Dynamical Systems

Dynamical systems is a branch of mathematics which studies processes which change. Such processes occur in all branches of science and examples of dynamical systems include the motion of stars, the change of stock markets, the variation of the world's weather, the rise and fall of populations, the reaction of chemicals and the motion of a simple pendulum. The central goal of the study of dynamical systems is to predict where the system under consideration is heading and where it will ultimately go (for example, one would like to know when the market goes up or down, whether it will be rainy or sunny, or if interacting populations become extinct). The study of dynamical systems, originated from differential or difference equations arising from many applied fields, has been one of the most fruitful fields of mathematical research in this century. Many profound results have been uncovered and applied to other branches of mathematics as well as to physics, chemistry, biology and economics.

In this course, we will utilize scalar maps and low-dimensional ordinary differential equations to demonstrate the main contents, methods and applications of dynamical systems. The course will be structured so that students are gradually introduced to more and more sophisticated ideas from analysis as the course proceeds. It starts with only a few elementary notions that can be explained using graphical methods or simple differential calculus. The concentration of the course is on the illustration of main ideas and results by graphical analysis and applications, rather than long calculations. Numerical experiments illustrating mathematical results will also be performed.

The contents of the course include phase portraits, orbits, stability and bifurcations, fixed points, periodic solutions, homoclinic orbits, periodic doubling and transition to chaos. Examples and applications include interacting populations, reaction kinetics, forced Van der Pol equation, damped Duffing and Lorenz equations.

The text is Dynamics and Bifurcations, by J. Hale and H. Kocak, Springer-Verlag, New York, 1991.