Almost-invariant Sets Research Articles

A Data-Driven Method for Computing Polyhedral Invariant Sets of Black-Box Switched Linear Systems

In this paper, we consider the problem of invariant set computation for black-box switched linear systems using merely a finite set of observations of system trajectories. In particular, this paper focuses on polyhedral invariant sets. We propose a data-driven method based on the one step forward reachable set. For formal verification of the proposed method, we introduce the concepts of $\lambda$-contractive sets and almost-invariant sets for switched linear systems. The convexity-preserving property of switched linear systems allows us to conduct contraction analysis on the computed set and derive a probabilistic contraction property. In the spirit of non-convex scenario optimization, we also establish a chance-constrained guarantee on set invariance. The performance of our method is then illustrated by numerical examples.

Scenario-Based Set Invariance Verification for Black-Box Nonlinear Systems

We consider the problem of set invariance verification in black-box nonlinear systems without analytic dynamical models. A data-driven set invariance verification approach relying on the observation of trajectories is proposed to determine almost-invariant sets, which are invariant almost everywhere except possibly in a small subset. With these observations, scenario optimization problems are formulated. We show that probabilistic invariance guarantees on the almost-invariant sets can be established. To get explicit expressions of such sets, a set identification procedure is designed by the use of a polynomial classifier. The practical performance of the proposed data-driven framework is illustrated by numerical examples.

Stability of the Malvinas Current.

Deterministic and probabilistic tools from nonlinear dynamics are used to assess enduring near-surface Lagrangian aspects of the Malvinas Current. The deterministic tools are applied to a multiyear record of velocities derived from satellite altimetry data, revealing a resilient cross-stream transport barrier. This is composed of shearless-parabolic Lagrangian coherent structures (LCSs), which, extracted over sliding time windows along the multiyear altimetry-derived velocity record, lie in near-coincidental position. The probabilistic tools are applied on a large collection of historical satellite-tracked drifter trajectories, revealing weakly communicating flow regions as basins of attraction for long-time asymptotic almost-invariant sets on either side of the altimetry-derived barrier. Shearless-parabolic LCSs are detected for the first time from altimetry data, and their significance is supported on satellite-derived ocean color data, which reveal shapes that quite closely resemble the peculiar V shapes, dubbed "chevrons," that have recently confirmed the presence of similar LCSs in the atmosphere of Jupiter. Finally, using available in situ velocity and hydrographic data, sufficient and necessary conditions for nonlinear symmetric stability are found to be satisfied, suggesting a duality between Lagrangian and Eulerian stability for the Malvinas Current.

Sparse eigenbasis approximation: Multiple feature extraction across spatiotemporal scales with application to coherent set identification

The output of spectral clustering is a collection of eigenvalues and eigenvectors that encode important connectivity information about a graph or a manifold. This connectivity information is often not cleanly represented in the eigenvectors and must be disentangled by some secondary procedure. We propose the use of an approximate sparse basis for the space spanned by the leading eigenvectors as a natural, robust, and efficient means of performing this separation. The use of sparsity yields a natural cutoff in this disentanglement procedure and is particularly useful in practical situations when there is no clear eigengap. In order to select a suitable collection of vectors we develop a new Weyl-inspired eigengap heuristic and heuristics based on the sparse basis vectors. We develop an automated eigenvector separation procedure and illustrate its efficacy on examples from time-dependent dynamics on manifolds. In this context, transfer operator approaches are extensively used to find dynamically disconnected regions of phase space, known as almost-invariant sets or coherent sets. The dominant eigenvectors of transfer operators or related operators, such as the dynamic Laplacian, encode dynamic connectivity information. Our sparse eigenbasis approximation (SEBA) methodology streamlines the final stage of transfer operator methods, namely the extraction of almost-invariant or coherent sets from the eigenvectors. It is particularly useful when used on domains with large numbers of coherent sets, and when the coherent sets do not exhaust the phase space, such as in large geophysical datasets.

An extended transfer operator approach to identify separatrices in open flows.

Vortices of coherent fluid volume are considered to have a substantial impact on transport processes in turbulent media. Yet, due to their Lagrangian nature, detecting these structures is highly nontrivial. In this respect, transfer operator approaches have been proven to provide useful tools: Approximating a possibly time-dependent flow as a discrete Markov process in space and time, information about coherent structures is contained in the operator's eigenvectors, which is usually extracted by employing clustering methods. Here, we propose an extended approach that couples surrounding filaments using "mixing boundary conditions" and focuses on the separation of the inner coherent set and embedding outer flow. The approach refrains from using unsupervised machine learning techniques such as clustering and uses physical arguments by maximizing a coherence ratio instead. We show that this technique improves the reconstruction of separatrices in stationary open flows and succeeds in finding almost-invariant sets in periodically perturbed flows.

