## Abstract

Let ρ_{0} be an invariant probability density of a deterministic dynamical system f and ρε the invariant probability density of a random perturbation of f by additive noise of amplitude ε. Suppose ρ_{0} is stochastically stable in the sense that ρε → _{0} as ε → 0. Through a systematic numerical study of concrete examples, I show that: The rate of convergence of ρε to 7rho;ε_{0} as ε → 0 is frequently governed by power laws: ∥ρε - ρ_{0}∥_{1} ∼ ε ^{γ} for some γ > 0. When the deterministic system / exhibits exponential decay of correlations, a simple heuristic can correctly predict the exponent γ based on the structure of ρ_{0}. The heuristic fails for systems with some 'intermittency', i.e. systems which do not exhibit exponential decay of correlations. For these examples, the convergence of ρε to ρ_{0} as ε → 0 continues to be governed by power laws but the heuristic provides only an upper bound on the power law exponent γ. Furthermore, this numerical study requires the computation of ∥ρε - ρ0∥ _{1} for 1.5-2.5 decades of ε and provides an opportunity to discuss and compare standard numerical methods for computing invariant probability densities in some depth.

Original language | English (US) |
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Pages (from-to) | 659-683 |

Number of pages | 25 |

Journal | Nonlinearity |

Volume | 18 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2005 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics