The transition to chaotic attractors with riddled basins

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

doi:10.1016/0167-2789(94)90047-7

Cited by 180 publications

(107 citation statements)

References 12 publications

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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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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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“…[90,91]. The dynamics of riddled basins was subsequently investigated in [92] using a more realistic physical model. A more extreme type of basin structure referred to as`intermingled basinsa in which the basins of more than one chaotic attractors are riddled, was also studied using both discrete maps [90] and a more realistic physical system [93].…”

Section: Example 2: Controlling Riddled Basinsmentioning

confidence: 99%

“…To assure that only small perturbations are applied, it is necessary to monitor the magnitude of the term in the denominator of Eq. (92). When "* ) DhG L " is below some small threshold, we set C L "0.…”

Section: Synchronization Of Spatiotemporal Chaotic Systems By Controlmentioning

confidence: 99%

The control of chaos: theory and applications

et al. 2000

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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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The transition to chaotic attractors with riddled basins

Cited by 180 publications

References 12 publications

Synchronization of chaotic systems

Synchronization of chaotic systems

The control of chaos: theory and applications

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

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