Dynamical Systems Theory
Oscar Castillo, Patricia Melín · Studies in fuzziness and soft computing · 2003
This chapter introduces the basic concepts of dynamical systems theory, and several basic mathematical methods for controlling chaos. The main goal of this chapter is to provide an introduction to and a summary to the theory of dynamical systems with particular emphasis on fractal theory, chaos theory, and chaos control. We first define what is meant by a dynamical system, then we define an attractor, and the concept of the fractal dimension of a geometrical object. Also, we define the Lyapunov exponents as a measure of the chaotic behavior of a dynamical system. On the other hand, the fractal dimension can be used to classify geometrical objects because it measures the complexity of an object. We finish the chapter by reviewing mathematical methods for controlling chaos in dynamic systems. These methods can be used to control a real dynamic system; however, due to efficiency and accuracy requirements we will be forced to use fuzzy logic to model the uncertainty, which is present when numerical simulations are performed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.