Optimal transport over nonlinear systems via infinitesimal generators on graphs

We present a set-oriented graph-based computational framework for continuous-time optimal transport over nonlinear dynamical systems. We recover provably optimal control laws for steering a given initial distribution in phase space to a final distribution in prescribed finite time for the case of non-autonomous nonlinear control-affine systems, while minimizing a quadratic control cost. The resulting control law can be used to obtain approximate feedback laws for individual agents in a swarm control application. Using infinitesimal generators, the optimal control problem is reduced to a modified Monge-Kantorovich optimal transport problem, resulting in a convex Benamou-Brenier type fluid dynamics formulation on a graph. The well-posedness of this problem is shown to be a consequence of the graph being strongly-connected, which in turn is shown to result from controllability of the underlying dynamical system. Using our computational framework, we study optimal transport of distributions where the underlying dynamical systems are chaotic, and non-holonomic. The solutions to the optimal transport problem elucidate the role played by invariant manifolds, lobe-dynamics and almost-invariant sets in efficient transport of distributions in finite time. Our work connects set-oriented operator-theoretic methods in dynamical systems with optimal mass transportation theory, and opens up new directions in design of efficient feedback control strategies for nonlinear multi-agent and swarm systems operating in nonlinear ambient flow fields.

Optimal perturbations for nonlinear systems using graph-based optimal transport

We formulate and solve a class of finite-time transport and mixing problems in the set-oriented framework. The aim is to obtain optimal discrete-time perturbations in nonlinear dynamical systems to transport a specified initial measure on the phase space to a final measure in finite time. The measure is propagated under system dynamics in between the perturbations via the associated transfer operator. Each perturbation is described by a deterministic map in the measure space that implements a version of Monge–Kantorovich optimal transport with quadratic cost. Hence, the optimal solution minimizes a sum of quadratic costs on phase space transport due to the perturbations applied at specified times. The action of the transport map is approximated by a continuous pseudo-time flow on a graph, resulting in a tractable convex optimization problem. This problem is solved via state-of-the-art solvers to global optimality. We apply this algorithm to a problem of transport between measures supported on two disjoint almost-invariant sets in a chaotic fluid system, and to a finite-time optimal mixing problem by choosing the final measure to be uniform. In both cases, the optimal perturbations are found to exploit the phase space structures, such as lobe dynamics, leading to efficient global transport. As the time-horizon of the problem is increased, the optimal perturbations become increasingly localized. Hence, by combining the transfer operator approach with ideas from the theory of optimal mass transportation, we obtain a discrete-time graph-based algorithm for optimal transport and mixing in nonlinear systems.

An early warning indicator for atmospheric blocking events using transfer operators.

The existence of persistent midlatitude atmospheric flow regimes with time-scales larger than 5-10 days and indications of preferred transitions between them motivates to develop early warning indicators for such regime transitions. In this paper, we use a hemispheric barotropic model together with estimates of transfer operators on a reduced phase space to develop an early warning indicator of the zonal to blocked flow transition in this model. It is shown that the spectrum of the transfer operators can be used to study the slow dynamics of the flow as well as the non-Markovian character of the reduction. The slowest motions are thereby found to have time scales of three to six weeks and to be associated with meta-stable regimes (and their transitions) which can be detected as almost-invariant sets of the transfer operator. From the energy budget of the model, we are able to explain the meta-stability of the regimes and the existence of preferred transition paths. Even though the model is highly simplified, the skill of the early warning indicator is promising, suggesting that the transfer operator approach can be used in parallel to an operational deterministic model for stochastic prediction or to assess forecast uncertainty.

A closing scheme for finding almost-invariant sets in open dynamical systems

We explore the concept of metastability or almost-invariance in open dynamical systems. In such systems, the loss of mass through a ``hole'' occurs in the presence of metastability. We extend existing techniques for finding almost-invariant sets in closed systems to open systems by introducing a closing operation that has a small impact on the system's metastability.

