Fractal structures in nonlinear dynamics

Abstract

In addition to the striking beauty inherent in their complex nature, fractals have become a fundamental ingredient of nonlinear dynamics and chaos theory since they were defined in the 1970s. Moreover, fractals have been detected in nature and in most fields of science, with even a certain influence in the arts. Fractal structures appear naturally in dynamical systems, in particular associated with the phase space. The analysis of these structures is especially useful for obtaining information about the future behavior of complex systems, since they provide fundamental knowledge about the relation between these systems and uncertainty and indeterminism. Dynamical systems are divided into two main groups: Hamiltonian and dissipative systems. The concepts of the attractor and basin of attraction are related to dissipative systems. In the case of open Hamiltonian systems, there are no attractors, but the analogous concepts of the exit and exit basin exist. Therefore basins formed by initial conditions can be computed in both Hamiltonian and dissipative systems, some of them being smooth and some fractal. This fact has fundamental consequences for predicting the future of the system. The existence of this deterministic unpredictability, usually known as final state sensitivity, is typical of chaotic systems, and makes deterministic systems become, in practice, random processes where only a probabilistic approach is possible. The main types of fractal basin, their nature, and the numerical and experimental techniques used to obtain them from both mathematical models and real phenomena are described here, with special attention to their ubiquity in different fields of physics.