arXiv subjects

Rafael de la Llave

Publications and source records attributed to Rafael de la Llave.

At least 55 records · Page 3Linked to original sources

arXiv2017

On the Global Dynamics of an Electroencephalographic Mean Field Model of the Neocortex

This paper investigates the global dynamics of a mean field model of the electroencephalogram developed by Liley et al., 2002. The model is presented as a system of coupled ordinary and partial differential equations with periodic boundary conditions. Existence, uniqueness, and regularity of weak and strong solutions of the model are established in appropriate function spaces, and the associated initial-boundary value problems are proved to be well-posed. Sufficient conditions are developed for the phase spaces of the model to ensure nonnegativity of certain quantities in the model, as required by their biophysical interpretation. It is shown that the semigroups of weak and strong solution operators possess bounded absorbing sets for the entire range of biophysical values of the parameters of the model. Challenges towards establishing a global attractor for the model are discussed and it is shown that there exist parameter values for which the constructed semidynamical systems do not possess a compact global attractor due to the lack of the asymptotic compactness property. Finally, using the theoretical results of the paper, instructive insights are provided into the complexity of the behavior of the model and computational analysis of the model.

math.DS↗

arXiv2017

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the `outer dynamics' along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined map, referred to as the `scattering map'. We find pseudo-orbits of the scattering map that keep advancing in some privileged direction. Then we use the recurrence property of the `inner dynamics', restricted to the normally hyperbolic invariant manifold, to return to those pseudo-orbits. Finally, we apply topological methods to show the existence of true orbits that follow the successive applications of the two dynamics. This method differs, in several crucial aspects, from earlier works. Unlike the well known `two-dynamics' approach, the method we present relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, as its invariant objects (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets) are not used at all. The method applies to unperturbed Hamiltonians of arbitrary degrees of freedom that are not necessarily convex. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, as well as to establish diffusion in generic systems. We include several applications, such as bridging large gaps in a priori unstable models in any dimension, and establishing diffusion in cases when the inner dynamics is a non-twist map.

math.DS↗

arXiv2016

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. As 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. We 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.

math.DS↗

arXiv2016

Numerical Computations and Computer Assisted Proofs of Periodic Orbits of the Kuramoto-Sivashinsky Equation

We present numerical results and computer assisted proofs of the existence of periodic orbits for the Kuramoto-Sivashinky equation. These two results are based on writing down the existence of periodic orbits as zeros of functionals. This leads to the use of Newton's algorithm for the numerical computation of the solutions and, with some a posteriori analysis in combination with rigorous interval arithmetic, to the rigorous verification of the existence of solutions. This is a particular case of the methodology developed in [19] for several types of orbits. An independent implementation, covering overlapping but different ground, using different functional setups appears in [33].

math.DS↗

arXiv2016

A Framework for the Numerical Computation and a Posteriori Verification of Invariant Objects of Evolution Equations

We develop a theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear PDEs. The invariant objects considered in this paper are equilibrium points, traveling waves, periodic orbits and invariant manifolds attached to fixed points or periodic orbits. The core of the study is writing down the invariance condition as a zero of an operator. These operators are in general not continuous, so one needs to smooth them by means of preconditioners before classical fixed point theorems can be applied. We develop in detail all the aspects of how to work with these objects: how to precondition the equations, how to work with the nonlinear terms, which function spaces can be useful, and how to work with them in a computationally rigorous way. In two companion papers, we present two different implementations of the tools developed in this paper to study periodic orbits.

math.DS↗

arXiv2016

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 \emph{a-posteriori} format, i.e., 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. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). The method of proof is based on an iterative method to solve a functional equation for the parameterization of the torus satisfying the invariance equations and for parametrization of directions invariant under the linearizatation. The iterative method does not use transformation theory or action-angle variables. It does not assume that the system is close to integrable. We first develop an abstract theorem. Then, we show how this abstract result applies to some concrete examples, including the scalar Boussinesq equation and the Boussinesq system so that we construct {\sl small amplitude} tori for the equations, which are even in the spatial variable. Note that the equations we use as examples are ill-posed.