An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems

The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. Barriers to transport, which mitigate mixing, are currently the subject of intense study. In the autonomous setting, the use of transfer operators (Perron–Frobenius operators) to identify invariant and almost-invariant sets has been particularly successful. In the nonautonomous (time-dependent) setting, coherent sets, a time-parameterised family of minimally dispersive sets, are a natural extension of almost-invariant sets. The present work introduces a new analytic transfer operator construction that enables the calculation of finite-time coherent sets (sets are that minimally dispersive over a finite time interval). This new construction also elucidates the role of diffusion in the calculation and we show how properties such as the spectral gap and the regularity of singular vectors scale with noise amplitude. The construction can also be applied to general Markov processes on continuous state space.

Topological chaos, braiding and bifurcation of almost-cyclic sets

In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., “Topological chaos and periodic braiding of almost-cyclic sets,” Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or “ghost rods” around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes’ flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.

A set oriented definition of finite-time Lyapunov exponents and coherent sets

The problem of phase space transport, which is of interest from both the theoretical and practical point of view, has been investigated extensively using geometric and probabilistic methods. Two important tools to study this problem that have emerged in recent years are finite-time Lyapunov exponents (FTLE) and the Perron–Frobenius operator. The FTLE measures the averaged local stretching around reference trajectories. Regions with high stretching are used to identify phase space transport barriers. One probabilistic method is to consider the spectrum of the Perron–Frobenius operator of the flow to identify almost-invariant densities. These almost-invariant densities are used to identify almost invariant sets. In this paper, a set-oriented definition of the FTLE is proposed which is applicable to phase space sets of finite size and reduces to the usual definition of FTLE in the limit of infinitesimal phase space elements. This definition offers a straightforward connection between the evolution of probability densities and finite-time stretching experienced by phase space curves. This definition also addresses some concerns with the standard computation of the FTLE. For the case of autonomous and periodic vector fields we provide a simplified method to calculate the set-oriented FTLE using the Perron–Frobenius operator. Based on the new definition of the FTLE we propose a simple definition of finite-time coherent sets applicable to vector fields of general time-dependence, which are the analogues of almost-invariant sets in autonomous and time-periodic vector fields. The coherent sets we identify will necessarily be separated from one another by ridges of high FTLE, providing a link between the framework of coherent sets and that of codimension one Lagrangian coherent structures. Our identification of coherent sets is applied to three examples.

Finite-time entropy: A probabilistic approach for measuring nonlinear stretching

Transport and mixing processes in dynamical systems are often difficult to study analytically and therefore a variety of numerical methods have been developed. Finite-time Lyapunov exponents (FTLEs) or related stretching indicators are frequently used as a means to estimate transport barriers. Alternatively, eigenvectors, singular vectors, or Oseledets vectors of numerical transfer operators find almost-invariant sets, finite-time coherent sets, or time-asymptotic coherent sets, respectively, which are minimally dispersed under the dynamics. While these families of approaches (geometric FTLEs and the probabilistic transfer operator) often give compatible results, a formal link is still missing; here we present a small step towards providing a mathematical link.We propose a new entropy-based methodology for estimating finite-time expansive behaviour along trajectories in autonomous and nonautonomous dynamical systems. We introduce the finite-time entropy (FTE) field as a simple and flexible way to capture nonlinear stretching directly from the entropy growth experienced by a small localised density evolved by the transfer operator. The FTE construction elucidates in a straightforward way the connection between the evolution of probability densities and the local stretching experienced.We develop an extremely simple and numerically efficient method of constructing an estimate of the FTE field. The FTE field is instantaneously calculable from a numerical transfer operator — a transition matrix of conditional probabilities that describes a discretised version of the dynamical system; once one has such a transition matrix, the FTE field may be computed “for free”. We also show (i) how to avoid long time integrations in autonomous and time-periodic systems, (ii) how to perform backward time computations by a fast matrix manipulation rather than backward time integration, and (iii) how to easily employ adaptive methods to focus on high-value FTE regions.

Instability of the isolated spectrum for W-shaped maps

AbstractIn this note we consider the W-shaped map $W_0=W_{s_1,s_2}$ with ${1}/{s_1}+{1}/{s_2}=1$ and show that the eigenvalue $1$ is not stable. We do this in a constructive way. For each perturbing map $W_a$ we show the existence of a ‘second’ eigenvalue $\lambda _a$, such that $\lambda _a\to 1$ as $a\to 0$, which proves instability of the isolated spectrum of $W_0$. At the same time, the existence of second eigenvalues close to 1 causes the maps $W_a$to behave in a metastable way. There are two almost-invariant sets, and the system spends long periods of consecutive iterations in each of them, with infrequent jumps from one to the other.

