Rafael de la Llave

Arnold diffusion in the planar elliptic restricted three-body problem: mechanism and numerical verification

We present a diffusion mechanism for time-dependent perturbations of autonomous Hamiltonian systems introduced in [25]. This mechanism is based on shadowing of pseudo-orbits generated by two dynamics: an `outer dynamics, given by homoclinic trajectories to a normally hyperbolic invariant manifold, and an `inner dynamics, given by the restriction to that manifold. On the inner dynamics the only assumption is that it preserves area. Unlike other approaches, [25] does not rely on the KAM theory and/or Aubry-Mather theory to establish the existence of diffusion. Moreover, it does not require to check twist conditions or non-degeneracy conditions near resonances. The conditions are explicit and can be checked by finite precision calculations in concrete systems. nAs an application, we study the planar elliptic restricted three-body problem. We present a rigorous theorem that shows that if some concrete calculations yield a non zero value, then for any sufficiently small, positive value of the eccentricity of the orbits of the main bodies, there are orbits of the infinitesimal body that exhibit a change of energy that is bigger than some fixed number, which is independent of the eccentricity. nWe verify numerically these calculations for values of the masses close to that of the Jupiter/Sun system. The numerical calculations are not completely rigorous, because we ignore issues of round-off error and do not estimate the truncations, but they are not delicate at all by the standard of numerical analysis. (Standard tests indicate that we get 7 or 8 figures of accuracy where 1 would be enough). The code of this verifications is available. We hope that some full computer assisted proofs will be obtained in a near future since there are packages (CAPD) designed for problems of this type.

Barriers to transport and mixing in volume-preserving maps with nonzero flux

Abstract We identify some geometric structures (secondary tori) that restrict transport and prevent mixing in perturbations of integrable volume-preserving systems with nonzero net flux. Unlike the customary KAM tori, secondary tori cannot be continued to the tori present in the integrable system but are generated by resonances and have a contractible direction. We also note that secondary tori persist under the addition of a net flux, which destroys all customary KAM tori. We introduce a remarkably simple algorithm to analyze the behavior of volume preserving maps and to obtain quantitative properties of the secondary tori. We then implement the algorithm and, after running it, present assertions regarding the distribution of the escape times of the unbounded orbits, the abundance of secondary tori, the size of the resonant regions, and the robustness of the tori under the addition of a mean flux.

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

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

Resonant Equilibrium Configurations in Quasi-periodic Media: Perturbative Expansions

We consider 1-D quasi-periodic Frenkel–Kontorova models. We study the existence of equilibria whose frequency (i.e. the inverse of the density of deposited material) is resonant with the frequencies of the substratum. We study perturbation theory for small potential. We show that there are perturbative expansions to all orders for the quasi-periodic equilibria with resonant frequencies. Under very general conditions, we show that there are at least two such perturbative expansions for equilibria for small values of the parameter. We also develop a dynamical interpretation of the equilibria in these quasi-periodic media. We show that equilibria are orbits of a dynamical system which has very unusual properties. We obtain results on the Lyapunov exponents of the dynamical systems, i.e. the phonon gap of the resonant quasi-periodic equilibria. We show that the equilibria can be pinned even if the gap is zero.

Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction

In this paper we consider lattice systems coupled by local interactions. We prove invariant manifold theorems for whiskered tori (we recall that whiskered tori are quasi-periodic solutions with exponentially contracting and expanding directions in the linearized system). The invariant manifolds we construct generalize the usual (strong) (un)stable manifolds and allow us to consider also non-resonant manifolds. We show that if the whiskered tori are localized near a collection of specific sites, then so are the invariant manifolds. We recall that the existence of localized whiskered tori has recently been proven for symplectic maps and flows in Fontich etxa0al. (J Diff Equ, 2012), but our results do not need that the systems are symplectic. For simplicity we will present first the main results for maps, but we will show that the result for maps imply the results for flows. It is also true that the results for flows can be proved directly following the same ideas.

Invariant Manifolds for Analytic Difference Equations

We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, and smooth dependence on parameters and study several singular limits, even if the difference equations do not define a dynamical system. This method also leads to efficient algorithms that we present with their implementations. The manifolds that we consider include not only the classical strong stable and unstable manifolds but also manifolds associated with nonresonant spaces. When the difference equations are the Euler--Lagrange equations of a discrete variational problem, we have sharper results. Note that, if the Legendre condition fails, the Euler--Lagrange equations cannot be treated as a dynamical system. If the Legendre condition becomes singular, the dynamical system may be singular while the difference equation remains regular. We present numerical applications to several examples in the physics literature: the Frenkel--Kontorova model with long-range ...

KAM theory for quasi-periodic equilibria in 1D quasi-periodic media: II. Long-range interactions

We consider Frenkel–Kontorova models corresponding to one-dimensional quasi-crystal with non-nearest neighbor and many-body interactions. We formulate and prove a KAM type theorem which establishes the existence of quasi-periodic solutions. The interactions we consider do not need to be of finite range or involve finitely many particles, but have to decay sufficiently fast with respect to the distance of the position of the atoms. The KAM theorem we present has an a posteriori format. We do not need to assume that the system is close to integrable. We just assume that there is an approximate solution for the functional equation which satisfies some non-degeneracy conditions.

An A Posteriori KAM Theorem for Whiskered Tori in Hamiltonian Partial Differential Equations with Applications to some Ill-Posed Equations

The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result has an a-posteriori format, that is, we show that if there is an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, then there is a true solution nearby. This allows, besides dealing with the quasi-integrable case, for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi-periodic solutions in degenerate situations. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). We emphasize that the non-degeneracy conditions required are just quantities evaluated on the approximate solution (no global assumptions on the system such as twist). Hence, they are readily verifiable in perturbation expansions. We will pay attention to the quantitative relations between the sizes of the approximation and the non-degeneracy conditions. This will allow us to prove what experts call small twist theorems (the non-degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation). The method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant. We also solve functional equations for bundles which imply that are invariant under the linearization. The iterative method does not use transformation theory nor action-angle variables. The main result does not assume that the system is close to integrable. More surprisingly, we do not need that the equations we study define an evolution for all initial conditions and are well posed. Even if the systems we study do not admit solutions for all initial conditions, we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi-periodic. We first develop an abstract theorem. Then, we show how this abstract result applies to some concrete examples. The examples considered in this paper are the scalar Boussinesq equation and the Boussinesq system (both are PDE models that aim to describe water waves in the long wave limit). For these equations we construct small amplitude time quasi-periodic solutions which are even in the spatial variable. The strategy for the abstract theorem is inspired by that in Fontich etxa0al. (Electron Res Announc Math Sci 16:9–22, 2009; J Differ Equ 246(8):3136–3213, 2009). The main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill-posed equations and which is stable under addition of unbounded perturbations. This requires that we assume smoothing properties. We also present very detailed bounds on the change of the splittings under perturbations.

Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions

A well-known result in complex dynamics shows that if the iterates of an analytic map are uniformly bounded in a complex domain, then the map is analytically conjugate to a linear map. We present a simple proof of this result in any dimension. We also present several generalizations and relations to other results in the literature.A well-known result in complex dynamics shows that if the iterates of an analytic map are uniformly bounded in a complex domain, then the map is analytically conjugate to a linear map. We present a simple proof of this result in any dimension. We also present several generalizations and relations to other results in the literature.

Global Melnikov Theory in Hamiltonian Systems with General Time-Dependent Perturbations

We consider a mechanical system consisting of n-penduli and a d-degree-of-freedom rotator. The phase space of the rotator defines a normally hyperbolic invariant manifold