Homoclinic Shadowing

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

doi:10.1007/s10884-005-3146-x

Cited by 20 publications

(21 citation statements)

References 26 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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“…When we solve the boundary value problem for the connecting orbit we contend only with the global dynamics. Compare this to related methods such as [38,33,22,11,10] where both the global and local dynamics are studied simultaneously using a single operator equation. Note also that if we formulate our short and long connecting operators using the linear approximation of the invariant manifolds by their eigenspaces (with rigorous error bounds), then our method also reduces to a single operator equation both for the local and global dynamics.…”

Section: Remarks 1 (A)mentioning

confidence: 99%

“…Since then many powerful and general techniques have emerged for rigorous computer assisted study of connecting orbits. For example [10,11,22,33,38] develop methods for validating the existence of connecting orbits of both continuous and discrete time dynamical systems by solving boundary value problems on non-compact intervals (the real line or the integers depending on wether the system is continuous or discrete). These methods exploit shadowing results based on the theory of exponential dichotomies to analyze the behavior of the orbits as time goes to plus or minus infinity (that is to control the connecting orbit in the neighborhood of an equilibria).…”

Section: Remarks 1 (A)mentioning

confidence: 99%

Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields

2014

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In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the NewtonKantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained.

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“…The problem appears to be equivalent to that of the existence of a homoclinic orbit in a four-dimensional diffeomorphism. In [13] shadowing techniques are used to prove the existence transversal homoclinic and heteroclinic orbits in higher dimensional maps. This shadowing technique includes a suitable numerical approximation.…”

Section: Existing Literature and Context Of The Current Workmentioning

confidence: 99%

Computation of Heteroclinic Arcs with Application to the Volume Preserving Hénon Family

James

Lomelí

2010

SIAM J. Appl. Dyn. Syst.

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Abstract. Let f : R 3 → R 3 be a diffeomorphism with p 0 , p 1 ∈ R 3 distinct hyperbolic fixed points. Assume that W u (p 0 ) and W s (p 1 ) are two dimensional manifolds which intersect transversally at a point q. Then the intersection is locally a one-dimensional smooth arcγ through q, and points onγ are orbits heteroclinic from p 0 to p 1 .We describe and implement a numerical scheme for computing the jets ofγ to arbitrary order. We begin by computing high order polynomial approximations of some functions P u , P s : R 2 → R 3 , and domain disksThen the intersection arcγ solves a functional equation involving P s and Pu. We develop an iterative numerical scheme for solving the functional equation, resulting in a high order Taylor expansion of the arcγ. We present numerical example computations for the volume preserving Hénon family, and compute some global invariant branched manifolds.

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Homoclinic Shadowing

Cited by 20 publications

References 26 publications

Homoclinic trajectories of non-autonomous maps

Homoclinic trajectories of non-autonomous maps

Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields

Computation of Heteroclinic Arcs with Application to the Volume Preserving Hénon Family

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