Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders

Bernard, Patrick; Калошин, В. А.; Zhang, Ke

doi:10.1007/s11511-016-0141-5

Cited by 52 publications

(96 citation statements)

References 59 publications

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“…See the related papers [19,40,44]. (5) Although the main application in this paper is on diffusion in a priori unstable systems, we expect that this method can be useful when applied to a priori stable systems, as well as to infinite-dimensional systems, once the existence of suitable normally hyperbolic invariant manifolds (called normally hyperbolic cylinders in [6,68,69,75]) and their homoclinic channels is established. See Remark 3.16.…”

Section: Brief Description Of the Main Resultsmentioning

confidence: 99%

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A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Gidea

Llave

Seara

2019

Comm Pure Appl Math

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We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map.” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic. We also mention several other applications where this mechanism is easy to verify (analytically or numerically), such as the planar elliptic restricted three‐body problem and the spatial circular restricted three‐body problem. Our method differs, in several crucial aspects, from earlier works. Unlike the well‐known “two‐dynamics” approach, the method we present here relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, such as on existence of its invariant objects (e.g., primary and secondary tori, lower‐dimensional hyperbolic tori, and their stable/unstable manifolds, Aubry‐Mather sets), which are not used at all. © 2019 Wiley Periodicals, Inc.

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Section: Brief Description Of the Main Resultsmentioning

confidence: 99%

“…then we can shadow orbits of the form y i C1 D i;iC1 .y i /, with y i 2 ƒ i and y i C1 2 ƒ i C1 , for i D 1; : : : ; n 1. Such scattering maps appear in the study of double resonances [6,68,69,77]. We hope to come back to this problem.…”

Section: Shadowing Of Pseudo-orbits Of the Scattering Mapmentioning

confidence: 95%

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Gidea

Llave

Seara

2019

Comm Pure Appl Math

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“…Therefore, the energy manifold is (2n − 1)-dimensional, while the invariant tori are ndimensional, so that they cannot act as barriers for the motion. This opens the problem of the so-called Arnold diffusion, the generic existence of which has been recently proven near a resonance of codimension one (see [3] and references therein).…”

Section: General Frameworkmentioning

confidence: 99%

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories

2017

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We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, that can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form, so as to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that is at basis of the analytic part of the Nekhoroshev's theorem, so as to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller. *

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“…Note that System (2) has been reduced to a first order system be adding the variables y =ẋ and v =u.…”

Section: Two-coupled Piezoelectric Oscillatorsmentioning

confidence: 99%

“…Unfortunately, theory for Arnold diffusion is still too restrictive to be applied in systems of the types (1) and (2), mainly due to the presence of dissipation, as it provides an extra obstacle to the growth of energy. In this work we present a first step on the study of Arnold diffusion in energy harvesting systems based on damped oscillators.…”

Section: Introductionmentioning

confidence: 99%

Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators

Granados

2017

Physica D: Nonlinear Phenomena

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Energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy. Widely extended are those based on piezoelectric materials, whose dynamics are Hamiltonian submitted to different sources of dissipation: damping and coupling. These dissipations bring the system to low energy regimes, which is not desired in long term as it diminishes the absorbed energy. To avoid or to minimize such situations, we propose that the coupling of two oscillators could benefit from theory of Arnold diffusion. Such phenomenon studies O(1) energy variations in Hamiltonian systems and hence could be very useful in energy harvesting applications. This article is a first step towards this goal. We consider two piezoelectric beams submitted to a small forcing and coupled through an electric circuit. By considering the coupling, damping and forcing as perturbations, we prove that the unperturbed system possesses a 4-dimensional Normally Hyperbolic Invariant Manifold with 5 and 4-dimensional stable and unstable manifolds, respectively. These are locally unique after the perturbation. By means of the parameterization method, we numerically compute parameterizations of the perturbed manifold, its stable and unstable manifolds and study its inner dynamics. We show evidence of homoclinic connections when the perturbation is switched on. * The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013. This work has also been partially supported by MINECO MTM2015-65715-P Spanish grant. We acknowledge the use of the UPC Dynamical Systems group's cluster for research computing (https://dynamicalsystems.upc.edu/en/computing/)

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Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders

Cited by 52 publications

References 59 publications

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories

Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators

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