Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map

We study numerically the "analyticity breakdown" transition in 1-dimensional quasi-periodic media. This transition corresponds physically to the transition between pinned down and sliding ground states. Mathematically, it corresponds to the solutions of a functional equation losing their analyticity properties.We implemented some recent numerical algorithms that are efficient and backed up by rigorous results so that we can compute with confidence even close to the breakdown.We have uncovered several phenomena that we believe deserve a theoretical explanation: A) The transition happens in a smooth surface. B) There are scaling relations near breakdown. C) The scaling near breakdown is very anisotropic. Derivatives in different directions blow up at different rates.Similar phenomena seem to happen in other KAM problems.

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“…For twist mappings it has been implemented in [CdlL09,CdlL10a,FM12]. For dissipative systems it has been implemented in [CC10,CF12].…”

Section: 3mentioning

confidence: 99%

The Analyticity Breakdown for Frenkel-Kontorova Models in Quasi-periodic Media: Numerical Explorations

Blass

Llave

2013

J Stat Phys

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“…Of course, the equation (1.3) will have to be supplemented by some normalization conditions, which ensure that the solutions are locally unique. We refer to [CdlL09,CdlL10a,dlLR91,CF12] for a method to find invariant curves that solve the invariance equation (1.3) numerically. Our main result, Theorem 12, shows that, if there is a solution of (1.3) for ε = 0 (the symplectic case), which satisfies some mild non-degeneracy conditions, we can find K ε and µ ε defined for a set G of ε.…”

Section: Introductionmentioning

confidence: 99%

Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian systems

2017

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Many problems in Physics are described by dynamical systems that are conformally symplectic (e.g., mechanical systems with a friction proportional to the velocity, variational problems with a small discount or thermostated systems). Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting.We provide all details for maps, but we present also the (somewhat minor) modifications needed to obtain a direct proof for the case of differential equations. We consider a family of conformally symplectic maps f µ,ε defined on a 2d-dimensional symplectic manifold M with exact symplectic form Ω; we assume that f µ,ε satisfies f * µ,ε Ω = λ(ε)Ω. We assume that the family depends on a d-dimensional parameter µ (called drift ) and also on a small scalar parameter ε. Furthermore, we assume that the conformal factor λ depends on ε, in such a way that for ε = 0 we have λ(0) = 1 (the symplectic case). We also assume thatWe study the domains of analyticity in ε near ε = 0 of perturbative expansions (Lindstedt series) of the parameterization of the quasi-periodic orbits of frequency ω (assumed to be Diophantine) and of the parameter µ. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the Lindstedt series are analytic in a domain in the complex ε plane, which is obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin. We state also a conjecture on the optimality of our results.The proof is based on the following procedure. To find a quasi-periodic solution, one solves an invariance equation for the embedding of the torus, depending on the parameters of the family. Assuming that the frequency of the torus satisfies a Diophantine condition, under mild non-degeneracy assumptions, using a Lindstedt procedure we construct an approximate solution to all orders of the invariance equation describing the KAM torus; the zeroth order Lindstedt series is provided by the solution of the invariance equation of the symplectic case. Starting from such approximate solution, we implement an a-posteriori KAM theorem to get the true solution of the invariance equation, and we show that the procedure converges. This allows also the study of monogenic and Withney extensions.2010 Mathematics Subject Classification. 70K43, 70K20, 37J40.

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“…This method is suitable for application to computer assisted proofs. (For examples of such applications see [3,7,14,19,22], amongst others.) We believe that our approach to Melnikov method (from sections 1.1, 1.2, and 3) could also be successfully combined with [2].…”

Section: Introductionmentioning

confidence: 99%

Beyond the Melnikov method: A computer assisted approach

Capiński

Zgliczyński

2017

Journal of Differential Equations

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We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds (NHIMs). The method is based on a new geometric proof of the normally hyperbolic invariant manifold theorem, which establishes the existence of a NHIM, together with its associated invariant manifolds and bounds on their first and second derivatives. We do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are 'small enough', as is the case in the classical Melnikov approach.

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Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map

Cited by 41 publications

References 26 publications

The Analyticity Breakdown for Frenkel-Kontorova Models in Quasi-periodic Media: Numerical Explorations

The Analyticity Breakdown for Frenkel-Kontorova Models in Quasi-periodic Media: Numerical Explorations

Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian systems

Beyond the Melnikov method: A computer assisted approach

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