SRB measures for a class of partially hyperbolic attractors in Hilbert spaces

Zeng, Lian; Liu, Peidong; Lu, Kening

doi:10.1016/j.jde.2016.04.006

Cited by 19 publications

(5 citation statements)

References 38 publications

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“…Examples where Assumption 4 is satisfied include: (i) ergodic flows on compact manifolds with Lebesgue absolutely continuous, fully supported, invariant measures, in which case U = V = B µ = M = X; (ii) certain classes of dissipative flows on potentially noncompact manifolds (e.g., the Lorenz 63 (L63) system on M = R 3 [45] studied in Section 8 ahead); and (iii) certain classes of dissipative partial differential equations possessing inertial manifolds and physical measures [44,46].…”

Section: Physical Measuresmentioning

confidence: 99%

Delay-Coordinate Maps and the Spectra of Koopman Operators

2019

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The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component of the dynamics. They factor the dynamics, which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a method through which these eigenfunctions can be obtained from a kernel integral operator, which also annihilates the continuous spectrum. We show that incorporating a large number of delay coordinates in constructing the kernel of that operator results, in the limit of infinitely many delays, in the creation of a map into the discrete spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions from high-dimensional data in systems with pure point or mixed spectra. arXiv:1706.08544v7 [math.DS] 5 Nov 2018Delay-coordinate maps and the spectra of Koopman operators 2 Ergodic theory, and in particular its operator-theoretic formulation [15,23], provides a natural framework to address these objectives. In this framework, the focus is on the action of the dynamical system on spaces of observables (functions of the state), as opposed to the dynamical flow itself. The advantage of this approach, first realized in the seminal work of Koopman [39], is that the action of a general dynamical system on spaces of observables is always linear. As a result, with appropriate regularity assumptions, the problem of identification and prediction of dynamically intrinsic coherent patterns can be formulated as an estimation problem for the spectrum of a linear evolution operator. In addition, for systems exhibiting ergodic behavior, spectral quantities such as eigenvalues and eigenfunctions can be statistically estimated from time-ordered data without prior knowledge of the state space geometry or the equations of motion. At the same time, spaces of observables are also infinite dimensional, so the issue of finitedimensional approximation of (potentially unbounded) operators becomes relevant.Starting from the techniques proposed in [22,50,52], the operator-theoretic approach to ergodic theory has stimulated the development of a broad range of techniques for data-driven modeling of dynamical systems. These methods employ either the Koopman [2,14,15,30,33,34,40,41,50,52,55,57,62,67] or the Perron-Frobenius (transfer) operators [21,22,[27][28][29], which are duals to one another in appropriate function spaces. The goal common to these techniques is to approximate spectral quantities for the operator in question, such as eigenvalues, eigenfunctions, and spectral projections, from measured values of observables along orbits of the dynamics. To that end, a diverse range of approaches has been employed, including state space partitions [21,22,[27][28][29], harmonic averaging [20,50,52], iterative methods [55,57], dictionary/basis representations [30,33,34,40,62,67], delay-coordinate embeddings [2,14,30,33], and spectral-mom...

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Section: Physical Measuresmentioning

confidence: 99%

Delay-Coordinate Maps and the Spectra of Koopman Operators

2019

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“…As stated in Section 5.2, for data-driven approximation purposes, we will formally assume that the measure µ is physical. While, to our knowledge, there are no results in the literature addressing the existence of physical measures (with appropriate modifications to account for the infinite state space dimension) specifically for the KS system, recent results [63,64] on infinite-dimensional dynamical systems that include the class of dissipative systems in which the KS system belongs indicate that analogs of the assumptions made in Section 5.2 should hold.…”

Section: Overview Of the Kuramoto Sivashinsky Modelmentioning

confidence: 99%

Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

et al. 2019

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We present a data-driven framework for extracting complex spatiotemporal patterns generated by ergodic dynamical systems. Our approach, called Vector-valued Spectral Analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking values in a space of functions (scalar fields) on a spatial domain. This operator is constructed by combining aspects of the theory of operator-valued kernels for machine learning with delay-coordinate maps of dynamical systems. In contrast to conventional eigendecomposition techniques, which decompose the input data into pairs of temporal and spatial modes with a separable, tensor product structure, the patterns recovered by VSA can be manifestly non-separable, requiring only a modest number of modes to represent signals with intermittency in both space and time. Moreover, the kernel construction naturally quotients out dynamical symmetries in the data, and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling decomposition of multiscale signals into dynamically intrinsic patterns. Application of VSA to the Kuramoto-Sivashinsky model demonstrates significant performance gains in efficient and meaningful decomposition over eigendecomposition techniques utilizing scalar-valued kernels.

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“…Our proof follows in outline the one sketched in [20], and is different than [16,15,9]. We mention that [9], as well as the very recent paper [10], both prove a similar result for mappings of Hilbert spaces. An important difference between Hilbert and Banach spaces is that the latter need not have good geometry.…”

Section: Introduction and Settingmentioning

confidence: 85%

“…We fix δ ≤ 1 4 δ ′ 1 small enough for the results in Section 3 to apply. Let ǫ 0 > 0 is as in Lemma 3.8, and let S be the stack of strong stable leaves defined as in (10)…”

Section: Theorem A: Precise Formulation and Outline Of Proofmentioning

confidence: 99%

Absolute Continuity of Stable Foliations for Mappings of Banach Spaces

Blumenthal

Young

2017

Commun. Math. Phys.

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We prove the absolute continuity of stable foliations for mappings of Banach spaces satisfying conditions consistent with time-t maps of certain classes of dissipative PDEs. This property is crucial for passing information from submanifolds transversal to the stable foliation to the rest of the phase space; it is also used in proofs of ergodicity. Absolute continuity of stable foliations is well known in finite dimensional hyperbolic theory. On Banach spaces, the absence of nice geometric properties poses some additional difficulties.

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SRB measures for a class of partially hyperbolic attractors in Hilbert spaces

Cited by 19 publications

References 38 publications

Delay-Coordinate Maps and the Spectra of Koopman Operators

Delay-Coordinate Maps and the Spectra of Koopman Operators

Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

Absolute Continuity of Stable Foliations for Mappings of Banach Spaces

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