International Journal of Non-Linear Mechanics

1982

DOI: 10.1016/0020-7462(82)90002-6

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Analyse numérique du comportement d'une solution presque périodique d'une équation différentielle périodique par une méthode des sections

Elaine Thoulouze-Pratt

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à Haro and R De La Llave1

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1983

1983

2006

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Cited by 7 publications

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“…Fourier methods and their spectral properties are used in [3]. Other earlier papers using similar ideas are [78,79].…”

Section: à Haro and R De La Llavementioning

confidence: 99%

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms

Haro¹,

2006

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper we develop several numerical algorithms for the computation of invariant manifolds in quasi-periodically forced systems. The invariant manifolds we consider are invariant tori and the asymptotic invariant manifolds (whiskers) to these tori.The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34].The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties.The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.

“…Fourier methods and their spectral properties are used in [3]. Other earlier papers using similar ideas are [78,79].…”

Section: à Haro and R De La Llavementioning

confidence: 99%

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms

Haro¹,

2006

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

In this paper we develop several numerical algorithms for the computation of invariant manifolds in quasi-periodically forced systems. The invariant manifolds we consider are invariant tori and the asymptotic invariant manifolds (whiskers) to these tori.The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34].The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties.The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.

Numerical Analysis of the Behaviour of an Almost Periodic Solution to a Periodic Differential Equation, An Example of Successive Bifurcations of Invariant Tori

Thoulouze-Pratt¹

1983

Rhythms in Biology and Other Fields of Application

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No abstract

Transitions torichaos through collisions with hyperbolic orbits

¹

1985

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No abstract

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