Noise induced dimension changing bifurcations

I B Schwartz, L Billings, D S Morgan, Ying-Cheng Lai

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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
Title of host publicationNoise in Complex Systems and Stochastic Dynamics III
EditorsLB Kish, K Lindenberg, Z Gingl
Place of PublicationBellingham, WA, USA
PublisherSPIE
Pages245-255
Number of pages11
ISBN (Print)0819458406
DOIs
Publication statusPublished - 2005
EventConference on Noise in Complex Systems and Stochastic Dynamics III - Austin
Duration: 24 May 200526 May 2005

Publication series

NameProceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE)
PublisherSPIE - Int. Soc. Optical Engineering
Volume 5845
ISSN (Print)0277-786X

Conference

ConferenceConference on Noise in Complex Systems and Stochastic Dynamics III
CityAustin
Period24/05/0526/05/05

Keywords

  • coupled chaotic systems
  • Lyapunov exponent
  • dynamics
  • oscillators
  • variability
  • slow

Cite this

Schwartz, I. B., Billings, L., Morgan, D. S., & Lai, Y-C. (2005). Noise induced dimension changing bifurcations. In LB. Kish, K. Lindenberg, & Z. Gingl (Eds.), Noise in Complex Systems and Stochastic Dynamics III (pp. 245-255). (Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE); Vol. 5845). Bellingham, WA, USA: SPIE. https://doi.org/10.1117/12.610050

Noise induced dimension changing bifurcations. / Schwartz, I B ; Billings, L ; Morgan, D S ; Lai, Ying-Cheng.

Noise in Complex Systems and Stochastic Dynamics III. ed. / LB Kish; K Lindenberg; Z Gingl. Bellingham, WA, USA : SPIE, 2005. p. 245-255 (Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE); Vol. 5845).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Schwartz, IB, Billings, L, Morgan, DS & Lai, Y-C 2005, Noise induced dimension changing bifurcations. in LB Kish, K Lindenberg & Z Gingl (eds), Noise in Complex Systems and Stochastic Dynamics III. Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), vol. 5845, SPIE, Bellingham, WA, USA, pp. 245-255, Conference on Noise in Complex Systems and Stochastic Dynamics III, Austin, 24/05/05. https://doi.org/10.1117/12.610050
Schwartz IB, Billings L, Morgan DS, Lai Y-C. Noise induced dimension changing bifurcations. In Kish LB, Lindenberg K, Gingl Z, editors, Noise in Complex Systems and Stochastic Dynamics III. Bellingham, WA, USA: SPIE. 2005. p. 245-255. (Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE)). https://doi.org/10.1117/12.610050
Schwartz, I B ; Billings, L ; Morgan, D S ; Lai, Ying-Cheng. / Noise induced dimension changing bifurcations. Noise in Complex Systems and Stochastic Dynamics III. editor / LB Kish ; K Lindenberg ; Z Gingl. Bellingham, WA, USA : SPIE, 2005. pp. 245-255 (Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE)).
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N2 - 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.

AB - 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.

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KW - variability

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