The parameterization method for invariant manifolds III: overview and applications

Cabré, Xavier; Fontich, Ernest; Llave, Rafael de la

doi:10.1016/j.jde.2004.12.003

Cited by 234 publications

(327 citation statements)

References 41 publications

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“…The idea of the parameterization method was developed in [CFdlL03a,CFdlL03b,CFdlL05], for invariant manifolds associated to non-resonant spectral components of the linearization at the fixed point. With this method one finds simultaneously a parameterization of the invariant manifold and the reduction of the dynamics on it.…”

Section: Baldomá and A Haromentioning

confidence: 99%

One dimensional invariant manifolds of Gevrey type in real-analytic maps

Baldomá¹,

Haro²

2008

DCDS-B

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Abstract. In this paper we study the basic questions of existence, uniqueness, differentiability, analyticity and computability of the one dimensional center manifold of a parabolic-hyperbolic fixed point of a real-analytic map. We use the parameterization method, reducing the dynamics on the center manifold to a polynomial. We prove that the asymptotic expansions of the center manifold are of Gevrey type. Moreover, under suitable hypothesis, we also prove that the asymptotic expansions correspond to a real-analytic parameterization of an invariant curve that goes to the fixed point. The parameterization is Gevrey at the fixed point, hence C ∞ .1. Introduction. Center manifold theory is very important in the analysis of degenerate fixed points and in bifurcation theory. The questions of existence, uniqueness, smoothness of the center manifold, and its applications, have been studied by many authors, among them [Pli64, Kel67, HPS77, Car81, Sij85]. These works show up the puzzling properties of center manifolds. In the study of degenerate fixed points, it is important to know the dynamical properties of the center manifol, what is known as reduction of the dynamics to the center manifold. In numerical applications, one can approximate the center manifold through power series expansions whose coefficients are recursively computed (see, for instance [Sim90, Har99, Jor99]). In order to bound the error of finite order approximations, it is important to know the rate of growth of the coefficient of the asymptotic expansions [Sim00].This paper consider the previous topics for real-analytic (local) diffeomorphisms with a fixed point whose linearization has 1 as a simple eigenvalue. Thus, we look for a one dimensional invariant manifold going to the fixed point, where is tangent to an eigenvector of 1. A particular case is when the fixed point is parabolic-hyperbolic, that is, the rest of eigenvalues are away of the unit circle, and the invariant manifold is a branch of the center manifold. In this setting, the goals of this paper are:(a) To give a parameterization of the invariant manifold for which the reduced dynamics is "simple".In particular, we show that it can be reduced to a polynomial. (b) To describe the asymptotic properties of the expansions of the invariant manifold. We prove that the expansions are of the Gevrey type, i.e., the coefficients of the expansions grow as a power of a factorial. (c) To prove, under suitable hypotheses, that the expansions correspond to a real-analytic invariant manifold that goes to the fixed point (a branch). This is proved under the hypothesis that the dynamics tangent to the manifold is attracting and the dynamics transversal to the manifold is not linearly attracting. We also prove that the manifold is of Gevrey type at the fixed point, hence C ∞ . (d) To give sufficient conditions of uniqueness of the invariant manifold. This is proved under the assumption that the dynamics on the manifold is attracting and the transversal dynamics is repelling.

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Section: Baldomá and A Haromentioning

confidence: 99%

One dimensional invariant manifolds of Gevrey type in real-analytic maps

Baldomá¹,

Haro²

2008

DCDS-B

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“…For this we follow the previous work of authors one and two with J.B. van den Berg and K. Mischaikow in [3,26] and exploit the so-called parameterization method for invariant manifolds [5,6,7]. This method facilitates the computation of polynomial approximations of the chart maps to any desired finite order and provides rigorous error bounds on the truncation errors.…”

Section: Remarks 1 (A)mentioning

confidence: 99%

“…We also discuss in detail how we obtain a real manifold when there are complex conjugate eigenvalues. For the full development in all generality see [5,6,7]. Recall that the general goal is to develop a rigorous computational method for the existence of a connecting orbit from p 1 to p 2 .…”

Section: Parameterization Of Invariant Manifoldsmentioning

confidence: 99%

“…After introducing the spaces we work on in Section 2.1, we summarize some analytic results used in the a-posteriori analysis in Section 2.2. One main ingredient for this analysis is the parametrization method for invariant manifolds [5,6,7] which we treat in Section 2.3. We finish this preparatory section by recalling some basic facts from the theory of ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

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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 case when L + N (., μ) is C k in a Banach space X has been treated in [38,39]. In a more general context, more recent results on local invariant manifolds for C k maps in Banach spaces can be found in references [40,41].…”

Section: Finite-dimensional Casementioning

confidence: 99%

Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

James

Sire

The Fermi-Pasta-Ulam Problem

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Abstract. Center manifold theory has been used in recent works to analyze small amplitude waves of different types in nonlinear (Hamiltonian) oscillator chains. This led to several existence results concerning traveling waves described by scalar advance-delay differential equations, pulsating traveling waves determined by systems of advance-delay differential equations, and time-periodic oscillations (including breathers) obtained as orbits of iterated maps in spaces of periodic functions. The Hamiltonian structure of the governing equations is not taken into account in the analysis, which heavily relies on the reversibility of the system. The present work aims at giving a pedagogical review on these topics. On the one hand, we give an overview of existing center manifold theorems for reversible infinite-dimensional differential equations and maps. We illustrate the theory on two different problems, namely the existence of breathers in Fermi-Pasta-Ulam lattices and the existence of traveling breathers (superposed on a small oscillatory tail) in semi-discrete KleinGordon equations. IntroductionThe seminal work of Fermi, Pasta and Ulam in 1955 [12] had a broad impact on the development of the theory of nonlinear lattices. The Fermi-PastaUlam (FPU) model consists of a chain of masses nonlinearly coupled to their nearest neighbors. For a general nearest-neighbors interaction potential V , the governing equations readwhere we note x n the displacement of the nth mass with respect to an equilibrium position (in this version of the model the chain is of infinite extent). The anharmonic interaction potential V satisfies V (0) = 0, V (0) = 1. System (6.1) is Hamiltonian, with total energy

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The parameterization method for invariant manifolds III: overview and applications

Cited by 234 publications

References 41 publications

One dimensional invariant manifolds of Gevrey type in real-analytic maps

One dimensional invariant manifolds of Gevrey type in real-analytic maps

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

Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

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