arXiv subjects

Rafael de la Llave

Publications and source records attributed to Rafael de la Llave.

At least 37 records · Page 2Linked to original sources

arXiv2021

Efficient and accurate KAM tori construction for the dissipative spin-orbit problem using a map reduction

We consider the dissipative spin-orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and "drift". Our goal is to construct quasi-periodic solutions with fixed frequency, satisfying appropriate conditions. With the goal of applying rigorous KAM theory, we compute such quasi-periodic solution with very high precision. To this end, we have developed a very efficient algorithm. The first step is to compute very accurately the return map to a surface of section (using a high order Taylor's method with extended precision). Then, we find an invariant curve for the return map using recent algorithms that take advantage of the geometric features of the problem. This method is based on a rapidly convergent Newton's method which is guaranteed to converge if the initial error is small enough. So, it is very suitable for a continuation algorithm. The resulting algorithm is quite efficient. We only need to deal with a one dimensional function. If this function is discretized in $N$ points, the algorithm requires $O(N \log N) $ operations and $O(N) $ storage. The most costly step (the numerical integration of the equation along a turn) is trivial to parallelize. The main goal of the paper is to present the algorithms, implementation details and several sample results of runs. We also present both a rigorous and a numerical comparison of the results of averaged and not averaged models.

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arXiv2021

Melnikov method for non-conservative perturbations of the three-body problem

We consider the planar circular restricted three-body problem (PCRTBP), as a model for the motion of a spacecraft relative to the Earth-Moon system. We focus on the Lagrange equilibrium points $L_1$ and $L_2$. There are families of Lyapunov periodic orbits around either $L_1$ or $L_2$, forming Lyapunov manifolds. There also exist homoclinic orbits to the Lyapunov manifolds around either $L_1$ or $L_2$, as well as heteroclinic orbits between the Lyapunov manifold around $L_1$ and the one around $L_2$. The motion along the homoclinic/heteroclinic orbits can be described via the scattering map, which gives the future asymptotic of a homoclinic orbit as a function of the past asymptotic. In contrast with the more customary Melnikov theory, we do not need to assume that the asymptotic orbits have a special nature (periodic, quasi-periodic, etc.). We add a non-conservative, time-dependent perturbation, as a model for a thrust applied to the spacecraft for some duration of time, or for some other effect, such as solar radiation pressure. We compute the first order approximation of the perturbed scattering map, in terms of fast convergent integrals of the perturbation along homoclinic/heteroclinic orbits of the unperturbed system. As a possible application, this result can be used to determine the trajectory of the spacecraft upon using the thrust.

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arXiv2021

Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation

Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation -- not just a formal power series -- bringing the Hamiltonian into its Birkhoff normal form.

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arXiv2021

Persistence and Smooth Dependence on Parameters of Periodic Orbits in Functional Differential Equations Close to an ODE or an Evolutionary PDE

We consider functional differential equations(FDEs) which are perturbations of smooth ordinary differential equations(ODEs). The FDE can involve multiple state-dependent delays or distributed delays (forward or backward). We show that, under some mild assumptions, if the ODE has a nondegenerate periodic orbit, then the FDE has a smooth periodic orbit. Moreover, we get smooth dependence of the periodic orbit and its frequency on parameters with high regularity. The result also applies to FDEs which are perturbations of some evolutionary partial differential equations(PDEs). The proof consists in solving functional equations satisfied by the parameterization of the periodic orbit and the frequency using a fixed point approach. We do not need to consider the smoothness of the evolution or even the phase space of the FDEs.

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arXiv2020

Gevrey estimates for asymptotic expansions of tori in weakly dissipative systems

We consider a singular perturbation for a family of analytic symplectic maps of the annulus possessing a KAM torus. The perturbation introduces dissipation and contains an adjustable parameter. By choosing the adjustable parameter, one can ensure that the torus persists under perturbation. Such models are common in celestial mechanics. In field theory, the adjustable parameter is called the counterterm and in celestial mechanics, the drift. It is known that there are formal expansions in powers of the perturbation both for the quasi-periodic solution and the counterterm. We prove that the asymptotic expansions for the quasiperiodic solutions and the counterterm satisfy Gevrey estimates. That is, the $n$-th term of the expansion is bounded by a power of $n!$. The Gevrey class (the power of $n!$) depends only on the Diophantine condition of the frequency and the order of the friction coefficient in powers of the perturbative parameter. The method of proof we introduce may be of interest beyond the problem considered here. We consider a modified Newton method in a space of power expansions. As it is custumary in KAM theory, each step of the method is estimated in a smaller domain. In contrast with the KAM results, the domains where we control the Newton method shrink very fast and the Newton method does not prove that the solutions are analytic. On the other hand, by examining carefully the process, we can obtain estimates on the coefficients of the expansions and conclude the series are Gevrey.

