Bifurcation to High-Dimensional Chaos

Harrison, Mary Ann F.; Lai, Ying–Cheng

doi:10.1142/s0218127400000967

Cited by 17 publications

(22 citation statements)

References 44 publications

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“…5 indicate that the titration algorithm detects chaos even when the dimension is as high as 20. In addition, we have performed analyses (not shown) of bifurcations towards high-dimensional chaos (''hyperchaos'') in two types of systems: discrete models of ecological dispersion (67,68) and networks of interconnected Rössler chaotic systems relevant to chaos communications (38,39,67,68). In both cases, the NL algorithm performed equally well.…”

Section: Resultsmentioning

confidence: 99%

Titration of chaos with added noise

Poon

Barahona

2001

Proc. Natl. Acad. Sci. U.S.A.

135

114

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Deterministic chaos has been implicated in numerous natural and man-made complex phenomena ranging from quantum to astronomical scales and in disciplines as diverse as meteorology, physiology, ecology, and economics. However, the lack of a definitive test of chaos vs. random noise in experimental time series has led to considerable controversy in many fields. Here we propose a numerical titration procedure as a simple ''litmus test'' for highly sensitive, specific, and robust detection of chaos in short noisy data without the need for intensive surrogate data testing. We show that the controlled addition of white or colored noise to a signal with a preexisting noise floor results in a titration index that: (i) faithfully tracks the onset of deterministic chaos in all standard bifurcation routes to chaos; and (ii) gives a relative measure of chaos intensity. Such reliable detection and quantification of chaos under severe conditions of relatively low signal-to-noise ratio is of great interest, as it may open potential practical ways of identifying, forecasting, and controlling complex behaviors in a wide variety of physical, biomedical, and socioeconomic systems.

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Section: Resultsmentioning

confidence: 99%

Titration of chaos with added noise

Poon

Barahona

2001

Proc. Natl. Acad. Sci. U.S.A.

135

114

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“…As a bifurcation parameter changes, an additional positive Lyapunov exponent can arise. The nature of chaotic driving stipulates that the second exponent becomes positive in a smooth fashion [17][18][19], a feature that is characteristic of the transition to chaos in random dynamical systems [22][23][24][25].…”

Section: Discussionmentioning

confidence: 99%

“…Since the weights vary smoothly with the bifurcation parameter [17], the second nontrivial Lyapunov exponent passes through zero smoothly. The generic feature associated with the transition in smooth dynamical systems is thus that high-dimensional chaos arises directly and smoothly from low-dimensional chaos [17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Transition to high-dimensional chaos in nonsmooth dynamical systems

Lai

2018

Phys. Rev. E

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We uncover a route from low-dimensional to high-dimensional chaos in nonsmooth dynamical systems as a bifurcation parameter is continuously varied. The striking feature is the existence of a finite parameter interval of periodic attractors in between the regimes of low-and high-dimensional chaos. That is, the emergence of high-dimensional chaos is preceded by the system's settling into a totally nonchaotic regime. This is characteristically distinct from the situation in smooth dynamical systems where high-dimensional chaos emerges directly and smoothly from low-dimensional chaos. We carry out an analysis to elucidate the underlying mechanism for the abrupt emergence and disappearance of the periodic attractors and provide strong numerical support for the typicality of the transition route in the pertinent two-dimensional parameter space. The finding has implications to applications where high-dimensional and robust chaos is desired.

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“…Hetero-chaos is not Hyper-chaos. Hetero-chaos should not be confused with "hyper-chaos" [28]. A hetero-chaotic attractor can have one or more positive Lyapunov exponents.…”

Section: Hetero-chaos Connects Many Phenomena Like Fluctuating Ementioning

confidence: 99%

Low-dimensional paradigms for high-dimensional hetero-chaos

Saiki

Sanjuán

Yorke

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some regions than others. Some regions of a chaotic attractor can be expanding in more dimensions than other regions. Imagine a situation where two such regions and each contains trajectories that stay in the region for all time while typical trajectories wander throughout the attractor. Such an attractor is "hetero-chaotic" (i.e. it has heterogeneous chaos) if furthermore arbitrarily close to each point of the attractor there are points on periodic orbits that have different unstable dimensions. This is hard to picture but we believe that most physical systems possessing a high-dimensional attractor are of this type. We have created simplified models with that behavior to give insight to real high-dimensional phenomena.Prediction and simulation for chaotic systems occur throughout science. Predictability is more difficult when the "chaotic attractor" is heterogeneous, i.e. if different regions of the chaotic attractor are unstable in more directions than in others. More precisely, when arbitrarily close to each point of the attractor there are different periodic points with different unstable dimensions, we say the chaos is heterogeneous and we call it hetero-chaos. Simple illustrative models of hetero-chaos have been lacking in the literature, and here we present the simplest examples we have found.

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Bifurcation to High-Dimensional Chaos

Cited by 17 publications

References 44 publications

Titration of chaos with added noise

Titration of chaos with added noise

Transition to high-dimensional chaos in nonsmooth dynamical systems

Low-dimensional paradigms for high-dimensional hetero-chaos

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