## Abstract

We introduce a variational method for analyzing limit cycle oscillators in R^{d} driven by Gaussian noise. This allows us to derive exact stochastic differential equations for the amplitude and phase of the solution, which are accurate over times of order (Cb∈^{-1}), where ∈ is the amplitude of the noise and b the magnitude of decay of transverse uctuations. Within the variational framework, different choices of the amplitude-phase decomposition correspond to different choices of the inner product space R^{d}. For concreteness, we take a weighted Euclidean norm, so that the minimization scheme determines the phase by projecting the full solution onto the limit cycle using Floquet vectors. Since there is coupling between the amplitude and phase equations, even in the weak noise limit, there is a small but nonzero probability of a rare event in which the stochastic trajectory makes a large excursion away from a neighborhood of the limit cycle. We use the amplitude and phase equations to bound the probability of it doing this: finding that the typical time the system takes to leave a neighborhood of the oscillator scales as exp(Cb∈^{-1}). We also show how the variational method provides a numerically tractable framework for calculating a stochastic phase, which we illustrate using a modified version of the Morris{Lecar model of a neuron.

Original language | English (US) |
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Pages (from-to) | 2205-2233 |

Number of pages | 29 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 17 |

Issue number | 3 |

DOIs | |

State | Published - 2018 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Analysis
- Modeling and Simulation

## Keywords

- Limit cycle
- Phase reduction
- Stochastic oscillator