An inflated dynamic Laplacian to track the emergence and disappearance of semi-material coherent sets

Abstract

Lagrangian methods continue to stand at the forefront of the analysis of time-dependent dynamical systems. Most Lagrangian methods have criteria that must be fulfilled by trajectories as they are followed throughout a given finite flow duration. This key strength of Lagrangian methods can also be a limitation in more complex evolving environments. It places a high importance on selecting a time window that produces useful results, and these results may vary significantly with changes in the flow duration. We show how to overcome this drawback in the tracking of coherent flow features. Finite-time coherent sets (FTCS) are material objects that strongly resist mixing in complicated nonlinear flows. Like other materially coherent objects, by definition they must retain their coherence properties throughout the specified flow duration. Recent work [Froyland and Koltai, CPAM, 2023] introduced the notion of semi-material FTCS, whereby a balance is struck between the material nature and the coherence properties of FTCS. This balance provides the flexibility for FTCS to come and go, merge and separate, or undergo other changes as the governing unsteady flow experiences dramatic shifts. The purpose of this work is to illustrate the utility of the inflated dynamic Laplacian introduced in [Froyland and Koltai, CPAM, 2023] in a range of dynamical systems that are challenging to analyse by standard Lagrangian means, and to provide an efficient meshfree numerical approach for the discretisation of the inflated dynamic Laplacian.

Figures (25)

• Figure 1: Left: The curve $\Gamma$ disconnects the manifold $M$; $M$ and $\Gamma$ are evolved by $\phi_t$. Right: The curve $\Gamma$ again disconnects $M$, now without touching the boundary of $M$. In this case, $A$ could be $A_1$ and the complement of $A$ (minus $\Gamma$) could be $A_2$.
• Figure 2: Left: The eigenfunction $f_2$ of $\Delta^D$ corresponding to $\lambda_2^D$; here the extreme values of $f_2$ -- shown in red and blue -- clearly highlight the two coherent sets. Right: The same eigenfunction evolved forward and plotted in spacetime. This plot demonstrates that the red and blue regions of the two-dimensional phase space remain largely dynamically disconnected and with relatively small evolving one-dimensional boundary. Both images are displayed with an absolute value cutoff of 0.25. The underlying flow is defined and further analysed in Section \ref{['sec:switching']}.
• Figure 3: If coherence in the system persists through the entire time interval $[0,\tau]$, a material coherent set provides a small dynamic Cheeger constant $h^D$. The top of the diagram shows a single disconnector $\Gamma\subset M$ (green dots at the very top of the figure). Upper left: by trivial copying in time we obtain the disconnector $\reflectbox{\hbox{o}rigin=c]{180}{$ L$}}_{\rm copy} = [0,\tau] \times \Gamma \subset\mathbb M_0$. We visualise one of the sets it disconnects $\mathbb M_0$ into in pale green. Upper right: by evolving $\reflectbox{\hbox{o}rigin=c]{180}{$ L$}}_{\rm copy}$ forward in time with the dynamics from time 0 to time $\tau$ we trace out the disconnector $\bigcup_{t\in[0,\tau]}\{t\}\times\phi_t(\Gamma) \subset \mathbb M_1$. The lower row of the diagram concerns the situation where there is a coherent set present only for part of the time interval, say a subinterval $[\tau_1,\tau_2]\subset [0,\tau]$. Lower right: following the dynamics, a coherent set appears at $\tau_1$ from a small expanding core, exists for a while, and then shrinks and dissipates completely at $\tau_2$. Lower left: We pull back the lower right image to time $t=0$ using the inverse of $\Phi$ to obtain the disconnector $\reflectbox{\hbox{o}rigin=c]{180}{$ L$}} \subset \mathbb M_0$.
• Figure 4: Illustration of $\mathbb M_0$ vs $\mathbb M_1$ and associated disconnectors for steady translational dynamics $\phi_tx = x+\alpha t$, with $t\in [0,\tau]$. Left: the disconnector $\reflectbox{\hbox{o}rigin=c]{180}{$ L$}}_{\rm copy}$ in $\mathbb M_0$ is independent of $\alpha$. Right: the velocity $\alpha$ of the translation $\phi_t$ changes the measure of the spacetime disconnector $\reflectbox{\hbox{o}rigin=c]{180}{$ L$}}'=\Phi(\reflectbox{\hbox{o}rigin=c]{180}{$ L$}}_{\rm copy})$ in $\mathbb M_1$, even though the translational dynamics has no effect on the coherence of the set.
• Figure 5: Third (second spatial) eigenfunction of the inflated dynamic Laplacian $\Delta_{G_{0,a}}$ with $a=0.3340, \varepsilon=0.0044$ (see Section \ref{['sec:dmapsDL']} for the role of the parameter $\varepsilon$ in the numerical discretisation of $\Delta_{G_{0,a}}$). An absolute value cutoff of 0.25 has been applied. The extreme values of the eigenfunction -- deep red is $+1$ and deep blue is $-1$ -- clearly highlight the two semi-material coherent sets. Left: $\mathbb M_0$. Material trajectories (straight lines in $\mathbb M_0$) tend to have constant eigenfunction values. The strong change in colour of some trajectories indicate that the sets are only almost-material. Right: $\mathbb M_1$. The red and blue regions in the co-evolved time-expanded phase space remain largely disconnected, indicating strong coherence. Figure \ref{['fig:DGswitch_tempvar']} highlights trajectories along which there is a nonmaterial change in the eigenfunction.

Theorems & Definitions (10)

• Proposition 3.1
• Conjecture 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Conjecture 3.5
• Theorem 3.6
• proof
• proof