Coherent sets for nonautonomous dynamical systems

Froyland, Gary; Lloyd, S. P.; Santitissadeekorn, Naratip

doi:10.1016/j.physd.2010.03.009

Cited by 110 publications

(141 citation statements)

References 45 publications

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“…However, the derivation will be presented in detail in the interests of coherence, since several adjustments are needed; such as the performance of the calculation at a movable pointx u (p) rather than a fixed location, the emphasis on locating the manifold rather than distances between manifolds, the lack of necessity of a homoclinic connection, the legitimacy of discarding higher-order terms, and the potential divergence of the Melnikov expression. This last issue is also related to the possible divergence of a boundary term in going from (20) to (21) in Holmes [32]. Most geometric Melnikov developments following formal perturbative analysis [26,4,32] discard the O(ε) terms in the Melnikov integral, which actually needs additional consideration since an integration over a noncompact interval needs to be performed.…”

Section: Proof Of Theorem 21 (Unstable Manifold's Normal Displacemenmentioning

confidence: 99%

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A Tangential Displacement Theory for Locating Perturbed Saddles and Their Manifolds

Balasuriya¹

2011

SIAM J. Appl. Dyn. Syst.

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The stable and unstable manifolds associated with a saddle point in two-dimensional non-areapreserving flows under general time-aperiodic perturbations are examined. An improvement to existing geometric Melnikov theory on the normal displacement of these manifolds is presented. A new theory on the previously neglected tangential displacement is developed. Together, these enable locating the perturbed invariant manifolds to leading order. An easily usable Laplace transform expression for the location of the perturbed time-dependent saddle is also obtained. The theory is illustrated with an application to the Duffing equation. 1. Introduction. Invariant manifolds are important entities in continuous dynamical systems, forming crucial flow organizers. Their movement under perturbations can alter the global flow structure. The original results of Melnikov [40] relate to the normal displacement of stable and unstable manifolds in a homoclinic structure in two-dimensional area-preserving flow, under a time-sinusoidal perturbation. The transverse zeroes of the so-called Melnikov function identify when the perturbed invariant manifolds intersect, leading to chaos via the Smale-Birkhoff theorem [26, 4, 57]. Extensions of the Melnikov method to higher dimensions [25, 44, 60, 54], time-aperiodicity and/or finite-time [41, 44, 58, 62, 8], subharmonic bifurcations [40, 57, 61, 59], nonhyperbolicity [54, 60, 63], and non-area-preservation [32] are available.While Melnikov methods can be used to determine how invariant manifolds move normal to the original manifolds, there has been no method in the literature in which the tangential movement is characterized. This study addresses this issue, arriving at a Melnikov-like function for the tangential displacement, under general time-dependent perturbations. The original two-dimensional flow is assumed to contain a saddle structure but need not be areapreserving. The displacement is expressed as a function of the original position p on the manifold and the time-slice t. Along the way, a similar quantification for the normal displacement is obtained, in which potential divergence issues in the Melnikov function and the legitimacy of ignoring higher-order terms are explicitly addressed. The normal and tangential results together permit the locating of the perturbed stable and unstable manifolds of the

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Section: Proof Of Theorem 21 (Unstable Manifold's Normal Displacemenmentioning

confidence: 99%

“…These form time-varying flow separators and specifically are the boundaries of Lagrangian coherent structures [30,47,28,29,46,52,20,23]. Their determination is therefore crucial to understanding fluid transport in oceanic dynamics but remains difficult due to the fact that theoretical tools in the genuinely timeaperiodic setting are lacking.…”

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confidence: 99%

A Tangential Displacement Theory for Locating Perturbed Saddles and Their Manifolds

Balasuriya¹

2011

SIAM J. Appl. Dyn. Syst.

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“…64 Hence, the connection between set-oriented statistical methods and topological methods in dynamical systems made in this paper provides an additional tool for analyzing There is work on spectral analysis of the "mixing matrix" 65 that is closely related to the identification of almost invariant sets and ACSs, and connections have been made between this mixing matrix analysis and the "strange eigenmode" that arises from spectral analysis of the continuous advection-diffusion operator. 66,67 The topological information available from analyzing trajectories of ACS suggests that a similar approach can be applied to the braiding of eigenvectors in these related methods.…”

Section: Discussionmentioning

confidence: 99%

Topological chaos, braiding and bifurcation of almost-cyclic sets

Grover

Ross

Stremler

et al. 2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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. When a body of fluid moves, whether it be in the atmosphere, an ocean, or a kitchen sink, there are often regions of fluid that move together for an extended period of time. As these coherent sets of fluid trace out trajectories in space and time, they can be thought of as "stirring" the surrounding fluid. The entanglement of these trajectories as they "braid" around each other is connected to the level of chaos present in the fluid system. When the sets return periodically to their initial positions in a twodimensional, time-dependent system, the entanglement of their trajectories can be used to predict a lower bound on the rate of stretching in the surrounding domain. We examine the trajectories of such "almost-cyclic sets" in a lid-driven cavity flow and demonstrate that this combination of topological analysis with set-oriented methods can be an effective means of predicting chaos. The characterization of the entanglement and associated prediction of stretching are achieved through application of the Thurston-Nielsen Classification Theorem, which in general classifies the topological complexity of homeomorphisms of punctured surfaces. While a rigorous lower bound on topological entropy is not available in the absence of exactly periodic braiding structures, our approach finds the "topological skeleton" that can be used to get an approximate rate of stretching.

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“…is obtained from the tensor transformation rule for the Eulerian homogeneous, isotropic diffusion tensor I 3×3 which is implicitly contained in (11). In this Lagrangian frame of reference, the original advection diffusion equation (11) becomes a diffusion equation, albeit with an inhomogeneous, anisotropic and time-dependent diffusion tensor field D. The data-driven detection methods used in this paper can be interpreted as discretization approaches to this Lagrangian diffusion equation (12).…”

Section: A Lagrangian Passive Scalar Transport Perspectivementioning

confidence: 99%

Lagrangian coherent sets in turbulent Rayleigh-Bénard convection

et al. 2019

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Coherent circulation rolls and their relevance for the turbulent heat transfer in a two-dimensional Rayleigh-Bénard convection model are analyzed. The flow is in a closed cell of aspect ratio four at a Rayleigh number Ra = 10 6 and at a Prandtl number Pr = 10. Three different Lagrangian analysis techniques based on graph Laplacians -distance spectral trajectory clustering, time-averaged diffusion maps and finite-element based dynamic Laplacian discretization -are used to monitor the turbulent fields along trajectories of massless Lagrangian particles in the evolving turbulent convection flow. The three methods are compared to each other and the obtained coherent sets are related to results from an analysis in the Eulerian frame of reference. We show that the results of these methods agree with each other and that Lagrangian and Eulerian coherent sets form basically a disjoint union of the flow domain. Additionally, a windowed time-averaging of variable interval length is performed to study the degree of coherence as a function of this additional coarse graining which removes small-scale fluctuations that cause trajectories to disperse quickly. Finally, the coherent set framework is extended to study heat transport.

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Coherent sets for nonautonomous dynamical systems

Cited by 110 publications

References 45 publications

A Tangential Displacement Theory for Locating Perturbed Saddles and Their Manifolds

A Tangential Displacement Theory for Locating Perturbed Saddles and Their Manifolds

Topological chaos, braiding and bifurcation of almost-cyclic sets

Lagrangian coherent sets in turbulent Rayleigh-Bénard convection

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