arXiv subjects

Marcel Guardia

Publications and source records attributed to Marcel Guardia.

At least 37 records · Page 2Linked to original sources

arXiv2022

Topological Shadowing Methods in Arnold Diffusion: Weak Torsion and Multiple Time Scales

Consider a symplectic map which possesses a normally hyperbolic invariant manifold of any even dimension with transverse homoclinic channels. We develop a topological shadowing argument to prove the existence of Arnold diffusion along the invariant manifold, shadowing some iterations of the inner dynamics carried by the invariant manifold and the outer dynamics induced by the stable and unstable foliations. In doing so, we generalise an idea of Gidea and de la Llave in [26], based on the method of correctly aligned windows and a so-called transversality-torsion argument. Our proof permits that the dynamics on the invariant manifold satisfy only a non-uniform twist condition, and, most importantly for applications, that the splitting of separatrices be small in certain directions and thus the associated drift in actions very slow; diffusion occurs in the directions of the manifold having non-small splitting. Furthermore we provide estimates for the diffusion time.

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arXiv2022

Arnold diffusion in Hamiltonian systems on infinite lattices

We consider a system of infinitely many penduli on an $m$-dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next-to-nearest neighbor or long range as long as it is strongly decaying. The transfer of energy is given by an Arnold diffusion mechanism which relies on the original V. I Arnold approach: to construct a sequence of hyperbolic invariant quasiperiodic tori with transverse heteroclinic orbits. We implement this approach in an infinite dimensional setting, both in the space of bounded $\mathbb{Z}^m$-sequences and in spaces of decaying $\mathbb{Z}^m$-sequences. Key steps in the proof are an invariant manifold theory for hyperbolic tori and a Lambda Lemma for infinite dimensional coupled map lattices with decaying interaction.

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arXiv2022

Sobolev norms explosion for the cubic NLS on irrational tori

We consider the cubic nonlinear Schrödinger equation on $2$-dimensional irrational tori. We construct solutions which undergo growth of Sobolev norms. More concretely, for every $s>0$, $s\neq 1$ and almost every choice of spatial periods we construct solutions whose $H^s$ Sobolev norms grow by any prescribed factor. Moreover, for a set of spatial periods with positive Hausdorff dimension we construct solutions whose Sobolev norms go from arbitrarily small to arbitrarily large. We also provide estimates for the time needed to undergo the norm explosion. Note that the irrationality of the space periods decouples the linear resonant interactions into products of $1$-dimensional resonances, reducing considerably the complexity of the resonant dynamics usually used to construct transfer of energy solutions. However, one can provide these growth of Sobolev norms solutions by using quasi-resonances relying on Diophantine approximation properties of the space periods.

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arXiv2021

Breakdown of homoclinic orbits to L3 in the RPC3BP (II). An asymptotic formula

The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies are coplanar, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In rotating coordinates, it can be modeled by a two degrees of freedom Hamiltonian, which has five critical points called the Lagrange points L1,.., L5. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and beyond the largest of the two. In this paper, we obtain an asymptotic formula for the distance between the stable and unstable manifolds of L3 for small values of the mass ratio $0<μ\ll 1$. In particular we show that L3 cannot have (one round) homoclinic orbits. If the ratio between the masses of the primaries $μ$ is small, the hyperbolic eigenvalues of L3 are weaker, by a factor of order $\sqrtμ$, than the elliptic ones. This rapidly rotating dynamics makes the distance between manifolds exponentially small with respect to $\sqrtμ$. Thus, classical perturbative methods (i.e the Melnikov-Poincaré method) can not be applied. The obtention of this asymptotic formula relies on the results obtained in the prequel paper on the complex singularities of the homoclinic of a certain averaged equation and on the associated inner equation. In this second paper, we relate the solutions of the inner equation to the analytic continuation of the parameterizations of the invariant manifolds of L3 via complex matching techniques. We complete the proof of the asymptotic formula for their distance showing that its dominant term is the one given by the analysis of the inner equation.

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arXiv2021

Breakdown of homoclinic orbits to L3 in the RPC3BP (I). Complex singularities and the inner equation

The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom Hamiltonian system with five critical points, L1,..,L5, called the Lagrange points. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and is located beyond the largest of the two. In this paper and its sequel, we provide an asymptotic formula for the distance between the one dimensional stable and unstable invariant manifolds of L3 when the ratio between the masses of the primaries $μ$ is small. It implies that L3 cannot have one-round homoclinic orbits. If the mass ratio $μ$ is small, the hyperbolic eigenvalues are weaker than the elliptic ones by factor of order $\sqrtμ$. This implies that the distance between the invariant manifolds is exponentially small with respect to $μ$ and, therefore, the classical Poincaré--Melnikov method cannot be applied. In this first paper, we approximate the RPC3BP by an averaged integrable Hamiltonian system which possesses a saddle center with a homoclinic orbit and we analyze the complex singularities of its time parameterization. We also derive and study the inner equation associated to the original perturbed problem. The difference between certain solutions of the inner equation gives the leading term of the distance between the stable and unstable manifolds of L3. In the sequel we complete the proof of the asymptotic formula for the distance between the invariant manifolds.

