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).

Escape from a chaotic attractor with fractal basin boundaries

SPIE Proceedings, 2003

We study fluctuational transitions between two co-existing chaotic attractors separated by a fractal basin boundary in a discrete dynamical system. It is shown that the mechanism of fluctuational transition through a fractal boundary is generic, and determined by a hierarchy of homoclinic original saddles. The most probable escape path from a chaotic attractor to the fractal boundary is found using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.

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.

Mixed basin boundary structures of chaotic systems

Physical review, 1999

Motivated by recent numerical observations on a four-dimensional continuous-time dynamical system, we consider different types of basin boundary structures for chaotic systems. These general structures are essentially mixtures of the previously known types of basin boundaries where the character of the boundary assumes features of the previously known boundary types at different points arbitrarily finely interspersed in the boundary. For example, we discuss situations where an everywhere continuous boundary that is otherwise smooth and differentiable at almost every point has an embedded uncountable, zero Lebesgue measure set of points at which the boundary curve is nondifferentiable. Although the nondifferentiable set is only of zero Lebesgue measure, the curve's fractal dimension may ͑depending on parameters͒ still be greater than one. In addition, we discuss bifurcations from such a mixed boundary to a ''pure'' boundary that is a fractal nowhere differentiable curve or surface and to a pure nonfractal boundary that is everywhere smooth.

Chaotic motions and fractal basin boundaries in spring-pendulum system

Nonlinear Analysis: Real World Applications, 2006

This study investigates the chaotic response of the spring-pendulum system. In this system besides of strange attractors, multiple regular attractors may co-exist for some values of system parameters, and it is important to study the global behavior of the system using the basin boundaries of the attractors. Here multiple scales method is used to distinguish the regions of stable and unstable attractors. Early studies show that there are unstable regions for the spring-pendulum system. In this study using bifurcation diagrams and Poincaré maps, it is shown that in some cases the response becomes quasiperiodic or chaotic for some deviations from external and internal resonance frequencies. Also it will be shown that the response is sensitive to the value of damping parameters, which may result in chaotic response. Results show that the jumping phenomena may occur when multiple regular attractors exist. Using basin boundaries of attractors it is also shown that in some regions these boundaries are fractal.

Population Dynamics with a Refuge: Fractal Basins and the Suppression of Chaos

Theoretical Population Biology, 2002

We consider the effect of coupling an otherwise chaotic population to a refuge. A rich set of dynamical phenomena is uncovered. We consider two forms of density dependence in the active population: logistic and exponential. In the former case, the basin of attraction for stable population growth becomes fractal, and the bifurcation diagrams for the active and refuge populations are chaotic over a wide range of parameter space. In the case of exponential density dependence, the dynamics are unconditionally stable (in that the population size is always positive and finite), and chaotic behavior is completely eradicated for modest amounts of dispersal. We argue that the use of exponential density dependence is more appropriate, theoretically as well as empirically, in a model of refuge dynamics.

Some comments on chaos and fractals

Technological Forecasting and Social Change, 1990

Some comments are made on a recent paper by Gordon and Greenspan [l]. Two interesting variants of the logistic difference equation in the context of innovation diffusion, the first with a superimposed periodic force term and the second with an additional lag of one generation, with their new features are discussed.

Fluctuational Transitions through a Fractal Basin Boundary

Physical Review Letters, 2003

Fluctuational transitions between two co-existing chaotic attractors, separated by a fractal basin boundary, are studied in a discrete dynamical system. It is shown that the mechanism for such transitions is determined by a hierarchy of homoclinic points. The most probable escape path from the chaotic attractor to the fractal boundary is found using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.

Fluctuational Escape from Chaotic Attractors

AIP Conference Proceedings, 2003

Abstract. Fluctuational transitions between two coexisting attractors are investigated. Two different systems are considered: the periodically driven nonlinear oscillator and the two-dimensional map introduced by Holmes. These two systems have smooth and fractal boundaries, respectively, separating their coexisting attractors. It is shown that, starting from a cycle embedded in the chaotic attractor, the periodically-driven oscillator escapes to a saddle cycle at the boundary of the basin of attraction, and does so through sequential ...

Scaling properties of saddle-node bifurcations on fractal basin boundaries

Physical Review E, 2003

We analyze situations where a saddle-node bifurcation occurs on a fractal basin boundary. Specifically, we are interested in what happens when a system parameter is slowly swept in time through the bifurcation. Such situations are known to be indeterminate in the sense that it is difficult to predict the eventual fate of an orbit that tracks the pre-bifurcation node attractor as the system parameter is swept through the bifurcation. In this paper we investigate the scaling of (1) the fractal basin boundary of the static (i.e., unswept) system near the saddle-node bifurcation, (2) the dependence of the orbit's final destination on the sweeping rate, (3) the dependence of the time it takes for an attractor to capture a swept orbit on the sweeping rate, and (4) the dependence of the final attractor capture probability on the noise level. With respect to noise, our main result is that the effect of noise scales with the 5/6 power of the parameter drift rate. Our approach is to first investigate all these issues using one-dimensional map models. The simplification of treatment inherent in one dimension greatly facilitates analysis and numerical experiment, aiding us in obtaining the new results listed above. Following our one-dimensional investigations, we explain that these results can be applied to two-dimensional systems. We show, through numerical experiments on a periodically forced second order differential equation example, that the scalings we have found also apply to systems that result in two dimensional maps.