Computing Invariant Manifolds by Integrating Fat Trajectories

Henderson, Michael E.

doi:10.1137/040602894

Cited by 45 publications

(37 citation statements)

References 26 publications

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“…A new method by Henderson [16] is based on integrating an individual trajectory together with a second-order approximation to the manifold along that trajectory. The surface is constructed as a collection of k-dimensional strips centered at such trajectories; the use of these (nonintersecting) strips provides uniform bounds on the spacing of the trajectories.…”

Section: Prior Methodsmentioning

confidence: 99%

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A Fast Method for Approximating Invariant Manifolds

Guckenheimer¹,

Vladimirsky²

2004

SIAM J. Appl. Dyn. Syst.

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Abstract. The task of constructing higher-dimensional invariant manifolds for dynamical systems can be computationally expensive. We demonstrate that this problem can be locally reduced to solving a system of quasi-linear PDEs, which can be efficiently solved in an Eulerian framework. We construct a fast numerical method for solving the resulting system of discretized nonlinear equations. The efficiency stems from decoupling the system and ordering the computations to take advantage of the direction of information flow. We illustrate our approach by constructing two-dimensional invariant manifolds of hyperbolic equilibria in R 3 and R 4 .

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Section: Prior Methodsmentioning

confidence: 99%

“…As a result, it became a de facto standard for testing methods for the invariant manifoldapproximation (e.g., compare with [22,15,7,16]). …”

Section: Stopping Criteriamentioning

confidence: 99%

A Fast Method for Approximating Invariant Manifolds

Guckenheimer¹,

Vladimirsky²

2004

SIAM J. Appl. Dyn. Syst.

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“…However, for discrete problems, the invariance equation is a functional equation and it would be interesting to develop an analog approach for difference equations. Finally, also [16] deals with differential equations by considering whole bundles of trajectories and describing an algorithm to control them in order to approximate invariant manifolds. Various illustrative examples on this extensive area can be found in the well-written and interesting survey paper [24], to which we also refer for a more complete overview of the corresponding vast literature.…”

Section: Introductionmentioning

confidence: 99%

Computation of nonautonomous invariant and inertial manifolds

Pötzsche

Rasmussen

2009

Numer. Math.

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We derive a numerical scheme to compute invariant manifolds for timevariant discrete dynamical systems, i.e., nonautonomous difference equations. Our universally applicable method is based on a truncated Lyapunov-Perron operator and computes invariant manifolds using a system of nonlinear algebraic equations which can be solved both locally using (nonsmooth) inexact Newton, and globally using continuation algorithms. Compared to other algorithms, our approach is quite flexible, since it captures time-dependent, nonsmooth, noninvertible or implicit equations and enables us to tackle the full hierarchy of strongly stable, stable and center-stable manifolds, as well as their unstable counterparts.Our results are illustrated using a test example and are applied to a population dynamical model and the Hénon map. Finally, we discuss a linearly implicit Euler-Bubnov-Galerkin discretization of a reaction diffusion equation in order to approximate its inertial manifold.

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“…This phenomenon of "geometric stiffness" is described in section 2. We note that a similar challenge already arises even for ODEs, and a number of algorithms have been developed to get around this difficulty [1,2,3,4,5]. In section 3 we show that a small-delay DDE can be approximated by the corresponding ODE system, thus making these prior methods directly applicable.…”

Section: Introductionmentioning

confidence: 99%

“…Non-transverse intersections of stable and unstable manifolds give rise to homoclinic and heteroclinic bifurcations. Several numerical methods for approximating higher-dimensional invariant manifolds of ODEs have been developed over the years [1,2,3,4,5]; a recent overview and comparison of these can be found in [6]. For delay differential equations (DDEs), an algorithm for computing one-dimensional invariant manifolds (in the Poincaré map) of periodic orbits has been introduced by Krauskopf and Green in [7,8].…”

Section: Introductionmentioning

confidence: 99%

Numerical Methods for Approximating Invariant Manifolds of Delayed Systems

Sahai¹,

Vladimirsky²

2009

SIAM J. Appl. Dyn. Syst.

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In this paper we develop methods for computing k−dimensional invariant manifolds of delayed systems for k ≥ 2. For small delays, we consider methods for approximating delay differential equations (DDEs) with ordinary differential equations (ODEs). Once these approximations are made, any existing method for computing invariant manifolds of ODEs can then be used directly. We derive bounds on errors incurred by the most natural of these approximations. For large delays, we extend to DDEs the method originally introduced by Krauskopf and Osinga [1] for invariant manifolds of ODEs. We test the convergence of the resulting algorithm numerically and further illustrate our approach by computing two-dimensional unstable manifolds of equilibria in the context of phase-conjugate feedback lasers.

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Computing Invariant Manifolds by Integrating Fat Trajectories

Cited by 45 publications

References 26 publications

A Fast Method for Approximating Invariant Manifolds

A Fast Method for Approximating Invariant Manifolds

Computation of nonautonomous invariant and inertial manifolds

Numerical Methods for Approximating Invariant Manifolds of Delayed Systems

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