Coherence resonance at noisy precursors of bifurcations in nonlinear dynamical systems

Abstract

A general mechanism of coherence resonance that occurs in noisy dynamical systems close to the onset of bifurcation is demonstrated through examples of period-doubling and torus-birth bifurcations. Near the bifurcation of a periodic orbit, noise produces the characteristic peaks of ''noisy precursors'' in the power spectrum. The signal-to-noise ratio evaluated at these peaks is maximal for a certain optimal noise intensity in a manner that resembles a stochastic resonance. ͓S1063-651X͑97͒06307-1͔ PACS number͑s͒: 05.40.ϩj, 05.20.Ϫy Nonlinear systems perturbed by noise have the potential to display a wide range of complex responses including, somewhat paradoxically, an enhancement of net order and coherence as noise levels increase. A distinguished example of this phenomenon is stochastic resonance ͑SR͒ ͓1͔ which has attracted considerable attention over the last decade ͑see for references the reviews ͓2͔͒. Conventional SR occurs in noisy dynamical systems when perturbed by a weak external periodic signal. For such systems, significant amplification of the weak periodic signal may occur solely by increasing the level of the noise intensity. The signal-to-noise ratio ͑SNR͒, and other appropriate measures of signal coherence, pass through a maximum at an optimal noise strength when the noise-controlled time scale of the system matches the period of the external signal.A similar effect of noise-induced coherence may also be observed in systems which lack an external signal, but whose intrinsic dynamics are controlled by noise intensity. In earlier studies ͓3,4͔ the noise-induced enhancement of coherence in underdamped nonlinear oscillators has been found. The noise-induced peak at zero frequency appeared in the vicinity of a pitchfork bifurcation ͓3͔, whereas the decrease of the width of a fluctuating peak in the power spectrum is shown for an underdamped oscillator, whose eigenfrequency possesses an extreme in energy, in ͓4͔. Recently, a noiseinduced coherent motion has been observed for autonomous systems in ͓5͔, where the effect of noise on a nonuniform limit cycle has been studied, and in ͓6͔, where a coherence resonance in a noise-driven excitable system has been reported. This group of phenomena can be called coherence resonance or ''internal'' SR, which underlines the fact that one can observe SR-like phenomena without an external periodic signal.In the present paper we study the response of nonlinear dynamical systems to noise excitation near the onset of dynamical instabilities of periodic orbits. Our starting point is the key paper of Wiesenfeld ͓7͔, which carefully elaborates the way in which noise controls the qualitative structure of the power spectrum. In brief, Wiesenfeld demonstrates that the power spectrum of a system observed after a bifurcation point can, nevertheless, be visible even before the bifurcation actually occurs if there is noise present. We thus observe a noisy precursor of the bifurcation.To follow this line of thought further, let us suppose that noise induces a peak of height H at ...

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“…This delicate balance between noise and nonlinear dynamics is known as autonomous stochastic resonance (Gang et al 1993;Longtin 1997) or coherence resonance (Pikovsky and Kurths 1997;Lindner and Schimansky-Geier 1999;Pradines et al 1999). Coherence resonance can be explained in terms of a noisy precursor to deterministic bifurcations such as Hopf bifurcations (Neiman et al 1997) or homoclinic and heteroclinic bifurcations (Stone and Holmes 1989;Stone and Armbruster 1999). One of the main features of this dynamical systems' concept is that stochastic forcing that acts upon nonlinear excitable systems induces a new nondeterministic timescale similar to the one identified in our stochastic sensitivity experiments.…”

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We demonstrate the existence of noise-induced periodicity (coherence [16,17], spatio-temporal arrays [18,19] and a few experimental systems [13,15,[20][21][22][23][24].

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