A Survey of Methods for Computing (Un)stable Manifolds of Vector Fields

Krauskopf, Bernd; Osinga, Hinke M.; Doedel, Eusebius J.; Henderson, Michael E.; Vladimirsky, Alexander; Dellnitz, Michael; Junge, Oliver

doi:10.1142/9789812774569_0004

Cited by 63 publications

(87 citation statements)

References 34 publications

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“…In multiple-time-scale systems, the fast exponential instability of Fenichel manifolds that are not attracting makes initial value solvers incapable of tracking these manifolds by forward integration. These issues prompt the use of boundary value methods combined with continuation as an alternate strategy for computing invariant manifolds [132,133]. We have used both strategies in this paper.…”

Section: Numerical Methods For Slow-fast Systemsmentioning

confidence: 99%

Mixed-Mode Oscillations with Multiple Time Scales

Desroches¹,

Guckenheimer

Krauskopf³

et al. 2012

SIAM Rev.

497

635

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Abstract. Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale "mechanism." Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

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Section: Numerical Methods For Slow-fast Systemsmentioning

confidence: 99%

Mixed-Mode Oscillations with Multiple Time Scales

Desroches¹,

Guckenheimer

Krauskopf³

et al. 2012

SIAM Rev.

497

635

View full text Add to dashboard Cite

show abstract

“…when shadowing solutions. More sophisticated methods include the computation of invariant manifolds, forming the boundaries of basins of attraction of attractors (Krauskopf et al, 2005); here, additional analysis of the parts with gradient-like flow is necessary. The cell mapping approach (Hsu, 1987) or set oriented methods (Dellnitz and Junge, 2002) divide the phase space into cells and compute the dynamics between these cells, see also for example (Osipenko, 2007); these ideas have also been used to compute complete Lyapunov functions, see Section 2.1.…”

Section: Introductionmentioning

confidence: 99%

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Argáez

Hafstein

Giesl

2017

Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

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Abstract:Ordinary differential equations arise in a variety of applications, including e.g. climate systems, and can exhibit complicated dynamical behaviour. Complete Lyapunov functions can capture this behaviour by dividing the phase space into the chain-recurrent set, determining the long-time behaviour, and the transient part, where solutions pass through. In this paper, we present an algorithm to construct complete Lyapunov functions. It is based on mesh-free numerical approximation and uses the failure of convergence in certain areas to determine the chain-recurrent set. The algorithm is applied to three examples and is able to determine attractors and repellers, including periodic orbits and homoclinic orbits.

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“…7,8 Several powerful methods exist for computing stable and unstable manifolds of specifically known invariant sets, such as fixed pints and periodic orbits. 23 The present paper provides a general approach that identifies influential manifolds all over the phase space, without the need to know the asymptotic behavior of the trajectories in the manifold. Even if periodic orbits are known to exist, their accurate numerical detection can pose a challenge.…”

Section: Discussionmentioning

confidence: 99%

Detecting invariant manifolds as stationary Lagrangian coherent structures in autonomous dynamical systems

Teramoto

Haller

Komatsuzaki

2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Normally hyperbolic invariant manifolds (NHIMs) are well-known organizing centers of the dynamics in the phase space of a nonlinear system. Locating such manifolds in systems far from symmetric or integrable, however, has been an outstanding challenge. Here, we develop an automated detection method for codimension-one NHIMs in autonomous dynamical systems. Our method utilizes Stationary Lagrangian Coherent Structures (SLCSs), which are hypersurfaces satisfying one of the necessary conditions of a hyperbolic LCS, and are also quasi-invariant in a well-defined sense. Computing SLCSs provides a quick way to uncover NHIMs with high accuracy. As an illustration, we use SLCSs to locate two-dimensional stable and unstable manifolds of hyperbolic periodic orbits in the classic ABC flow, a three-dimensional solution of the steady Euler equations. Invariant manifolds play an important role in transport phenomena that range from chemical reactions through fluid mixing to celestial mechanics. Such manifolds, however, are often challenging to locate in multi-dimensional systems that are far from integrable and possess no special symmetries. Among these invariant manifolds, normally hyperbolic invariant manifolds are particularly important due to their persistence under small perturbations. This renders them robust with respect to modeling errors and numerical inaccuracies. Here, we develop a method to detect codimension-one, normally hyperbolic invariant manifolds in general autonomous systems. We approximate such manifolds as zero level sets of a scalar function derived from the recent variational theory of hyperbolic Lagrangian Coherent Structures (LCSs). This approximation converges exponentially fast to a true invariant manifold as longer and longer flow-map samples are used in its construction. We illustrate this method on the classic steady ABC flow, revealing some of its two-dimensional invariant manifolds at a previously unseen level of detail.

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A Survey of Methods for Computing (Un)stable Manifolds of Vector Fields

Cited by 63 publications

References 34 publications

Mixed-Mode Oscillations with Multiple Time Scales

Mixed-Mode Oscillations with Multiple Time Scales

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Detecting invariant manifolds as stationary Lagrangian coherent structures in autonomous dynamical systems

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