Noise induced dimension changing bifurcations

Ira B. Schwartz, Lora Billings, David S. Morgan, Ying Cheng Lai

Research output: Contribution to journalConference articleResearchpeer-review

Abstract

The transition to chaos is a fundamental and widely studied problem in deterministic nonlinear dynamics. Well known routes to chaos, which include the period-doubling bifurcation route, the intermittency route, the quasiperiodic route, and the crisis route, describe transitions to low-dimensional chaotic attractors with one positive Lyapunov exponent. Transitions to high-dimensional chaotic attractors with multiple positive Lyapunov exponents have just started being addressed. In stochastic systems, transitions to chaotic-like behavior are less well characterized. Global analysis coupled with stochastic transport probability can explain emergent behavior in which stable and unstable manifolds may interact with noise to cause "stochastic chaos". Another stochastic route may induce chaotic signatures through a dimension changing bifurcation, whereby the topological dimension changes when the amplitude of the noise goes beyond a critical parameter. In this paper we present a theory of how the Lyapunov exponents may scale with the noise amplitude in general systems. A physical class of multiscale dynamical systems will be presented to show that noise may induce low dimensional chaos, or for other parameters, may induce chaos that bifurcates to an attractor contained in a high topological dimension. We present a numerical bifurcation analysis of the resulting system, illustrating the mechanism for the onset of high dimensional chaos. By computing the constrained invariant sets, we reveal the transition from low dimensional to high dimensional chaos. Applications include both deterministic and stochastic bifurcations.

Original languageEnglish
Article number35
Pages (from-to)245-255
Number of pages11
JournalProceedings of SPIE - The International Society for Optical Engineering
Volume5845
DOIs
StatePublished - 12 Dec 2005
EventNoise in Complex Systems and Stochastic Dynamics III - Austin, TX, United States
Duration: 24 May 200526 May 2005

Fingerprint

Chaos theory
chaos
Chaos
Bifurcation
routes
Lyapunov Exponent
High-dimensional
Chaotic Attractor
exponents
Stochastic Bifurcation
Emergent Behavior
Stable and Unstable Manifolds
Period-doubling Bifurcation
Stochastic systems
period doubling
Global Analysis
Intermittency
intermittency
Bifurcation Analysis
Invariant Set

Cite this

Schwartz, Ira B. ; Billings, Lora ; Morgan, David S. ; Lai, Ying Cheng. / Noise induced dimension changing bifurcations. In: Proceedings of SPIE - The International Society for Optical Engineering. 2005 ; Vol. 5845. pp. 245-255.
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Noise induced dimension changing bifurcations. / Schwartz, Ira B.; Billings, Lora; Morgan, David S.; Lai, Ying Cheng.

In: Proceedings of SPIE - The International Society for Optical Engineering, Vol. 5845, 35, 12.12.2005, p. 245-255.

Research output: Contribution to journalConference articleResearchpeer-review

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