fractal sets often have the structure of a Cantor set on curves, planes, or other higher dimensional surfaces. Thus, not only the dynamics but also the topology of the attractor has a structure, but a complicated one.

Finally, mathematicians have been able to identify key aspects of the features of these complicated dynamics. However, there are systems where the theory suggests the explanation of dynamics observed but it is difficult or impossible to prove the dynamics observed. Computers have been used in various ways from experimentation to numerical proofs to study dynamical systems.

In the last forty years, we have progressed in understanding, both theoretically and through models, the complexities which are possible for deterministic dynamical systems that occur mathematically and in applications.