Noise induced dimension changing bifurcations

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

doi:10.1117/12.610050

23 May 2005 Noise induced dimension changing bifurcations (Invited Paper)

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

Proceedings Volume 5845, Noise in Complex Systems and Stochastic Dynamics III; (2005) https://doi.org/10.1117/12.610050
Event: SPIE Third International Symposium on Fluctuations and Noise, 2005, Austin, Texas, United States

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.

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Ira B. Schwartz, Lora Billings, David S. Morgan, and Ying Cheng Lai "Noise induced dimension changing bifurcations (Invited Paper)", Proc. SPIE 5845, Noise in Complex Systems and Stochastic Dynamics III, (23 May 2005); https://doi.org/10.1117/12.610050

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KEYWORDS

Chaos

Stochastic processes

Oscillators

Complex systems

Mechanics

Systems modeling

Dynamical systems

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