math.DS↗

arXiv2015

Resonant equilibrium configurations in quasi-periodic media: KAM theory

We develop an a-posteriori KAM theory for the equilibrium equations for quasi-periodic solutions in a quasi-periodic Frenkel-Kontorova model when the frequency of the solutions resonates with the frequencies of the substratum. The KAM theory we develop is very different both in the methods and in the conclusions from the more customary KAM theory for Hamiltonian systems or from the KAM theory in quasi-periodic media for solutions with frequencies which are Diophantine with respect to the frequencies of the media. The main difficulty is that we cannot use transformations (as in the Hamiltonian case) nor Ward identities (as in the case of frequencies Diophantine with those of the media). The technique we use is to add an extra equation that ensures the linearization of the equilibrium equation factorizes. To solve the extra equation requires an extra counterterm. We compare this phenomenon with other phenomena in KAM theory. It seems that this technique could be used in several other problems. As a conclusion, we obtain that the perturbation expansions developed in the previous paper \cite{SuZL15} converge when the potentials are in a codimension one manifold in a space of potentials. The method of proof also leads to efficient (low storage requirements and low operation count) algorithms to compute the quasi-periodic solutions.

math.DS↗

arXiv2015

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.

math.DS↗

arXiv2015

A continuous family of equilibria in ferromagnetic media are ground states

We show that a foliation of equilibria (a continuous family of equilibria whose graph covers all the configuration space) in ferromagnetic models are ground states. The result we prove is very general, and it applies to models with long range interactions and many body. As an application, we consider several models of networks of interacting particles including models of Frenkel-Kontorova type on $\mathbb{Z}^d$ and one-dimensional quasi-periodic media. The result above is an analogue of several results in the calculus variations (fields of extremals) and in PDE's. Since the models we consider are discrete and long range, new proofs need to be given. We also note that the main hypothesis of our result (the existence of foliations of equilibria) is the conclusion (using KAM theory) of several recent papers. Hence, we obtain that the KAM solutions recently established are minimizers when the interaction is ferromagnetic (and transitive).

math-ph↗

arXiv2015

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

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 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 $\mathcal M$ 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 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.

math.DS↗

arXiv2015

Response solutions for quasi-periodically forced, dissipative wave equations

We consider several models of nonlinear wave equations subject to very strong damping and quasi-periodic external forcing. This is a singular perturbation, since the damping is not the highest order term. We study the existence of response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Under very general non-resonance conditions on the frequency, we show the existence of asymptotic expansions of the response solution; moreover, we prove that the response solution indeed exists and depends analytically on $\varepsilon$ (where $\varepsilon$ is the inverse of the coefficient multiplying the damping) for $\varepsilon$ in a complex domain, which in some cases includes disks tangent to the imaginary axis at the origin. In other models, we prove analyticity in cones of aperture $π/2$ and we conjecture it is optimal. These results have consequences for the asymptotic expansions of the response solutions considered in the literature. The proof of our results relies on reformulating the problem as a fixed point problem, constructing an approximate solution and studying the properties of iterations that converge to the solutions of the fixed point problem.

math.AP↗

arXiv2014

Barriers to Transport and Mixing in Volume-Preserving Maps with Nonzero Flux

In this paper we identify the geometric structures that restrict transport and mixing in perturbations of integrable volume-preserving systems with nonzero net flux. Unlike KAM tori, these objects cannot be continued to the tori present in the integrable system but are generated by resonance and have a contractible direction. We introduce a remarkably simple algorithm to analyze the behavior of these maps and obtain quantitative properties of the tori. In particular, we present assertions regarding the distribution of the escape times of the unbounded orbits, the abundance of tori, and the size of the resonant regions.

