An obstruction method for the destruction of invariant curves

A representative model of a return map near homoclinic bifurcation is studied. This model is the so-called fattened Arnold map, a diffeomorphism of the annulus. The dynamics is extremely rich, involving periodicity, quasiperiodicity and chaos.The method of study is a mixture of analytic perturbation theory, numerical continuation, iteration to an attractor and experiments, in which the guesses are inspired by the theory. In turn the results lead to fine-tuning of the theory. This approach is a natural paradigm for the study of complicated dynamical systems.By following generic bifurcations, both local and homoclinic, various routes to chaos and strange attractors are detected. Here, particularly, the 'large' strange attractors which wind around the annulus are of interest. Furthermore, a global phenomenon regarding Arnold tongues is important. This concerns the accumulation of tongues on lines of homoclinic bifurcation. This phenomenon sheds some new light on the occurrence of infinitely many sinks in certain cases, as predicted by the theory.

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“…It is suspected that some obstruction phenomenon as in the conservative case (see [42]) plays an important role. …”

Section: Loss Of Smoothness and Destruction Of Invariant Circlesmentioning

confidence: 99%

Towards global models near homoclinic tangencies of dissipative diffeomorphisms

1998

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“…Therefore, the estimate (25) is obtained after observing that the first Fourier coefficient is given by equation (26). Finally, the limit lim ε→0γ …”

Section: Higher Order Unfolding Of Curvesmentioning

confidence: 99%

“…Given a numerical method for the continuation of invariant curves, it is specially interesting to verify if the method is valid up to the breakdown threshold corresponding to the critical invariant curve (see [9,16,26]). These critical curves are specially important objects that organize the long-term behavior of a given dynamical system, because of their role as "last barriers" or "bottlenecks" to chaos (see [12]).…”

Section: Introductionmentioning

confidence: 99%

Numerical computation of rotation numbers of quasi-periodic planar curves

Luque

Villanueva

2009

Physica D: Nonlinear Phenomena

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Recently, a new numerical method has been proposed to compute rotation numbers of analytic circle diffeomorphisms, as well as derivatives with respect to parameters, that takes advantage of the existence of an analytic conjugation to a rigid rotation. This method can be directly applied to the study of invariant curves of planar twist maps by simply projecting the iterates of the curve onto a circle. In this work we extend the methodology to deal with general planar maps. Our approach consists in computing suitable averages of the iterates of the map that allow to obtain a new curve for which the direct projection onto a circle is well posed. Furthermore, since our construction does not use the invariance of the quasiperiodic curve under the map, it can be applied to more general contexts. We illustrate the method with several examples.

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“…where V (2) denotes the coefficient of the mean-value of the quadratic part, in the representation :…”

Section: Renormalization-group Transformationmentioning

confidence: 99%

“…The study of the break-up of invariant tori is thus an important issue to understand the onset of chaos. For two degrees of freedom, there are several numerical methods to determine the threshold of the break-up of invariant tori : for instance, Greene's criterion [1], obstruction method [2], converse KAM [3,4], frequency map analysis [5][6][7][8], or renormalization-group methods [9][10][11]. In this article, we propose to compute this threshold for a one-parameter family of Hamiltonians with three degrees of freedom and for a specific frequency vector, by two techniques : by frequency map analysis and by renormalization.…”

Section: Introductionmentioning

confidence: 99%

Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems

Chandre

Laskar

Benfatto

et al. 2001

Physica D: Nonlinear Phenomena

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We consider a class of Hamiltonians with three degrees of freedom that can be mapped into quasi-periodically driven pendulums. The purpose of this paper is to determine the threshold of the break-up of invariant tori with a specific frequency vector. We apply two techniques : the frequency map analysis and renormalizationgroup methods. The renormalization transformation acting on a Hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the Hamiltonian and a rescaling of phase space around the considered torus. We give numerical evidence that the critical coupling at which the renormalization transformation starts to diverge is the same as the value given by the frequency map analysis for the break-up of invariant tori. Furthermore, we obtain by these methods numerical values of the threshold of the break-up of the last invariant torus.

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An obstruction method for the destruction of invariant curves

Cited by 42 publications

References 6 publications

Towards global models near homoclinic tangencies of dissipative diffeomorphisms

Towards global models near homoclinic tangencies of dissipative diffeomorphisms

Numerical computation of rotation numbers of quasi-periodic planar curves

Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems

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