Transversal connecting orbits from shadowing

Coomes, Brian A.; Koçak, Hüseyin; Palmer, Kenneth J.

doi:10.1007/s00211-007-0065-2

Cited by 13 publications

(13 citation statements)

References 19 publications

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“…The opposite question, whether pseudo orbits lead to exact orbits -so called shadowing results -are discussed in detail in Pilyugin (1999). Shadowing techniques for homoclinic and heteroclinic orbits, converging towards periodic orbits, in discrete and continuous time, are developed in Coomes et al (2005) and Coomes et al (2007).…”

Section: Introductionmentioning

confidence: 99%

Homoclinic trajectories of non-autonomous maps

Hüls

2011

Journal of Difference Equations and Applications

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For non-autonomous difference equations of the form x n+1 = f (x n , λ n ), n ∈ we consider homoclinic trajectories. These are pairs of trajectories that converge in both time directions towards each other. Assuming hyperbolicity, we derive a numerical method to compute homoclinic trajectories in two steps. In the first step one trajectory is approximated by the solution of a boundary value problem and precise error estimates are given. In particular, influences of parameters λ n with |n| large are discussed in detail. A second trajectory that is homoclinic to the first one is computed in a subsequent step as follows. We transform the original system into a topologically equivalent form having zero as an n-independent fixed point. Applying the boundary value ansatz to the transformed system, we obtain a non-autonomous homoclinic orbit, converging towards the origin, cf. Hüls (2006). Transforming back to the original coordinates leads to the desired homoclinic trajectories. The numerical method and the validity of the error estimates are illustrated by examples.

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Section: Introductionmentioning

confidence: 99%

Homoclinic trajectories of non-autonomous maps

Hüls

2011

Journal of Difference Equations and Applications

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“…Let us mention that this is far from being the first time that rigorous numerics are used to study connecting orbits, and that there is a rapidly growing literature on the subject, see e.g. [2,15,22,54,61,67,77].…”

Section: Rigorous A-posteriori Validation Methodsmentioning

confidence: 99%

Rigorous validation of stochastic transition paths

Breden

Kuehn

2019

Journal de Mathématiques Pures et Appliquées

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Global dynamics in nonlinear stochastic systems is often difficult to analyze rigorously. Yet, many excellent numerical methods exist to approximate these systems. In this work, we propose a method to bridge the gap between computation and analysis by introducing rigorous validated computations for stochastic systems. The first step is to use analytic methods to reduce the stochastic problem to one solvable by a deterministic algorithm and to numerically compute a solution. Then one uses fixed-point arguments, including a combination of analytical and validated numerical estimates, to prove that the computed solution has a true solution in a suitable neighbourhood. We demonstrate our approach by computing minimum-energy transition paths via invariant manifolds and heteroclinic connections. We illustrate our method in the context of the classical Müller-Brown test potential.

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“…Suppose there are d s complex conjugate pairs of eigenvalues and n s − 2d s real ones. Furthermore, assume that we have ordered the eigenvalues λ s in such a way that 9) and note that Σ is an involution on C ns . For this reason we shall frequently write z := Σ(z).…”

Section: )mentioning

confidence: 99%

Validated computations for connecting orbits in polynomial vector fields

Berg

Sheombarsing

2020

Indagationes Mathematicae

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In this paper we present a computer-assisted procedure for proving the existence of transverse heteroclinic orbits connecting hyperbolic equilibria of polynomial vector fields. The idea is to compute high-order Taylor approximations of local charts on the (un)stable manifolds by using the Parameterization Method and to use Chebyshev series to parameterize the orbit in between, which solves a boundary value problem. The existence of a heteroclinic orbit can then be established by setting up an appropriate fixed-point problem amenable to computer-assisted analysis. The fixed point problem simultaneously solves for the local (un)stable manifolds and the orbit which connects these. We obtain explicit rigorous control on the distance between the numerical approximation and the heteroclinic orbit. Transversality of the stable and unstable manifolds is also proven.

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Transversal connecting orbits from shadowing

Cited by 13 publications

References 19 publications

Homoclinic trajectories of non-autonomous maps

Homoclinic trajectories of non-autonomous maps

Rigorous validation of stochastic transition paths

Validated computations for connecting orbits in polynomial vector fields

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