Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics (original) (raw)

The transition to chaotic attractors with riddled basins

Physica D: Nonlinear Phenomena, 1994

Recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby (the basin is "riddled" with holes). Here we consider the dynamics near the transition to this situation as a parameter is varied. Using a simple analyzable model, we obtain the characteristic behaviors near this transition. Numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered. 1 This situation is to be distinguished from the case where the basin is a solid volume with a fractal boundary (e.g., see Ref. [31).

Catastrophic bifurcation from riddled to fractal basins

Physical Review E, 2001

Most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry. In realistic applications of chaotic systems, however, there exists no perfect symmetry. The aim of this paper is to examine the consequences of symmetry-breaking on riddling. In particular, we consider smooth deterministic perturbations that destroy the existence of invariant subspace, and identify, as a symmetry-breaking parameter is increased from zero, two distinct bifurcations. In the first case, the chaotic attractor in the invariant subspace is transversely stable so that the basin is riddled. We find that a bifurcation from riddled to fractal basins can occur in the sense that an arbitrarily small amount of symmetry breaking can replace the riddled basin by fractal basins. We call this a catastrophe of riddling. In the second case, where the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system, the presence of a symmetry breaking, no matter how small, can immediately create fractal basins in the vicinity of the original invariant subspace. This is a smooth-fractal basin boundary metamorphosis. We analyze the dynamical mechanisms for both catastrophes of riddling and basin boundary metamorphoses, derive scaling laws to characterize the fractal basins induced by symmetry breaking, and provide numerical confirmations. The main implication of our results is that while riddling is robust against perturbations that preserve the system symmetry, riddled basins of chaotic attractors in the invariant subspace, on which most existing works are focused, are structurally unstable against symmetry-breaking perturbations.

Extreme fractal structures in chaotic mechanical systems: riddled basins of attraction

Journal of Physics: Conference Series, 2010

Chaotic dynamical systems with certain phase space symmetries may exhibit riddled basins of attraction, which can be viewed as extreme fractal structures in the sense that, regardless of how small is the uncertainty in the determination of an initial condition, we cannot decrease the fraction of such points that are certain to converge to a given attractor. We investigate a mechanical system exhibiting riddled basins of attraction: a particle under a two-dimensional potential with friction and time-periodic forcing. The verification of riddling is made by checking its mathematical requirements through computation of finite-time Lyapunov exponents as well as by scaling laws describing the fine structure of basin filaments densely intertwined in phase space.

Transient chaos as the backbone of dynamics on strange attractors beyond crisis

1994

We show that chaotic attractors at and above internal crisis points can be naturally decomposed into nonattracting invariant chaotic sets connected by weak intermittent heteroclinic couplings. These basic component sets are used to obtain the dynamical multifractal spectrum characterising the asymptotic and the finite time dynamics on the entire attractor.

On chaos, fractals and turbulence

Physica Scripta, 1993

Based on a recently proposed generic model for fractal basin boundaries we show the interrelation between chaotic dynamics and the form of fractal basin boundaries. For the case of transient chaos, intermittent fractality is shown. The explicit expression of the basin boundary is discussed on the background of typical turbulent phenomena, like log normal probability density distributions or multifractal scaling exponents. The precision of the evaluation of multifractal exponents is discussed on the background of finite statistical resolution.

Scaling behavior of chaotic systems with riddled basins

Physical Review Letters, 1993

Recently it has been shown that there are chaotic attractors whose basins are such that every point in the attractor's basin has pieces of another attractor's basin arbitrarily nearby (the basin is "riddled" with holes). Here we report quantitative theoretical results for such basins and compare with numerical experiments on a simple physical model.

Fractal structures in nonlinear dynamics

Reviews of Modern Physics, 2009

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.