A Dynamical Meaning of Fractal Dimension

Pelikan, Steve

doi:10.2307/2000239

Cited by 3 publications

(1 citation statement)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similarly for the uncertainty dimension we obtain P-I via In [5] and [14] it has been conjectured that db = du for basin boundaries in typical dynamical systems. (There is no appropriate way to rigorously express the idea of "almost all" dynamical systems, but by typical we mean that systems encountered in practical situations will be of this type.)…”

Section: Lemmasupporting

confidence: 60%

Basic sets: Sets that determine the dimension of basin boundaries

et al. 1988

View full text Add to dashboard Cite

In this paper we consider the question of how many possible dimensions a basin boundary can have. We conjecture that the number of possible dimension values is at most the number of some well defined asymptotic sets (called basic sets) on the basin boundary. It should be noticed that the dimension of a basic set also has a dynamical meaning. The conjecture will be proved for a class of Axiom A systems (namely two dimensional diffeomorphisms and one dimensional chaotic maps). In addition, we will give numerical evidence for a physical example.

show abstract

Section: Lemmasupporting

confidence: 60%

Basic sets: Sets that determine the dimension of basin boundaries

et al. 1988

View full text Add to dashboard Cite

show abstract

Fractal structures in nonlinear dynamics

2009

View full text Add to dashboard Cite

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.

show abstract