Effects of Moderate Noise on a Limit Cycle Oscillator: Counterrotation and Bistability

Abstract

The effects of noise on the dynamics of nonlinear systems is known to lead to many counterintuitive behaviors. Using simple planar limit cycle oscillators, we show that the addition of moderate noise leads to qualitatively different dynamics. In particular, the system can appear bistable, rotate in the opposite direction of the deterministic limit cycle, or cease oscillating altogether. Utilizing standard techniques from stochastic calculus and recently developed stochastic phase reduction methods, we elucidate the mechanisms underlying the different dynamics and verify our analysis with the use of numerical simulations. Lastly, we show that similar bistable behavior is found when moderate noise is applied to the more biologically realistic FitzHugh-Nagumo model.Limit cycle oscillators have been widely used to model various natural phenomena [1][2][3]. As such, they have been the subject of extensive study in the field of nonlinear science. In recent years, the effects of noise on the dynamics of limit cycle oscillators has received much interest [4][5][6][7][8]. When the noise is weak, formal reduction methods can be performed to reduce the dimensionality of the system to a so-called phase equation [7,8]. In this case, one can analytically show that both the magnitude (and correlation time for colored noise) of the added noise shift the mean frequency of oscillations away from the natural frequency of the limit cycle. On the other hand, when the noise is large, the trajectories of the system appear completely random and bear no resemblance to the deterministic limit cycle behavior.The case of moderate noise applied to a limit cycle oscillator has received less attention. It is known that moderate noise can cause stochastic resonance in systems close to a bifurcation to limit cycle oscillations [9,10], and in systems where limit cycles arise as the result of periodic forcing [4]. However, an exploration of how moderate noise interacts with the underlying deterministic dynamics of a system displaying limit cycle behavior has not yet been undertaken and is the purpose of the current Letter. We find that an oscillator subject to moderate noise can display numerous interesting and counterintuitive behaviors. In some cases, the addition of noise causes the phase to behave like a bistable switch, while in other cases noise can act to completely eliminate oscillations, and even cause the trajectories to rotate in the opposite direction of the deterministic limit cycle.Bistability in the amplitude of the limit cycle has been shown to occur in the Stuart-Landau (SL) system when it is subjected to periodic forcing [11], or specially constructed stochastic forcing [12]. It is also known that coupling limit cycle oscillators together can eliminate oscillations [13,14]. However, we show that these phenomena can occur in a planar limit cycle system when each component is subjected to additive white noise.For simplicity, we consider a radially symmetric deterministic system. One such symmetric oscillator that we employ is the ...

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