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arXiv2020

KAM theory for some dissipative systems

Dissipative systems play a very important role in several physical models, most notably in Celestial Mechanics, where the dissipation drives the motion of natural and artificial satellites, leading them to migration of orbits, resonant states, etc. Hence the need to develop theories that ensure the existence of structures such as invariant tori or periodic orbits and device efficient computational methods. In this work we concentrate on the existence of invariant tori for the specific case of dissipative systems known as "conformally symplectic" systems, which have the property that they transform the symplectic form into a multiple of itself. To give explicit examples of conformally symplectic systems, we will present two different models: a discrete system, the standard map, and a continuous system, the spin-orbit problem. In both cases we will consider the conservative and dissipative versions, that will help to highlight the differences between the symplectic and conformally symplectic dynamics. For such dissipative systems we will present a KAM theorem in an a-posteriori format. The method of proof is based on extending geometric identities originally developed in [39] for the symplectic case. Besides leading to streamlined proofs of KAM theorem, this method provides a very efficient algorithm which has been implemented. Coupling an efficient numerical algorithm with an a-posteriori theorem, we have a very efficient way to provide rigorous estimates close to optimal. Indeed, the method gives a criterion (the Sobolev blow up criterion) that allows to compute numerically the breakdown. We will review this method as well as an extension of J. Greene's method and present the results in the conservative and dissipative standard maps. Computing close to the breakdown, allows to discover new mathematical phenomena such as the "bundle collapse mechanism".

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arXiv2020

Numerical computation of periodic orbits and isochrons for state-dependent delay perturbation of an ODE in the plane

We present algorithms and their implementation to compute limit cycles and their isochrons for state-dependent delay equations (SDDE's) which are perturbed from a planar differential equation with a limit cycle. Note that the space of solutions of an SDDE is infinite dimensional. We compute a two parameter family of solutions of the SDDE which converge to the solutions of the ODE as the perturbation goes to zero in a neighborhood of the limit cycle. The method we use formulates functional equations among periodic functions (or functions converging exponentially to periodic). The functional equations express that the functions solve the SDDE. Therefore, rather than evolving initial data and finding solutions of a certain shape, we consider spaces of functions with the desired shape and require that they are solutions. The mathematical theory of these invariance equations is developed in a companion paper, which develops "a posteriori" theorems. They show that, if there is a sufficiently approximate solution (with respect to some explicit condition numbers), then there is a true solution close to the approximate one. Since the numerical methods produce an approximate solution, and provide estimates of the condition numbers, we can make sure that the numerical solutions we consider approximate true solutions. In this paper, we choose a systematic way to approximate functions by a finite set of numbers (Taylor-Fourier series) and develop a toolkit of algorithms that implement the operators -- notably composition -- that enter into the theory. We also present several implementation results and present the results of running the algorithms and their implementation in some representative cases.

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arXiv2020

Parameterization method for state-dependent delay perturbation of an ordinary differential equation

We consider state-dependent delay equations (SDDE) obtained by adding delays to a planar ordinary differential equation with a limit cycle. These situations appear in models of several physical processes, where small delay effects are added. Even if the delays are small, they are very singular perturbations since the natural phase space of an SDDE is an infinite dimensional space. We show that the SDDE admits solutions which resemble the solutions of the ODE. That is, there exist a periodic solution and a two parameter family of solutions whose evolution converges to the periodic solution. Even if the phase space of the SDDE is naturally a space of functions, we show that there are initial values which lead to solutions similar to that of the ODE. The method of proof bypasses the theory of existence, uniqueness, dependence on parameters of SDDE. We consider the class of functions of time that have a well defined behavior (e.g. periodic, or asymptotic to periodic) and derive a functional equation which imposes that they are solutions of the SDDE. These functional equations are studied using methods of functional analysis. We provide a result in "a posteriori" format: Given an approximate solution of the functional equation, which has some good condition numbers, we prove that there is true solution close to the approximate one. Thus, we can use the result to validate the results of numerical computations. The method of proof leads also to practical algorithms. In a companion paper, we present the implementation details and representative results. One feature of the method presented here is that it allows to obtain smooth dependence on parameters for the periodic solutions and their slow stable manifolds without studying the smoothness of the flow (which seems to be problematic for SDDEs, for now the optimal result on smoothness of the flow is $C^1$).

