Extracting Coherent Sets from Generators of Mather Semigroups in Aperiodically Driven Flows

Core Concepts

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.

Abstract

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.

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Key Insights Distilled From

Extracting coherent sets in aperiodically driven flows from generators of Mather semigroups

by Robi... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.19274.pdf

Deeper Inquiries

How can the methods presented in this work be extended to more general driving dynamics on the parameter space Θ, beyond the quasi-periodic case considered in the examples

The methods presented in the work can be extended to more general driving dynamics on the parameter space Θ by considering a broader class of vector fields and flows. Instead of restricting the analysis to quasi-periodic driving dynamics, one can explore the behavior of nonautonomous systems driven by ergodic flows with different characteristics. This extension would involve studying the spectral properties of the Mather semigroup generator G for a wider range of driving dynamics, beyond the specific examples considered in the paper. By analyzing the spectral properties of G in the context of different driving dynamics, one can gain insights into the long-term behavior and transport characteristics of the system under various driving scenarios.

What are the connections between the spectral properties of the Mather semigroup generator G and the Lyapunov spectrum of the transfer operator cocycle Pt θ, and how can these connections be further explored

The connections between the spectral properties of the Mather semigroup generator G and the Lyapunov spectrum of the transfer operator cocycle Pt θ provide valuable insights into the dynamics of the system. The Lyapunov spectrum characterizes the exponential growth rates of perturbations along different directions in the system, while the spectral properties of G reveal the behavior of coherent sets under the influence of the driving dynamics. By further exploring these connections, one can investigate how the stability and mixing properties of the system relate to the formation and persistence of coherent sets. This exploration can lead to a deeper understanding of the interplay between spectral analysis and coherent structure extraction in nonautonomous systems.

Can the coherent sets extracted from the Mather semigroup be related to other notions of coherent structures, such as Lagrangian Coherent Structures, in a rigorous way

The coherent sets extracted from the Mather semigroup can be related to other notions of coherent structures, such as Lagrangian Coherent Structures (LCS), in a rigorous way by examining their transport properties and mixing behavior. Coherent sets represent time-dependent regions in the physical space that exhibit little mixing with their surroundings, similar to the characteristics of LCS. By analyzing the coherence and persistence of these sets in relation to the dynamics of the system, one can establish connections between coherent sets and LCS. Understanding how coherent sets evolve and interact with the flow field can provide valuable insights into the organization of transport processes and the identification of key dynamical features in nonautonomous systems.