An experimental approach to discrete dynamical systems with Mathematica

Abstract

Experiment with discrete time dynamical systems, may represent an important educational resource in the investigation of the properties of dynamical systems and their potential applications to disciplines such as economics. As an illustration of this possibility teaching, this paper provides a brief introduction to the dynamics of discretetime dynamic systems using examples assisted by the symbolic language Mathematica. Such systems are essentially iterated maps. In the first part, we construct orbits of points under iteration of real and complex functions. If x is a real number or a complex number, then the orbit of x under f is the sequence {x, f (x), f (f (x)), ...}. These sequences may be convergent or sequences that tend to infinity. In particular, to test this behavior in complex sequences will require the concept of derivative of a complex function. In a second part, we use the concepts reviewed in the first to build Julia sets, these sets are obtained by assigning colors to a rectangular grid points according to the behavior of their orbits under the studied complex function, the colors are assigned according the classification of the points. The pattern obtained, the Julia set is a fractal. However, the image obtained is always an approximation.