Research Barbara Gentz

Stochastic Resonance and Noise-Induced Synchronization

Since its introduction in 1981 as a model for the periodic appearance of the ice ages, stochastic resonance has been observed in a large number of physical and biological systems, including ring lasers, electronic circuits and the sensory system of crayfish. The overdamped motion of a particle in a double-well potential serves as a simple model to illustrate the mechanism. The particle is subject to two different kinds of perturbation: small deterministic periodic driving and additive noise.

If the double-well potential is given by

then the motion of our particle is described by

We may as well view this as the motion of a Brownian particle in a periodically modulated asymmetric potential. Note the following:

	The periodic driving is assumed to have too small an amplitude to allow for transitions between the potential wells in the absence of noise.
	Without periodic forcing, additive noise will cause the particle to jump from one potential well to the other at random times.
	When both perturbations are combined, however, and their amplitudes suitably tuned, the particle will flip back and forth between the wells in a close to periodic way. This effect is known as noise-induced synchronization.

Thus additive noise can significantly enhance the weak periodic forcing, by producing large amplitude oscillations of the system.


	Noise-induced synchronization: A typical sample path of a Brownian particle in a periodically modulated asymmetric potential for suitably tuned modulation and noise intensity. The upper and lower black curve show the locations of the wells of the modulated potential, while the middle curve shows the location of the saddle between the wells.

We give a mathematically rigorous description of the behaviour of individual paths. There exists a threshold value for the noise intensity such that for noise intensities below the threshold, paths are concentrated near one potential well, and have an exponentially small probability to jump to the other well, while above the threshold, transitions between the wells occur with probability exponentially close to 1. We determine the power-law dependence of the critical noise intensity on frequency and amplitude of the driving. The transition zones are localised in time near the instants of maximal forcing.

We also investigated the motion of a Brownian particle in a periodically modulated symmetric potential. Here an additional feature arises: For suitably tuned modulation and noise intensity, each time the potential barrier is low, the particle chooses at random one of the potential wells. Thus with probability close to 1/2, a transition between the wells occurs. Again transition zones are localised in time. A typical sample path is shown below.


	Noise-induced synchronization: A typical sample path of a Brownian particle in a periodically modulated symmetric potential for suitably tuned modulation and noise intensity. The upper and lower black curve show the locations of the wells of the modulated potential, while the straight line in the middle shows the location of the saddle between the wells.

Selected Links of General Interest

	Ice ages (On-line exhibit by the Illinois State Museum)
	Paleoclimatology Slide Sets by the National Oceanic & Atmospheric Administration (NOAA)
	Stochastic Resonance and Signal Detection by Eric C. Toolson

Related Publications

	Nils Berglund and Barbara Gentz A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential Ann. Appl. Probab. 12, 1419–1470 (2002)
	Nils Berglund and Barbara Gentz Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems J. Phys. A: Math. Gen. 35, 2057–2091 (2002)
	Nils Berglund and Barbara Gentz Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations Stoch. Dyn. 2, 327–356 (2002)
	Nils Berglund and Barbara Gentz Noise-Induced Phenomena in Slow-Fast Dynamical Systems. A Sample-Paths Approach Probability and its Applications, Springer-Verlag (2005)

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Last modified: 29 October 2006, Barbara Gentz