Scaling behavior of chaotic systems with riddled basins

Ott, Edward; Sommerer, John C.; Alexander, Robert R.; Kan, Ittai; Yorke, James A.

doi:10.1103/physrevlett.71.4134

Cited by 155 publications

(85 citation statements)

References 6 publications

(14 reference statements)

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“…In some cases, this is most likely a global attractor since numerical calculations show that synchronization transition takes place also at this point. However, in some cases, a basin of the synchronized attractor is riddled with basin of another chaotic attractor [5]. We suggests that such a coexistence of synchronized and nonsynchronized chaotic attractors might be a low-dimensional analog of the so-called stable chaos that was found in some spatially extended systems [6].…”

Section: Introductionmentioning

confidence: 99%

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Synchronization and partial synchronization of linear maps

Lipowski

Droz

2005

Physica A: Statistical Mechanics and its Applications

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We study synchronization of low-dimensional (d = 2, 3, 4) chaotic piecewise linear maps. For Bernoulli maps we find Lyapunov exponents and locate the synchronization transition, that numerically is found to be discontinuous (despite continuously vanishing Lyapunov exponent(s)). For tent maps, a limit of stability of the synchronized state is used to locate the synchronization transition that numerically is found to be continuous. For nonidentical tent maps at the partial synchronization transition, the probability distribution of the synchronization error is shown to develop highly singular behavior. We suggest that for nonidentical Bernoulli maps (and perhaps some other discontinuous maps) partial synchronization is merely a smooth crossover rather than a well defined transition. More subtle analysis in the d = 4 case locates the point where the synchronized state becomes stable. In some cases, however, a riddled basin attractor appears, and synchronized and chaotic behaviors coexist. We also suggest that similar riddling of a basin of attractor might take place in some extended systems where it is known as stable chaos.

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Section: Introductionmentioning

confidence: 99%

“…Actually, it is already known that basins of synchronized chaotic attractors are sometimes riddled with basins of some other attractors, even when transversal Lyapunov exponents are negative [5]. As a result, only some initial conditions will lead the system to the synchronized state, and the other will lead to the other nonsynchronized attractor.…”

Section: )mentioning

confidence: 99%

Synchronization and partial synchronization of linear maps

Lipowski

Droz

2005

Physica A: Statistical Mechanics and its Applications

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“…At this stage, if there is a coexisting attractor that the system will eventually be attracted to as the parameter changes, the basin of attraction for the chaotic attractor becomes riddled with points from the basin of attraction of the coexisting attractor. [29][30][31] The second change occurs at another parameter value the attractor becomes a chaotic saddle, no longer a true attractor. And the third change in behavior as the parameter continues to be varied is that the saddle becomes a chaotic repeller.…”

Section: E Attractor Bubbling and Riddled Basins Of Attractionmentioning

confidence: 99%

Synchronization of chaotic systems

Pecora

Carroll

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

431

326

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We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.

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“…More generally dynamically invariant subspaces often arise as a result of symmetries and in models displaying basin riddling or intermittent behaviour 4,5,1,24,25,26,6]. They play an organisational role and the dynamics restricted to the invariant subspace may drastically a ect the dynamics of the ambient system.…”

Section: Introductionmentioning

confidence: 99%

Random and deterministic perturbation of a class of skew-product systems

Broomhead¹,

Hadjiloucas²,

Nicol³

1999

Dynamics and Stability of Systems

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This paper concerns the stability properties of skew-products T (x y) = ( f (x) g (x y)) in which ( f X ) is an ergodic map of a compact metric space X and g : X R n ! R n is continuous. We assume that the skew-product has a negative maximal Lyapunov exponent in the bre.We study the orbit stability and stability of mixing of T (x y) = ( f (x) g (x y)) under deterministic and random perturbation of g. We s h o w t h a t s u c h systems are stable in the sense that for any > 0 there is a pairing of orbits of the perturbed and unperturbed system such that paired orbits stay within a distance of each other except for a fraction of the time.We also show that such systems preserve ergodicity and higher order mixing properties under deterministic perturbation of g and perturbation of g by i.i.d. additive n o i s e . Furthermore we s h o w that the invariant measure for the perturbed system is continuous (in the Hutchinson metric) as a function of the size of the perturbation to g (Lipschitz topology) and the noise distribution. Our results have applications to the stability of Iterated Function Systems which \contract on average".

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Scaling behavior of chaotic systems with riddled basins

Cited by 155 publications

References 6 publications

Synchronization and partial synchronization of linear maps

Synchronization and partial synchronization of linear maps

Synchronization of chaotic systems

Random and deterministic perturbation of a class of skew-product systems

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