Fourth Granada Lectures in Computational Physics pp 298-298 | Cite as

# Resonance Phenomena Induced by Correlated Parametric Noise

## Abstract

The phenomenon of stochastic resonance (SR) has been of broad interest in many different fields and has been observed experimentally in different areas going from physics to biology. Standard SR is the result of the cooperative effect of noise and a periodic driving force acting upon a bistable system, in which the response of the system is enhanced for a particular value of the noise strength. Recently it has been proved numerically that SR can also occur in the absence of an external periodic force as a consequence of the intrinsic dynamics of the nonlinear system. In this work we study the free external force discrete system *X* _{ n+1} = *p* _{ n } *X* _{ n }(1 − *X* _{ n − 1}) where the control parameter is subject to a parametric noise *p* _{ n } = *p* + *ζ* _{ n } being *ζ* _{ n } an Ornstein-Uhlenbeck process with zero mean and exponential correlation, *τ*. In the vicinity of a Hopf bifurcation (*p* > *pH*) the system shows an stochastic resonancelike SNR response, but this appears as a function of *τ* instead of the usually considered noise intensity, *σ*. There is a particular correlation time, *τ* _{ r }, for which the signal to noise ratio (SNR) shows a maximum. It is observed that for *τ* _{ r } the frequency of the system’s time series is as similar as possible to the corresponding deterministic frequency (*ω* _{ d }(*p*)). This noise induced stochastic enhancement is observed when *p* is close to the bifurcation value and is not obtained when the SNR response is studied as a function of *σ*. This resonant behavior is qualitatively understood as determined by the time of permanence of the system around the limit cycle which imposes a compromise between *τ* and the deterministic convergence time to the stable oscillatory state. Our results show that there is a particular *τ* for which this coherence is optimized, indicating that the system selects in mean a particular orbit. This idea is reinforced by the results obtained for *p* > *pH* in which *τ* _{ r } appear shifted towards smaller values as also happens with the deterministic convergence time. We think that this behavior could be observed in other systems where a multiplicative colored random perturbation moves the control parameter through a Hopf bifurcation region. Preliminary results indicate the same kind of resonant response in other nonlinear discrete systems.

## Keywords

Hopf Bifurcation Noise Intensity Stochastic Resonance Parametric Noise Resonance Phenomenon## References

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