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Peter Thomas; Department of Mathematics, Applied Mathematics, and Statistics; Case Western Reserve ³Ô¹ÏÍø

A Universal Description of Stochastic Oscillators

Many systems in physics, chemistry and biology exhibit oscillationsÌýwith a pronounced random component. Such stochastic oscillations can emerge viaÌýdifferent mechanisms, for example linear dynamics of a stable focus withÌýfluctuations, limit-cycle systems perturbed by noise, or excitable systems inÌýwhich random inputs lead to a train of pulses. Despite their diverse origins,Ìýthe phenomenology of random oscillations can be strikingly similar. In jointÌýwork with Alberto Perez (Universidad Complutense de Madrid), Benjamin Lindner (Humboldt ³Ô¹ÏÍø, Berlin), and Boris Gutkin (ENS Paris), we introduce aÌýnonlinear transformation of stochastic oscillators to a new complex-valuedÌýfunction $Q^*_1(\mbx)$ that greatly simplifies and unifies the mathematicalÌýdescription of the oscillator's spontaneous activity, its response to anÌýexternal time-dependent perturbation, and the correlation statistics ofÌýdifferent oscillators that are weakly coupled. The function $Q^*_1(\mbx)$ is theÌýeigenfunction of the Kolmogorov backward operator with the least negative (butÌýnon-vanishing) eigenvalue $\lambda_1=\mu_1+i\omega_1$. The resulting powerÌýspectrum of the complex-valued function is exactly given by a Lorentz spectrumÌýwith peak frequency $\omega_1$ and half-width $\mu_1$; its susceptibility withÌýrespect to a weak external forcing is given by a simple one-pole filter,Ìýcentered around $\omega_1$; and the cross-spectrum between two coupledÌýoscillators can be easily expressed by a combination of the spontaneous powerÌýspectra of the uncoupled systems and their susceptibilities. Our approach makesÌýqualitatively different stochastic oscillators comparable, provides simpleÌýcharacteristics for the coherence of the random oscillation, and Ìýgives aÌýframework for the description of weakly coupled stochastic oscillators.