A rigorous partial justification of Greene's criterion

Falcolini, Corrado; Llave, Rafael de la

doi:10.1007/bf01049722

Cited by 57 publications

(55 citation statements)

References 25 publications

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“…9,16,17 Furthermore, using the facts that irrational rotations are uniquely ergodic and the directions are one-dimensional, it is shown in Ref. 18 that if the bundles are continuous the limits in Eq. ͑5͒ are reached uniformly in and the bundles are analytic if the system is analytic.…”

Section: Setup Of the Problemmentioning

confidence: 99%

Manifolds on the verge of a hyperbolicity breakdown

Haro

Llave

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study numerically the disappearance of normally hyperbolic invariant tori in quasiperiodic systems and identify a scenario for their breakdown. In this scenario, the breakdown happens because two invariant directions of the transversal dynamics come close to each other, losing their regularity. On the other hand, the Lyapunov multipliers associated with the invariant directions remain more or less constant. We identify notable quantitative regularities in this scenario, namely that the minimum angle between the two invariant directions and the Lyapunov multipliers have power law dependence with the parameters. The exponents of the power laws seem to be universal. © 2006 American Institute of Physics. ͓DOI: 10.1063/1.2150947͔Quasiperiodically forced systems occur in many situations in physics, mathematics, engineering, etc. In many cases, the external quasiperiodic perturbations induce quasiperiodic motions, which correspond to invariant tori. It is important to understand when these invariant tori persist under perturbations, and to identify the mechanisms of their breakdown. It has been known for a long time that the persistence of a torus is related to the exponential growth of the linearization along certain directions (normal hyperbolicity), and that normal hyperbolicity may be lost because of bifurcations such as saddle-node and period doubling, among others. The common feature of these standard bifurcations is that some Lyapunov multipliers approach 1, while the invariant directions remain smooth. In this paper we propose a new mechanism, in which two invariant directions of the linearized dynamics come close to each other, losing their regularity, and the corresponding Lyapunov multipliers remain more or less constant, away from each other and away from 1. Hence, the phenomenon is described by two observables: the minimum angle between the invariant directions and the Lyapunov multipliers. We also identify notable universal power laws of these observables.

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Section: Setup Of the Problemmentioning

confidence: 99%

Manifolds on the verge of a hyperbolicity breakdown

Haro

Llave

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…We can find the integrable period-two orbit (U ± , T ± ) of the renormalization operator (19) by requiring that R(U ± , T ± ) = (U ∓ , T ∓ ). This two-cycle is given by the following pairs of maps:…”

Section: B Simple Periodic Orbit Of Rmentioning

confidence: 99%

“…Several theorems have been proved that lend mathematical support to the criterion. [19,20] The numerical search for periodic orbits is difficult because, in principle, it is a two-dimensional root finding problem. However, the task is considerably simplified for reversible maps, [17,21] which are maps that can be factored as M = I 1 •I 0 , where I 0,1 are involution maps that satisfy I 2 1 = I 2 0 = 1.…”

Section: A Periodic Orbits and Residue Criterionmentioning

confidence: 99%

“…(8), i.e. using the bifurcation curves for [32], [31], [20] and [19] (see Tables I and II). Now one can justify a posteriori the use of Eq.…”

Section: Finding 1/γ 2 -Bifurcation Curvementioning

confidence: 99%

“…(12), were computed from the variation in numerical values of residues at a c for the three or four highest period orbits found (e.g. for C 1 , using residues of [13], [19], [25] and [31]). …”

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confidence: 99%

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Renormalization and destruction of 1/γ2 tori in the standard nontwist map

Apte

Wurm

Morrison

2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Extending the work of del-Castillo-Negrete, Greene, and Morrison, Physica D 91, 1 (1996) and 100, 311 (1997) on the standard nontwist map, the breakup of an invariant torus with winding number equal to the inverse golden mean squared is studied. Improved numerical techniques provide the greater accuracy that is needed for this case. The new results are interpreted within the renormalization group framework by constructing a renormalization operator on the space of commuting map pairs, and by studying the fixed points of the so constructed operator.In recent years, area-preserving maps that violate the twist condition locally in phase space have been the object of interest in several studies in physics and mathematics. These nontwist maps show up in a variety of physical models. An important problem from the physics point of view is the understanding of the breakup of invariant tori, which show remarkable resilience in the region where the twist condition is violated, called shearless tori. In terms of the physical system modelled, these tori represent transport barriers, and their breakup corresponds to the transition to global chaos. Mathematically, nontwist maps present a challenge since the standard proofs of celebrated theorems in the theory of area-preserving maps rely heavily on the twist condition. In this paper, we study the breakup of the shearless torus with winding number 1/γ 2 , where γ is the golden mean. This torus serves as a test case for improved techniques we developed. At the point of breakup the shearless torus exhibits universal scaling behavior which leads to a renormalization group interpretation.

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Perturbative Series Expansions: Theoretical Aspects and Numerical Investigations

Biasco¹,

Celletti²

Lecture Notes in Control and Information Science

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Perturbation theoryi si ntroducedb ym eans of models borrowedf romC elestial Mechanics, namely the two-bodyand three-body problems.Suchmodels allowo ne to introduce in as imple way the concepts of integrable and nearlyintegrable systems, whichcan be conveniently investigated usingHamiltonian formalism.Afterdiscussingthe problemofthe convergenceofperturbativeseries expansions, we introduce the basic notions of KAM theory,w hicha llows (under quite general assumptions) to state the persistenceo fi nvariantt ori. Thev aluea tw hichs uchs urfacesb reak-down canbed etermined by means of numerical algorithms. Among the others, we reviewt hree methods to which we refera sG reene, Padéa nd Lyapunov. We presents ome concretea pplications to discrete models of the three differentt echniques, in order to provide complementary informationa bout the break-down of invariant tori. 1I ntroductionThe dynamics of theplanets and satellites is ruled by Newton's law, according to whicht he gravitational force is proportional to the product of them asses of theinteracting bodiesand it is inversely proportional to thesquare of their distance.T he description of thet rajectoriess panned by the celestial bodies starts with the simplest model in whichone considers only theattraction exerted by the Sun, neglecting all contributions duet oo ther planets or satellites. Such model is known as the two-bodyp roblem andi ti sf ullyd escribed by Kepler's laws, according to whicht he motion is represented by ac onic.C onsider,f or example,t he trajectory of an asteroid moving on an elliptico rbit around the Sun. In the two-bodya pproximation thes emimajor axis and the eccentricityo ft he ellipse arefi xed in time. However, suche xample represents

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A rigorous partial justification of Greene's criterion

Cited by 57 publications

References 25 publications

Manifolds on the verge of a hyperbolicity breakdown

Manifolds on the verge of a hyperbolicity breakdown

Renormalization and destruction of 1/γ2 tori in the standard nontwist map

Perturbative Series Expansions: Theoretical Aspects and Numerical Investigations

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