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arXiv2020

KAM estimates for the dissipative standard map

From the beginning of KAM theory, it was realized that its applicability to realistic problems depended on developing quantitative estimates on the sizes of the perturbations allowed. In this paper we present results on the existence of quasi-periodic solutions for conformally symplectic systems in non-perturbative regimes. We recall that, for conformally symplectic systems, finding the solution requires also to find a "drift parameter". We present a proof on the existence of solutions for values of the parameters which agree with more than three figures with the numerically conjectured optimal values. The verification of the estimates presented here is not completely rigorous since we do not control the round-off error. Nevertheless, running with different precision shows very little difference in the results. Given the high precision of the calculation and the simplicity of the estimates, this does not seem to affect the results. A full verification should be done implementing interval arithmetic. We make available the approximate solutions, the highly efficient algorithms to generate them (incorporating high precision based on the MPFR library) and the routines used to verify the applicability of the theorem.

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arXiv2019

Expansions in the delay of quasi-periodic solutions for state dependent delay equations

We consider several models of State Dependent Delay Differential Equations (SDDEs), in which the delay is affected by a small parameter. This is a very singular perturbation since the nature of the equation changes. Under some conditions, we construct formal power series, which solve the SDDEs order by order. These series are quasi-periodic functions of time. This is very similar to the Lindstedt procedure in celestial mechanics. Truncations of these power series can be taken as input for a-posteriori theorems, that show that near the approximate solutions there are true solutions. In this way, we hope that one can construct a catalogue of solutions for SDDEs, bypassing the need of a systematic theory of existence and uniqueness for all initial conditions.

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arXiv2019

Response solutions to the quasi-periodically forced systems with degenerate equilibrium: A simple proof of a result of W. Si and J. Si. and extensions

We give a simple proof of the existence of response solutions in some quasi-periodically forced systems with a degenerate fixed points. The same questions were answered in \cite{ss18} using two versions of KAM theory. Our method is based on reformulating the existence of response solutions as a fixed point problem in appropriate spaces of smooth functions. By algebraic manipulations, the fixed point problem is transformed into a contraction. Compared to the KAM method, the present method does not incur a loss of regularity. That is, the solutions we obtain have the same regularity as the forcing. Moreover, the method here applies when problems are only finitely differentiable. It also weakens slightly the non-degeneracy conditions. Since the method is based on the contraction mapping principle, we also obtain automatically smooth dependence on parameters and, when studying complex versions of the problem we discover the new phenomenon of monodromy. We also present results for higher dimensional systems, but for higher dimensional systems, the concept of degenerate fixed points is much more subtle than in one dimensional systems. To illustrate the power of the method, we also consider two problems not studied in \cite{ss18}: the forcing with zero average and second order oscillators. We show that in the zero average forcing case, the solutions are qualitatively different, but the second order oscillators is remarkably similar.

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arXiv2019

Global effect of non-conservative perturbations on homoclinic orbits

We study the effect of time-dependent, non-conservative perturbations on the dynamics along homoclinic orbits to a normally hyperbolic invariant manifold. We assume that the unperturbed system is Hamiltonian, and the normally hyperbolic invariant manifold is parametrized via action-angle coordinates. The homoclinic excursions can be described via the scattering map, which gives the future asymptotic of an orbit as a function of the past asymptotic. We provide explicit formulas, in terms of convergent integrals, for the perturbed scattering map expressed in action-angle coordinates. We illustrate these formulas in the case of a perturbed rotator-pendulum system.

