Temporal localization of limit cycles in a noise-driven chemical oscillator

We study the influence of multiplicative random external perturbation on the Selkov model exhibiting limit cycles. The dynamics are described by stochastic differential equations. The numerically computed ensemble averages of the concentrations exhibit temporal localization of the limit cycles for small amplitudes of the noise. The localization time is smaller when the amplitude of the noise is higher. To understand this behavior, a stochastic complex Ginzburg–Landau equation is derived using normal form transformations on the stochastic evolution equations. In the case of weak noise, the equations for the radial and the phase components of the amplitude separate and can be solved exactly. The solution is used to compute the damping factors, which were found to be Gaussian. We also show that asymptotically, the radial part fluctuates near the average value and the phase is found to be a Gaussian random variable with time-dependent mean and variance and may be responsible for the temporal localization of the limit cycle.