Onset of resonance in chaotically driven systems
TL;DR: This analysis reveals the existence and identification of a characteristic natural frequency associated with the dynamics of the system confirming that classical resonance is responsible for chaotic resonance behavior.
About: This article is published in Applied Mathematics Letters.The article was published on 2000-11-01 and is currently open access. It has received 1 citations till now.
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TL;DR: The first-order SNR shows that chaotic resonance can correspond directly to stochastic resonance and is derived from the two-state model and the chaos-induced transition rate.
Abstract: We consider the resonant effects of chaotic fluctuations on a strongly damped particle in a bistable potential driven by weak sinusoidal perturbation. We derive analytical expressions of chaos-induced transition rate between the neighboring potential wells based on the inhomogeneous Smoluchowski equation. Our first-order analysis reveals that the transition rate has the form of the Kramers escape rate except for a perturbed prefactor. This modification to the prefactor is found to arise from the statistical asymmetry of the chaotic noise. By means of the two-state model and the chaos-induced transition rate, we arrive at an analytical expression of the signal-to-noise ratio (SNR). Our first-order SNR shows that chaotic resonance can correspond directly to stochastic resonance.
13 citations
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TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.
16,554 citations
TL;DR: A prototype equation to the Lorenz model of turbulence contains just one (second-order) nonlinearity in one variable as mentioned in this paper, which allows for a "folded" Poincare map (horseshoe map).
3,334 citations
TL;DR: In this article, simple dissipative dynamical systems exhibiting a transition from a stable periodic behavior to a chaotic one were studied, where the inverse coherence time grows continuously from zero to zero due to the random occurrence of widely separated bursts in the time record.
Abstract: We study some simple dissipative dynamical systems exhibiting a transition from a stable periodic behavior to a chaotic one. At that transition, the inverse coherence time grows continuously from zero due to the random occurrence of widely separated bursts in the time record.
1,753 citations
TL;DR: In certain nonlinear systems, including electronic circuits and biological sensory apparatus, the presence of noise can in fact enhance the detection of weak signals, called stochastic resonance.
Abstract: Noise in dynamical systems is usually considered a nuisance. But in certain nonlinear systems, including electronic circuits and biological sensory apparatus, the presence of noise can in fact enhance the detection of weak signals. This phenomenon, called stochastic resonance, may find useful application in physical, technological and biomedical contexts.
1,588 citations
TL;DR: A detailed theoretical and numerical study of stochastic resonance, based on a rate equation approach, results in an equation for the output signal-to-noise ratio as a function of the rate at which noise induces hopping between the two states.
Abstract: The concept of stochastic resonance has been introduced previously to describe a curious phenomenon in bistable systems subject to both periodic and random forcing: an increase in the input noise can result in an improvement in the output signal-to-noise ratio. In this paper we present a detailed theoretical and numerical study of stochastic resonance, based on a rate equation approach. The main result is an equation for the output signal-to-noise ratio as a function of the rate at which noise induces hopping between the two states. The manner in which the input noise strength determines this hopping rate depends on the precise nature of the bistable system. For this reason, the theory is applied to two classes of bistable systems, the double-well (continuous) system and the two-state (discrete) system. The theory is tested in detail against digital simulations.
1,231 citations