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arXiv2021

Oscillatory Motions and Parabolic Manifolds at Infinity in the Planar Circular Restricted Three Body Problem

Consider the Restricted Planar Circular 3 Body Problem with both realistic mass ratio and Jacobi constant for the Sun-Jupiter pair. We prove the existence of all possible combinations of past and future final motions. In particular, we obtain the existence of oscillatory motions. All the constructed trajectories cross the orbit of Jupiter but avoid close encounters with it. The proof relies on the method of correctly aligned windows and is computer assisted.

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arXiv2021

Symbolic Dynamics in the Elliptic Isosceles Restricted Three Body Problem

The elliptic isosceles restricted three body problem (REI3BP) models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies called the primaries. The primaries of masses $m_{1}=m_{2}$ move along a degenerate Keplerian elliptic collision orbit (on a line) under their gravitational attraction, whereas the third, massless particle, moves on the plane perpendicular to their line of motion and passing through the center of mass of the primaries. By symmetry, the component of the angular momentum $G$ of the massless particle along the direction of the line of the primaries is conserved. We show the existence of symbolic dynamics in the REI3BP for large $G$ by building a Smale horseshoe on a certain subset of the phase space. As a consequence we deduce that the REI3BP possesses oscillatory motions, namely orbits which leave every bounded region but return infinitely often to some fixed bounded region. The proof relies on the existence of transversal homoclinic connections associated to an invariant manifold at infinity. Since the distance between the stable and unstable manifolds of infinity is exponentially small, Melnikov theory does not apply.

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arXiv2020

Chaotic Resonant Dynamics and Exchanges of Energy in Hamiltonian PDEs

The aim of this note is to present the recent results in [16] where we provide the existence of solutions of some nonlinear resonant PDEs on the 2-dimensional torus exchanging energy among Fourier modes in a \emph{chaotic-like} way. We say that a transition of energy is \emph{chaotic-like} if either the choice of activated modes or the time spent in each transfer can be chosen randomly. We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations. The key point of the construction of the special solutions is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form of those equations.

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arXiv2020

Chaotic-like transfers of energy in Hamiltonian PDEs

We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations on $T^2$ and we prove the existence of different types of solutions which exchange energy between Fourier modes in certain time scales. This exchange can be considered \emph{chaotic-like} since either the choice of activated modes or the time spent in each transfer can be chosen randomly. The key point of the construction of those orbits is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form truncation of those equations.

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arXiv2018

Critical velocity in kink-defect interaction models: rigorous results

In this work we study a model of interaction of kinks of the sine-Gordon equation with a weak defect. We obtain rigorous results concerning the so-called critical velocity derived in [7] by a geometric approach. More specifically, we prove that a heteroclinic orbit in the energy level $0$ of a $2$-dof Hamiltonian $H_ε$ is destroyed giving rise to heteroclinic connections between certain elements (at infinity) for exponentially small (in $ε$) energy levels. In this setting Melnikov theory does not apply because there are exponentially small phenomena.

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arXiv2018

Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation

We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long timescales, they exhibit a strong form of transverse instability in Sobolev spaces $H^s(\mathbb{T}^2)$ ($0<s<1$). More precisely, we construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the $H^s$ topology and whose $H^s$ norm can grow by any given factor. This work is partly motivated by the problem of infinite energy cascade for 2D NLS, and seems to be the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed.

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arXiv2017

Growth of Sobolev norms for the analytic NLS on $\mathbb T^2$

We consider the completely resonant defocusing non-linear Schrödinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix $s>1$. We show the existence of solutions of this equation which achieve arbitrarily large growth of $H^s$ Sobolev norms. We also give estimates for the time required to attain this growth.