nlin.CD↗

arXiv2014

Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices

We construct quasi-periodic and almost periodic solutions for coupled Hamiltonian systems on an infinite lattice which is translation invariant. The couplings can be long range, provided that they decay moderately fast with respect to the distance. For the solutions we construct, most of the sites are moving in a neighborhood of a hyperbolic fixed point, but there are oscillating sites clustered around a sequence of nodes. The amplitude of these oscillations does not need to tend to zero. In particular, the almost periodic solutions do not decay at infinity. We formulate an invariance equation. Solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation. We show that, if there is an approximate solution of the invariance equation that satisfies some non-degeneracy conditions, there is a true solution close by. The proof of this \emph{a-posteriori} theorem is based on a Nash-Moser iteration, which does not use transformation theory. Simpler versions of the scheme were developed in E. Fontich, R. de la Llave,Y. Sire \emph{J. Differential. Equations.} {\bf 246}, 3136 (2009). One technical tool, important for our purposes, is the use of weighted spaces that capture the idea that the maps under consideration are local interactions. Using these weighted spaces, the estimates of iterative steps are similar to those in finite dimensional spaces. In particular, the estimates are independent of the number of nodes that get excited. Using these techniques, given two breathers, we can place them apart and obtain an approximate solution, which leads to a true solution nearby. By repeating the process infinitely often, we can get solutions with infinitely many frequencies which do not tend to zero at infinity.

math.DS↗

arXiv2013

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 et. al. (submitted), 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 tha 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.

math-ph↗

arXiv2013

Perturbations of geodesic flows by recurrent dynamics

We consider a geodesic flow on a compact manifold endowed with a Riemannian (or Finsler, or Lorentz) metric satisfying some generic, explicit conditions. We couple the geodesic flow with a time-dependent potential, driven by an external flow on some other compact manifold. If the external flow satisfies some very general recurrence condition, and the potential satisfies some explicit conditions that are also very general, we show that the coupled system has orbits whose energy grows at a linear rate with respect to time. This growth rate is optimal. We also show the existence of symbolic dynamics. The existence of orbits whose energy grows unboundedly in time is related to Arnold's diffusion problem. A particular case of this phenomenon is obtained when the external dynamical system is quasi-periodic, of rationally independent frequency vector, not necessarily Diophantine, thus extending earlier results. We also recover `Mather's acceleration theorem' for time-periodic perturbations. Since the general class of perturbations that we consider in this paper are not characterized by a frequency of motion, we are not able to use KAM or Aubry-Mather or averaging theory. We devise a new mechanism to construct orbits whose energy evolves in a prescribed fashion. This mechanism intertwines the inner dynamics restricted to some normally hyperbolic manifold with the outer dynamics corresponding to two distinct choices of homoclinic excursions to that manifold. As this mechanism uses very coarse information on the dynamics, it is potentially of wider applicability.

math.DS↗

arXiv2013

Instability of high dimensional Hamiltonian Systems: Multiple resonances do not impede diffusion

We consider models given by Hamiltonians of the form $$H(I,ϕ,p,q,t;ε) = h(I) + \sum_{j = 1}^n \pm(\frac{1}{2} p_j^2 + V_j(q_j)) + εQ(I,ϕ,p,q,t;ε)$$ where $I,ϕ$ are d-dimensional actions and angles, $p,q$ are n-dimensional real conjugated variables, and $t$ is an angle. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in previous papers by the athors. All these models present the large gap problem. We show that, for $ε$ small enough, under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables $I$ perform rather arbitrary excursions in a domain of size O(1). This domain includes resonance lines and, hence, large gaps among $d$-dimensional KAM tori. The method of proof follows closely the strategy of [DLS(=Delshams, Llave and Seara) 2006]. The main new phenomenon that appears when the dimension $d$ of the center directions is larger than one, is the existence of multiple resonances. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions $I$, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from [DLS 2006]. On the technical details of the proofs, we have taken advantage of the theory of the scattering map [DLS 2008], not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [DLS 2006].

math.DS↗

arXiv2013

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. 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.

math-ph↗

arXiv2012

Tracing KAM tori in presymplectic dynamical systems

We present a KAM theorem for presymplectic dynamical systems. The theorem has a " a posteriori " format. We show that given a Diophantine frequency $ω$ and a family of presymplectic mappings, if we find an embedded torus which is approximately invariant with rotation $ω$ such that the torus and the family of mappings satisfy some explicit non-degeneracy condition, then we can find an embedded torus and a value of the parameter close to to the original ones so that the torus is invariant under the map associated to the value of the parameter. Furthermore, we show that the dimension of the parameter space is reduced if we assume that the systems are exact.

math.DS↗