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arXiv2019

Global persistence of Lyapunov-subcenter-manifolds as spectral submanifolds under dissipative perturbations

For a nondegenerate analytic system with a conserved quantity, a classic result by Lyapunov guarantees the existence of an analytic manifold of periodic orbits tangent to any two-dimensional, elliptic eigenspace of a fixed point satisfying nonresonance conditions. These two dimensional manifolds are referred as Lyapunov Subcenter Manifolds (LSM). Numerical and experimental observations in the nonlinear vibrations literature suggest that LSM's often persist under autonomous, dissipative perturbations. These perturbed manifolds are useful since they provide information of the asymptotics of the convergence to equilibrium. In this paper, we formulate and prove precise mathematical results on the persistence of LSM under dissipation. We show that, under mild non-degeneracy conditions on the perturbation, for small enough dissipation, there are analytic invariant manifolds of the perturbed system that approximate (in the analytic sense) the LSM in a fixed neighborhood. We also provide examples that show that some non-degeneracy conditions on the perturbations are needed for the results to hold true. We also study the dependence of the manifolds on the dissipation parameter. If $\varepsilon$ is the dissipation parameter, we show that the manifolds are real analytic in $(-\varepsilon_0, \varepsilon_0) \setminus \{0\} $ and $C^\infty $ in $(-\varepsilon_0, \varepsilon_0)$. We construct explicit asymptotic expansions in powers of $\varepsilon$ (which presumably do not converge). Finally, we present applications of our results to several mechanical systems.

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arXiv2019

Response solutions to quasi-periodically forced systems, even to possibly ill-posed PDEs, with strong dissipation and any frequency vectors

We consider several models (including both multidimensional ordinary differential equations (ODEs) and partial differential equations (PDEs), possibly ill-posed), subject to very strong damping and quasi-periodic external forcing. We study the existence of response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Under some regularity assumptions on the nonlinearity and forcing, without any arithmetic condition on the forcing frequency $ω$, we show that the response solutions indeed exist. Moreover, the solutions we obtained possess optimal regularity in $\varepsilon$ (where $\varepsilon$ is the inverse of the coefficients multiplying the damping) when we consider $\varepsilon$ in a domain that does not include the origin $\varepsilon=0$ but has the origin on its boundary. We get that the response solutions depend continuously on $\varepsilon$ when we consider $\varepsilon $ tends to $0$. However, in general, they may not be differentiable at $\varepsilon=0$. In this paper, we allow multidimensional systems and we do not require that the unperturbed equations under consideration are Hamiltonian. One advantage of the method in the present paper is that it gives results for analytic, finitely differentiable and low regularity forcing and nonlinearity, respectively. As a matter of fact, we do not even need that the forcing is continuous. Notably, we obtain results when the forcing is in $L^2$ space and the nonlinearity is just Lipschitz as well as in the case that the forcing is in $H^1$ space and the nonlinearity is $C^{1 + \text{Lip}}$. In the proof of our results, we reformulate the existence of response solutions as a fixed point problem in appropriate spaces of smooth functions.

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arXiv2019

Whiskered KAM Tori of Conformally Symplectic Systems

We investigate the existence of whiskered tori in some dissipative systems, called \sl conformally symplectic \rm systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family $f_μ$ of conformally symplectic maps which depend on a drift parameter $μ$. We fix a Diophantine frequency of the torus and we assume to have a drift $μ_0$ and an embedding of the torus $K_0$, which satisfy approximately the invariance equation $f_{μ_0} \circ K_0 - K_0 \circ T_ω$ (where $T_ω$ denotes the shift by $ω$). We also assume to have a splitting of the tangent space at the range of $K_0$ into three bundles. We assume that the bundles are approximately invariant under $D f_{μ_0}$ and that the derivative satisfies some "rate conditions". Under suitable non-degeneracy conditions, we prove that there exists $μ_\infty$, $K_\infty$ and splittings, close to the original ones, invariant under $f_{μ_\infty}$. The proof provides an efficient algorithm to construct whiskered tori. Full details of the statements and proofs are given in [CCdlL18].