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arXiv2017

Random Iteration of Cylinder Maps and diffusive behavior away from resonances

In this paper we propose a model of random compositions of cylinder maps, which in the simplified form is as follows: let $(θ,r)\in \mathbb T\times \mathbb R=\mathbb A$ and \[ f_{\pm 1}: \left(\begin{array}{c}θ\\r\end{array}\right) \longmapsto \left(\begin{array}{c}θ+r+\varepsilon u_{\pm 1}(θ,r) \\ r+\varepsilon v_{\pm 1}(θ,r) \end{array}\right), \] where $u_\pm$ and $v_\pm$ are smooth and $v_\pm$ are trigonometric polynomials in $θ$ such that $\int v_\pm(θ,r)\,dθ=0$ for each $r$. We study the random compositions \[ (θ_n,r_n)=f_{ω_{n-1}}\circ \dots \circ f_{ω_0}(θ_0,r_0), \] where $ω_k =\pm 1$ with equal probability. We show that under non-degeneracy hypotheses and away from resonances for $n\sim \varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to a stochastic diffusion process with explicitly computable drift and variance. In the case $u_\pm(θ)=v_\pm(θ)$ are trigonometric polynomials of zero average we prove a vertical central limit theorem, namely, for $n\sim \varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to the normal distribution $\mathcal N(0,σ^2)$ with $σ^2=\frac14\int (v_+(θ)-v_-(θ))^2\,dθ$.} The considered random model up to higher order terms in $\varepsilon$ is conjugate to a restrictions to a Normally Hyperbolic Invariant Lamination of the generalized Arnold example. Combining the result of this paper with [8,23,28] we show formation of stochastic diffusive behaviour for the generalized Arnold example.

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arXiv2015

Oscillatory orbits in the restricted elliptic planar three body problem

The restricted planar elliptic three body problem models the motion of a massless body under the Newtonian gravitational force of the two other bodies, the primaries, which evolve in Keplerian ellipses. A trajectory is called oscillatory if it leaves every bounded region but returns infinitely often to some fixed bounded region. We prove the existence of such type of trajectories for any values for the masses of the primaries provided they make almost circular orbits.

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arXiv2015

Secular instability in the spatial three-body problem

Consider the spatial three-body problem, in the regime where one body revolves far away around the other two, in space, the masses of the bodies being arbitrary but fixed; in this regime, there are no resonances in mean motions. The so-called secular dynamics governs the slow evolution of the Keplerian ellipses. We show that it contains a horseshoe and all the chaotic dynamics which goes along with it, corresponding to motions along which the eccentricity of the inner ellipse undergoes large, random excursions. The proof goes through the surprisingly explicit computation of the homoclinic solution of the first order secular system, its complex singularities and the Melnikov potential.

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arXiv2014

Orbits of nearly integrable systems accumulating to KAM tori

Consider a sufficiently smooth nearly integrable Hamiltonian system of two and a half degrees of freedom in action-angle coordinates \[ H_ε(φ,I,t)=H_0(I)+εH_1(φ,I,t), φ\in T^2,\ I\in U\subset R^2,\ t\in T=R/Z. \] Kolmogorov-Arnold-Moser Theorem asserts that a set of nearly full measure in phase space consists of three dimensional invariant tori carrying quasiperiodic dynamics. In this paper we prove that for a class of nearly integrable Hamiltonian systems there is an orbit which contains these KAM tori in its $ω$-limit set. This implies that the closure of the orbit has almost full measure in the phase space. As byproduct, we show that KAM tori are Lyapunov unstable. The proof relies in the recent developments in the study of Arnold diffusion in nearly integrable systems Bernard-Kaloshin-Zhang, Kaloshin-Zhang12. It is a combination of geometric and variational techniques.

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arXiv2013

Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem

We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun--Jupiter--asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio $μ=10^{-3}$ and a small eccentricity $e_0>0$. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept almost constant. The proof relies on the careful analysis of the circular problem, which has a hyperbolic structure, but for which diffusion is prevented by KAM tori. We verify certain non-degeneracy conditions numerically. Based on the work of Treschev, it is natural to conjecture that diffusion time for this problem is $\sim \frac{-\ln (μe_0)}{μ^{3/2} e_0}$. We expect our instability mechanism to apply to realistic values of $e_0$ and we give heuristic arguments in its favor. If so, the applicability of Nekhoroshev theory to the three-body problem as well as the long time stability become questionable. It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. Our mechanism gives a possible explanation of their existence. To relate the existence of Kirkwood gaps with Arnold diffusion, we state a conjecture on its existence for a typical $\eps$-perturbation of the product of a pendulum and a rotator. Namely, we predict that a positive conditional measure of initial conditions concentrated in the main resonance exhibits Arnold diffusion on time scales $\frac{- \ln \eps}{\eps^{2}}$.

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arXiv2013

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation with a convolution potential

Fix $s>1$. Colliander, Keel, Staffilani, Tao and Takaoka proved in \cite{CollianderKSTT10} the existence of solutions of the cubic defocusing nonlinear Schrödinger equation in the two torus with $s$-Sobolev norm growing in time. In this paper we generalize their result to the cubic defocusing nonlinear Schrödinger equation with a convolution potential. Moreover, we show that the speed of growth is the same as the one obtained for the cubic defocusing nonlinear Schrödinger equation in \cite{GuardiaK12}. The results we obtain can deal with any potential in $H^{s_0}(\TT^2)$, $s_0>0$.

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