Noise induced hops

The influence of noise on a dynamical system may have two counteracting effects. On the one hand if the underlying deterministic systems is already oscillatory, like a limit cycle oscillator or a chaotic oscillator, one expects these oscillations to become less regular due to the influence of the noise. On the other hand oscillatory behavior can also be generated by the noise in systems which deterministically do not show any oscillations. A prominent example are excitable systems but also the noise induced hopping between the attractors in a bistable system can be considered as oscillations [1]. 

The mechanism of the one-way transition process is easily deduced from an examination of Fig. 1 [13]. Suppose the system is initially in the limit cycle state L2. Note that the basin boundary Bi intersects L2 thus, if a noise event occurs while the phase point is on L2 and to the left of Bi, it will find itself in the basin of FI and tend to evolve to FI in the absence of other noise events. Alternatively, if it is to the right of Bi when the noise acts, it will evolve to LI. The transition occurs by a shift of the basin boundary under the action of the parametric noise process (moving boundary mechanism) [13]. A similar argument shows that once a phase point is near FI or F2 it can never escape since both of these fixed points lie to the left of Bi and B2. Hence, under the stochastic dynamics the fixed point region will appear fuzzy as a result of noise-induced hops between FI and F2 likewise there will be noise-induced hops between LI and L2 leading to a fuzzy limit cycle. 

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