Abstract:

We present here an open courseware on Dynamical Systems, accessible through the World Wide Web, with several java applets, maple experiments and algorithms, and tutorials, which allow students to learn the basic concepts and properties of Real and Complex Discrete Dynamical Systems. With the help of this course, the student will be able to implement several algorithms to visualize the behavior of, both Real and Complex, dynamical systems, detecting the existence of chaos. In Complex Dynamical System the student will learn different methods to obtain Julia sets and the Mandelbrot set.
The course starts with a deep analysis of the logistic family of quadratic dynamical systems. This family is a simple model for population dynamics and gives examples of different behaviors of one-dimensional dynamical systems, arriving to the concept of chaos. Once introduced this fundamental notion, it is shown that chaos can appear in rather simple one-dimensional systems as the shift and tent map. Next, two-dimensional systems are studied. We begin with linear maps and then we explore paradigmatic 2-dimensional chaotic systems as the baker map, Smale horseshoe or Henon attractor. Finally, we consider Dynamical Systems in the Complex plane and introduce Julia and Mandelbrot sets, studying its dynamical properties.
The courseware includes web pages presenting the theoretical notions of the course, including some proofs, and illustrating them with numerous graphics and animations. Applets and tutorials allow students to experiment, making graphical analysis, generating orbits for quadratics maps, drawing periodic orbit for the logistic, shift and tent maps and checking the chaotic behavior of the dynamic for these maps. Also several Maple experiments and procedures are shown: transition to chaos, location of invariant intervals for polynomials, drawing of orbits in the phase space, graphical analysis of the orbits, analysis of the global behavior of a family of maps, experimentation of the global behavior of the graphs and finally, visualization of the Cantor set which appears when there is not an invariant interval,… Finally, for each of the units of the course, we have included a selection of exercises for the student to solve and algorithms to program.
This courseware is used as an interactive learning tool in the course "Introduction to Discrete Dynamical Systems" taught at the Computer Science School of the Technical University of Madrid. The courseware with all its contents can be accessed at the following internet address: http://ocw.upm.es/matematica-aplicada/introduccion-a-los-sistemas-dinamicos

Keywords:

Open courseware, applets, e-learning, chaos.