MEASURABLE DYNAMICS ANALYSIS OF TRANSPORT IN THE GULF OF MEXICO DURING THE OIL SPILL

On April 20, 2010, an oil well cap explosion below the Deepwater Horizon, an off-shore oil rig in the Gulf of Mexico, started the worst human-caused submarine oil spill ever. Though an historic tragedy for the marine ecosystem, the unprecedented monitoring of the spill in real time by satellites and increased modeling of the natural oceanic flows has provided a wealth of data, allowing analysis of the flow dynamics governing the spread of the oil. In this work, we present the results of two computational analyses describing the mixing, mass transport, and flow dynamics of the oil dispersion in the Gulf of Mexico over the first 100 days of the spill. Transfer operator methods are used to determine the spatial partitioning of regions of homogeneous dynamics into almost-invariant sets, and Finite Time Lyapunov Exponents are used to compute pseudo-barriers to the mixing of the oil between these regions. The two methods give complementary results, generating a comprehensive description of the oil flow dynamics over time.

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron–Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds and demonstrate L1 convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.

Estimating Long-Term Behavior of Flows without Trajectory Integration: The Infinitesimal Generator Approach

The long-term distributions of trajectories of a flow are described by invariant densities, i.e. fixed points of an associated transfer operator. In addition, global slowly mixing structures, such as almost-invariant sets, which partition phase space into regions that are almost dynamically disconnected, can also be identified by certain eigenfunctions of this operator. Indeed, these structures are often hard to obtain by brute-force trajectory-based analyses. In a wide variety of applications, transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system. The computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories. In this paper, we propose to directly work with the infinitesimal generator instead of the operator, completely avoiding trajectory integration. We propose two different discretization schemes; a cell based discretization and a spectral collocation approach. Convergence can be shown in certain circumstances. We demonstrate numerically that our approach is much more efficient than the operator approach, sometimes by several orders of magnitude.

Coherent sets for nonautonomous dynamical systems

We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets, metastable sets, persistent patterns, or strange eigenmodes, and have proved to be important in a variety of applications. In this current work, we explain how to extend existing autonomous approaches to the nonautonomous setting. We call the new time-dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent. The new methods are illustrated via detailed examples in both discrete and continuous time.

Coherent structures and isolated spectrum for Perron–Frobenius cocycles

AbstractWe present an analysis of one-dimensional models of dynamical systems that possess ‘coherent structures’: global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron–Frobenius cocycles. We prove that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron–Frobenius cocycle has at most finitely many isolated points. Moreover, we develop a strengthened version of the Multiplicative Ergodic Theorem for non-invertible matrices and construct an invariant splitting into Oseledets subspaces. We detail examples of cocycles of expanding maps with isolated Lyapunov spectrum and calculate the Oseledets subspaces, which lead to an identification of the underlying coherent structures. Our constructions generalize the notions of almost-invariant and almost-cyclic sets to non-autonomous dynamical systems and provide a new ensemble-based formalism for coherent structures in one-dimensional non-autonomous dynamics.

Almost-invariant sets and invariant manifolds — Connecting probabilistic and geometric descriptions of coherent structures in flows

We study the transport and mixing properties of flows in a variety of settings, connecting the classical geometrical approach via invariant manifolds with a probabilistic approach via transfer operators. For non-divergent fluid-like flows, we demonstrate that eigenvectors of numerical transfer operators efficiently decompose the domain into invariant regions. For dissipative chaotic flows such a decomposition into invariant regions does not exist; instead, the transfer operator approach detects almost-invariant sets. We demonstrate numerically that the boundaries of these almost-invariant regions are predominantly comprised of segments of co-dimension 1 invariant manifolds. For a mixing periodically driven fluid-like flow we show that while sets bounded by stable and unstable manifolds are almost-invariant, the transfer operator approach can identify almost-invariant sets with smaller mass leakage. Thus the transport mechanism of lobe dynamics need not correspond to minimal transport. The transfer operator approach is purely probabilistic; it directly determines those regions that minimally mix with their surroundings. The almost-invariant regions are identified via eigenvectors of a transfer operator and are ranked by the corresponding eigenvalues in the order of the sets’ invariance or “leakiness”. While we demonstrate that the almost-invariant sets are often bounded by segments of invariant manifolds, without such a ranking it is not at all clear which intersections of invariant manifolds form the major barriers to mixing. Furthermore, in some cases invariant manifolds do not bound sets of minimal leakage. Our transfer operator constructions are very simple and fast to implement; they require a sample of short trajectories, followed by eigenvector calculations of a sparse matrix.