arXiv subjects

Rafael de la Llave

Publications and source records attributed to Rafael de la Llave.

At least 19 recordsLinked to original sources

arXiv2026

Existence and Regularity of Stable Resonant Spectral Submanifolds and Linearization Maps

We prove the existence of $C^{r,1-}$-regular stable invariant manifolds and linear conjugacies for analytic maps near a hyperbolic fixed point in the presence of resonances. The regularity exponent $r$ depends on the minimal resonant index, while the Hölder exponent can be chosen arbitrarily close to $1$, i.e., $1-\varepsilon$ for any $\varepsilon>0$. As a consequence, we obtain the stable version of the Hartman conjecture for analytic systems with semisimple linearization: in the fully stable case the local conjugacy can be chosen $C^{1,1-\varepsilon}$ for every $\varepsilon>0$.\\ Our approach introduces a new class of functional expansions based on logarithmic polynomials, which enables the invariance equation to be solved explicitly and algorithmically to arbitrary order. The existence results are obtained via a fixed-point argument in a suitable Banach space. We further present several analytic examples that both illustrate the applicability of the theorem while demonstrating the necessity of its assumptions.

math.DS↗

arXiv2025

Geometric, topological and dynamical properties of conformally symplectic systems, normally hyperbolic invariant manifolds, and scattering maps

Conformally symplectic diffeomorphisms $f:M \rightarrow M$ transform a symplectic form $ω$ on a manifold M into a multiple of itself, $f^* ω= ηω$. We assume $ω$ is bounded, as some of the results may fail otherwise. We show that there are deep interactions between the topological properties of the manifold, the dynamical properties of the map, and the geometry of invariant manifolds. We show that, when the symplectic form is not exact, the possible conformal factors $η$ are related to topological properties of the manifold. For some manifolds the conformal factors are restricted to be algebraic numbers. We also find relations between dynamical properties (relations between growth rate of vectors and $η$) and symplectic properties. Normally hyperbolic invariant manifolds (NHIM) and their (un)stable manifolds are important landmarks that organize long-term dynamical behaviour. We prove that a NHIM is symplectic if and only if the rates satisfy certain pairing rules and if and only if the rates and the conformal factor satisfy certain (natural) inequalities. Homoclinic excursions to NHIMs are quantitatively described by scattering maps. These maps give the trajectory asymptotic in the future as a function of the trajectory asymptotic in the past. We prove that the scattering maps are symplectic even if the dynamics is dissipative. We also show that if the symplectic form is exact, then the scattering maps are exact, even if the dynamics is not exact. We give a variational interpretation of scattering maps in the conformally symplectic setting. We also show that similar properties of NHIMs and scattering maps hold in the case when $ω$ is presymplectic. In dynamical systems with many rates (e.g., quasi-integrable systems near multiple resonances), pre-symplectic geometries appear naturally.

math.DS↗

arXiv2025

Simultaneous computation of whiskered tori and their whiskers in Hamiltonian systems using flow maps

We consider autonomous Hamiltonian systems and present an algorithm to compute at the same time partially hyperbolic invariant tori (whiskered tori), as well as high-order expansions of their stable and unstable manifolds. Such whiskered tori have been shown to be important for transport phenomena in phase space. For instance, by following their invariant manifolds one could obtain zero-cost trajectories in space mission design. We present in detail the case when the (un)stable directions are one-dimensional. The strategy to compute tori and their invariant manifolds is based on the parameterization method. We formulate a functional equation for a parameterization of both the torus and its whiskers expressing that they are invariant. This equation is naturally discretized in Fourier-Taylor series or, equivalently, in a grid of Taylor series. Using a return map, we are reduced to study functions of n - 1 variables where n is the number of degrees of freedom (the phase space is 2n dimensional). Then, we implement a Newton-like method that converges quadratically. They key advantage of our approach is that, using geometric identities coming from the Hamiltonian nature of the problem, the algorithm has small storage requirements and a low operation count per step which is highly efficient. The simultaneous computation of the torus and the whiskers improves the efficiency and the stability of the algorithm. We present implementations and extensive numerical experiments in the Circular Restricted Three Body Problem.

math.DS↗

arXiv2025

Persistence of hyperbolic solutions of ODE's under functional perturbations: Applications to the motion of relativistic charged particles

We rigorously construct a variety of orbits for certain delay differential equations, including the electrodynamic equations formulated by Wheeler and Feynman in 1949. These equations involve delays and advances that depend on the trajectory itself, making it unclear how to formulate them as evolution equations in a conventional phase space. Despite their fundamental significance in physics, their mathematical treatment remains limited. Our method applies broadly to various functional differential equations that have appeared in the literature, including advanced/delayed equations, neutral or state-dependent delay equations, and nested delay equations, under appropriate regularity assumptions. Rather than addressing the notoriously difficult problem of proving the existence of solutions for all the initial conditions in a set, we focus on the direct construction of a diverse collection of solutions. This approach is often sufficient to describe physical phenomena. For instance, in certain models, we establish the existence of families of solutions exhibiting symbolic dynamics. Our method is based on the assumption that the system is, in a weak sense, close to an ordinary differential equation (ODE) with "hyperbolic" solutions as defined in dynamical systems. We then derive functional equations to obtain space-time corrections. As a byproduct of the method, we obtain that the solutions constructed depend very smoothly on parameters of the model. Also, we show that many formal approximations currently used in physics are valid with explicit error terms. Several of the relations between different orbits of the ODE persist qualitatively in the full problem.