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arXiv2019

Existence of whiskered KAM tori of conformally symplectic systems

We study the existence of whiskered tori in a family $f_μ$ of conformally symplectic maps depending on parameters $μ$. Whiskered tori are tori on which the motion is a rotation, but they have as many expanding/contracting directions as allowed by the preservation of the geometric structure. Our main result is formulated in an "a-posteriori" format. We fix $ω$ satisfying Diophantine conditions. We assume that we are given 1) a value of the parameter $μ_0$, 2) an embedding of the torus $K_0$ into the phase space, approximately invariant under $f_{μ_0}$ in the sense that $f_{μ_0} \circ K_0 - K_0 \circ T_ω$ is small, 3) a splitting of the tangent space at the range of $K_0$, into three bundles which are approximately invariant under $D f_{μ_0}$ and such that the derivative satisfies "rate conditions" on each of the components. Then, if some non-degeneracy conditions are satisfied, we show that there is another parameter $μ_\infty$, an embedding $K_\infty$ and splittings close to the original ones which are invariant under $f_{μ_\infty}$. We also bound $|μ_\infty - μ_0|$, $\|K_\infty - K_0 \|$ and the distance of the initial and final splittings in terms of the initial error. The proof of the main theorem consists in describing an iterative process that takes advantage of cancellations coming from the geometry. Then, we show that the process converges to a true solution when started from an approximate enough solution. The iterative process leads to an efficient algorithm that is quite practical to implement. As an application, we study the singular problem of effects of small dissipation on whiskered tori. We develop formal expansions in the perturbative parameter and use them as input for the a-posteriori theorem. This allows to obtain lower bounds for the domain of analyticity of the tori as function of the perturbative parameter.

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arXiv2018

On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in \cite[Section 5.11]{Federer74} to illustrate the need of certain definitions in the calculus of variations. The Almgren-Federer example, besides its intended goal of illustrating subtle aspects of geometric measure theory, is also a problem in the theory of geodesics. Hence, we wrote an exposition of the beautiful ideas of Almgren and Federer from the point of view of geodesics. In the language of geodesics, Almgren-Federer example constructs metrics in $\mathbb{S}^1\times \mathbb{S}^2$, with the property that none of the Tonelli geodesics (geodesics which minimize the length in a homotopy class) are Class-A minimizers in the sense of Morse (any finite length segment in the universal cover minimizes the length between the end points; this is also sometimes given other names). In other words, even if a curve is a minimizer of length among all the curves homotopic to it, by repeating it enough times, we get a closed curve which does not minimize in its homotopy class. In that respect, the example is more dramatic than a better known example due to Hedlund of a metric in $\mathbb{T}^3$ for which only 3 Tonelli minimizers (and their multiples) are Class-A minimizers. For dynamics, the example also illustrates different definitions of `integrable' and clarifies the relation between minimization and hyperbolicity and its interaction with topology.

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arXiv2017

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

We consider a mechanical system consisting of $n$ penduli and a $d$-dimensional generalized rotator subject to a time-dependent perturbation. The perturbation is not assumed to be either Hamiltonian, or periodic or quasi-periodic. The strength of the perturbation is given by a parameter $ε\in\mathbb{R}$. For all $|ε|$ sufficiently small, the augmented flow has a $(2d + 1)$-dimensional normally hyperbolic locally invariant manifold $\tildeΛ_ε$. We define a Melnikov vector, which gives the first order expansion of the displacement of the stable and unstable manifolds of $\tildeΛ_0$ under the perturbation. We provide an explicit formula for the Melnikov vector in terms of convergent improper integrals of the perturbation along homoclinic orbits of the unperturbed system. We show that if the perturbation satisfies some explicit non-degeneracy conditions, then the stable and unstable manifolds of $\tildeΛ_ε$, $W^s(\tildeΛ_ε)$ and $W^u(\tildeΛ_ε)$, respectively, intersect along a transverse homoclinic manifold, and, moreover, the splitting of $W^s(\tildeΛ_ε)$ and $W^u(\tildeΛ_ε)$ can be explicitly computed, up to the first order, in terms of the Melnikov vector. This implies that the excursions along some homoclinic trajectories yield a non-trivial increase of order $O(ε)$ in the action variables of the rotator, for all sufficiently small perturbations. The formulas that we obtain are independent of the unperturbed motions and give, at the same time, the effects on periodic, quasi-periodic, or general orbits. When the perturbation is Hamiltonian, we express the effects of the perturbation, up to the first order, in terms of a Melnikov potential. In addition, if the perturbation is periodic, we obtain that the non-degeneracy conditions on the Melnikov potential are generic.

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