The parameterization method for one- dimensional invariant manifolds of  higher dimensional parabolic fixed points

Baldomá, Inmaculada; Fontich, Ernest; Llave, Rafael de la; Martín, Pau

doi:10.3934/dcds.2007.17.835

Cited by 30 publications

(54 citation statements)

References 17 publications

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“…We remark that the first component of the three-dimensional scheme iteration defined by g as in ( 14), agrees with the difference equation ( 1) when k = 2, proposed in [37], while the family defined by g as in ( 15) is related to the difference equation (2) proposed in [8].…”

Section: Introduction and Main Resultssupporting

confidence: 81%

“…where the coefficients are fixed in such a way that x n is proved to be a solution of equation (2). In our work, we answer to the problem of determining the complete asymptotic expansion of x n , for one dimensional difference equations having a parabolic equilibrium point.…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

“…However, it is worth mentioning the existence and uniqueness results in some cases. See [1,2,3,15,17,29], for instance. To develop algorithms for the computation of local approximations of invariant manifolds of parabolic fixed points is also of interest.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic Dynamics of a Difference Equation with a Parabolic Equilibrium

Coll

Gasull

Prohens

2020

Qual. Theory Dyn. Syst.

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The aim of this work is the study of the asymptotic dynamical behaviour, of solutions that approach parabolic fixed points in difference equations. In one dimensional difference equations, we present the asymptotic development for positive solutions tending to the fixed point. For higher dimensions, through the study of two families of difference equations in the two and three dimensional case, we take a look at the asymptotic dynamic behaviour. To show the existence of solutions we rely on the parametrization method.

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Section: Introduction and Main Resultssupporting

confidence: 81%

Section: Introduction and Main Resultsmentioning

confidence: 97%

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Asymptotic Dynamics of a Difference Equation with a Parabolic Equilibrium

Coll

Gasull

Prohens

2020

Qual. Theory Dyn. Syst.

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“…One-dimensional manifolds of fixed points with linear part equal to the identity are studied in [2] using the parameterization method. Higher-dimensional manifolds in the same setting are considered in [1] using a generalized version of the method of McGehee, and in [4,5] using the parameterization method, where applications to Celestial Mechanics are given.…”

Section: Introductionmentioning

confidence: 99%

Differentiable invariant manifolds of nilpotent parabolic points

Cufí-Cabré¹,

Fontich²

2021

DCDS

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We consider a map F of class C r with a fixed point of parabolic type whose differential is not diagonalizable, and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Concretely, we show that under suitable conditions on the coefficients of F , there exist invariant curves of class C r away from the fixed point, and that they are analytic when F is analytic. The differentiability result is obtained as an application of the fiber contraction theorem. We also provide an algorithm to compute an approximation of a parameterization of the invariant curves and a normal form of the restricted dynamics of F on them.

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“…The parameterization method has wide application outside the scope of the present work; see [10,11,12,6,26,27,21,31,33] for theoretical developments, as well as [13,14,25,7,37] for additional numerical applications.…”

mentioning

confidence: 99%

Rigorous A Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps

James

Mischaikow²

2013

SIAM J. Appl. Dyn. Syst.

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This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a posteriori theorems for these polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover, we represent the truncation error as an analytic function, so that the derivatives of the truncation error can be bounded using classical estimates of complex analysis. As an application of these ideas we combine the approximate manifolds and rigorous bounds with a standard Newton-Kantorovich argument in order to obtain a kind of "analytic-shadowing" result for connecting orbits between fixed points of discrete time dynamical systems. A feature of this method is that we obtain the transversality of the connecting orbit automatically. Examples of the manifold computation are given for invariant manifolds which have dimension between two and ten. Examples of the a posteriori error bounds and the analytic-shadowing argument for connecting orbits are given for dynamical systems in dimension three and six.

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The parameterization method for one- dimensional invariant manifolds of higher dimensional parabolic fixed points

Cited by 30 publications

References 17 publications

Asymptotic Dynamics of a Difference Equation with a Parabolic Equilibrium

Asymptotic Dynamics of a Difference Equation with a Parabolic Equilibrium

Differentiable invariant manifolds of nilpotent parabolic points

Rigorous A Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps

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