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

TL;DR

This work develops a spectral framework for extracting time-dependent coherent sets in aperiodically driven flows with small diffusion by formulating an augmented Mather semigroup on the skew-product space. Coherence is characterized via eigenfunctions, approximate eigenfunctions, and spectral projections of the augmented generator $\mathbf{G}$, with a robust spectral-mapping relation $\sigma(\mathbf{G}) = \{\lambda + \eta i: \lambda \in \Sigma(\Phi), \ \eta \in \mathbb{R}\}$ that links base dynamics to transport structures. The method uses a Fourier Galerkin discretization for quasi-periodic driving, enabling trajectory-free computation of coherent families $A_\theta^t$ across all driving states, and is demonstrated on translated and oscillating gyres as well as a shear flow, with coherence quantified by escape rates and cumulative survival probabilities. The results provide a principled route to identify and quantify persistent transport patterns in nonautonomous flows, with potential applications in environmental and engineering transport scenarios where driving is ergodic and diffusion cannot be neglected. Overall, the augmented-generator approach offers a rigorous, computable pathway to reveal and analyze time-dependent coherence under complex driving, expanding the toolbox for nonautonomous transport analysis.

Abstract

Coherent sets are time-dependent regions in the physical space of nonautonomous flows that exhibit little mixing with their neighborhoods, robustly under small random perturbations of the flow. They thus characterize the global long-term transport behavior of the system. We propose a framework to extract such time-dependent families of coherent sets for nonautonomous systems with an ergodic driving dynamics and (small) Brownian noise in physical space. Our construction involves the assembly and analysis of an operator on functions over the augmented space of the associated skew product that, for each fixed state of the driving, propagates distributions on the corresponding physical-space fibre according to the dynamics. This time-dependent operator has the structure of a semigroup (it is called the Mather semigroup), and we show that a spectral analysis of its generator allows for a trajectory-free computation of coherent families, simultaneously for all states of the driving. Additionally, for quasi-periodically driven torus flows, we propose a tailored Fourier discretization scheme for this generator and demonstrate our method by means of three examples of two-dimensional flows.

Figures (11)

• Figure 1: Sketch of the temporal evolution of the coherent set $A_\theta^t=\{\mathbf{f}(\phi^t\theta) \geq 0\} \subset M$, where $\mathbf{f}$ is an eigenfunction of the Mather semigroup with real eigenvalue. The distribution $\mathbf{f}(\theta)$ is mapped to (a multiple of) $\mathbf{f}(\phi^t\theta)$ under the cocycle $\mathcal{P}_\theta^t$ over the driving $\phi^t: \Theta \to \Theta$.
• Figure 2: A schematic representation of the spectrum of the generator $\mathbf{G}$ (left), and the spectrum of the Mather operator $\mathbf{M}^1$ (right). The unit circle $S^1$ is marked as a dashed circle. In this example, the Sacker--Sell spectrum consists of four intervals, one of which is a single point.
• Figure 3: Left: Stream function $\psi(x) = \frac{1}{2\pi}\cos(2\pi x_1) \cos(2\pi x_2)$. Right: The corresponding vector field $v_\text{aut}$.
• Figure 4: Translated gyre example. Left: 100 eigenvalues of the discrete generator $\Gamma_S$ around the origin. Right: In blue, the simulated survival probability of the coherent sets extracted from an eigenfunction $\mathbf{f}$ to an eigenvalue $z$ using method (CS3) with threshold $q=1$. The black lines show the curves $e^{\lambda t}$ (dashed) and $e^{2\lambda t}$ (solid) for comparison, where $\lambda$ is the real-part of $z$. The computed cumulative survival probability is $C(A_{\theta}^\bullet)= 6.070$.
• Figure 5: Translated gyre example: Evolution of the coherent sets, marked in black, determined using method (CS3) with threshold $q = 1$. The snapshots are taken at time $t=0,5,10$. The top row depicts a particle simulation. Particles that started inside $A_{\theta}^0$ and have remained there until time $t$ are colored in blue. Particles that have started inside $A_{\theta}^0$ but were outside the coherent set at some time $0<s\leq t$ are colored in red. Other particles are colored in grey. The bottom row locates the coherent set with respect to the position of the gyres, as indicated by the stream function at the respective time. A video of the particle simulation for the time interval $[0,10]$ can be found at https://github.com/RobinChemnitz/MatherCoherent.

Theorems & Definitions (35)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• Remark 2.5
• Theorem 2.6
• proof
• Definition 3.1
• Theorem 3.2
• proof