Основные понятия
The core message of this article is to propose a framework for extracting time-dependent families of coherent sets in nonautonomous systems with an ergodic driving dynamics and small Brownian noise, by analyzing the spectrum and spectral subspaces of the generator of the associated Mather semigroup.
Аннотация
The article introduces a framework for extracting coherent sets in nonautonomous flows driven by an ergodic dynamics on a parameter space Θ, with small Brownian noise in the physical space M. The key components are:
The transfer operators Pt
θ that evolve distributions on M according to the Fokker-Planck equation associated with the stochastic differential equation (SDE) on M, driven by the parameter dynamics on Θ.
The Mather semigroup Mt that evolves functions on the augmented space Θ × M, by propagating the fibres of the functions according to the transfer operators Pt
θ.
The generator G of the Mather semigroup, whose spectrum and spectral subspaces are used to extract coherent sets. The spectrum of G is characterized in terms of the Sacker-Sell spectrum of the transfer operator cocycle Pt
θ.
Rigorous bounds on the coherence of the extracted families of coherent sets, measured by escape rates and cumulative survival probabilities, are derived from the spectral properties of G.
For the case of quasi-periodically driven torus flows, a tailored Fourier discretization scheme for the generator G is proposed, and the method is demonstrated through three numerical examples.