math.DS↗

arXiv2024

KAM Theory for almost-periodic equilibria in one dimensional almost-periodic media

We consider one dimensional chains of interacting particles subjected to one dimensional almost-periodic media. We formulate and prove two KAM type theorems corresponding to both short-range and long-range interactions respectively. Both theorems presented have an a posteriori format and establish the existence of almost-periodic equilibria. The new part here is that the potential function is given by some almost-periodic function with infinitely many incommensurate frequencies. In both cases, we do not need to assume that the system is close to integrable. We will show that if there exists an approximate solution for the functional equations, which satisfies some appropriate non-degeneracy conditions, then a true solution nearby is obtained. This procedure may be used to validate efficient numerical computations. Moreover, to well understand the role of almost-periodic media which can be approximated by quasi-periodic ones, we present a different approach -- the step by step increase of complexity method -- to the study of the above results of the almost-periodic models.

math.DS↗

arXiv2023

Rapid GPU-Assisted Search and Parameterization-Based Refinement and Continuation of Connections between Tori in Periodically Perturbed Planar Circular Restricted 3-Body Problems

When the planar circular restricted 3-body problem (PCRTBP) is periodically perturbed, as occurs in many useful astrodynamics models, most unstable periodic orbits persist as whiskered tori. Intersections between stable and unstable manifolds of such tori provide natural heteroclinic pathways enabling spacecraft to greatly modify their orbits without using propellant. However, the 2D Poincaré sections used in PCRTBP studies no longer work to find these intersections. Thus, in this study, we develop new fast methods to search for and compute such heteroclinics. First, the dynamics are used to restrict the intersection search to only certain manifold subsets, greatly reducing the required computational effort. Next, we present a massively parallel procedure for carrying out this search by representing the manifolds as discrete meshes and adapting methods from computer graphics collision detection algorithms. Implementing the method in Julia and OpenCL, we obtain a 5-7x speedup by leveraging GPUs versus CPU-only execution. Finally, we show how to use manifold parameterizations to refine the approximate intersections found in the mesh search to very high accuracy, as well as to numerically continue the connections through families of tori; the families' Whitney differentiability enables interpolation of needed parameterizations. The ability to very rapidly find a heteroclinic intersection between tori of fixed frequencies thus allows the systematic exploration of intersections for tori of nearby frequencies as well, yielding a variety of potential zero-fuel spacecraft trajectories. We demonstrate the tools on the Jupiter-Europa planar elliptic RTBP.

math.DS↗

arXiv2023

4th Body-Induced Secondary Resonance Overlapping Inside Unstable Resonant Orbit Families: a Jupiter-Ganymede 4:3 + Europa Case Study

The overlapping of mean-motion resonances is useful for low or zero-propellant space mission design, but while most related prior work uses a planar CRTBP model, tours of multi-moon systems require using resonances affected by two moons. In this case study, we investigate Jupiter-Ganymede unstable 4:3 resonant orbits in a concentric circular restricted 4-body Jupiter-Europa-Ganymede model. We show that despite their high order, secondary resonances between the 4:3 orbits and Europa have a large effect, including 11/34, 12/37, 23/71, and 25/77. Computing newly generated objects inside the secondary resonances definitively confirms their overlap, which causes a complete structural change of the higher-energy unstable 4:3 orbits whose manifolds are most useful for low-TOF orbit transfers. We believe this phenomenon is general, with major implications for resonant orbit use in tour design.

astro-ph.EP↗

arXiv2023

Accurate computations up to break-down of quasi-periodic attractors in the dissipative spin-orbit problem

We consider a Celestial Mechanics model: the spin-orbit problem with a dissipative tidal torque, which is a singular perturbation of a conservative system. The goal of this paper is to show that it is possible to compute quasi-periodic attractors accurately and reliably for parameter values extremely close to the breakdown. Therefore, it is possible to obtain information on mathematical phenomena at breakdown. The method we use incorporates the same time numerical and rigorous improvements. Among them (i) the formalism is based on studying the time-one map of the spin-orbit problem (which reduces the dimensionality of the problem) and has mathematical advantages; (ii) very accurate integration of the ODE (high order Taylor methods implemented with extended precision) for the map at its jets; (iii) a very efficient KAM method for maps which computes the attractor and its tangent spaces ( quadratically convergent step with low storage requirements, and low operation count); (iv) the algorithms are backed by a rigorous a-posteriori KAM Theorem, which establishes that if the algorithm, produces a very approximate solution of functional equation with reasonable condition numbers. then there is a true solution nearby; and (v) the continuation algorithm is guaranteed to reach arbitrarily close to the border of existence if it is given enough computer resources. As a byproduct of the accuracy that we maintain till breakdown, we study several scale invariant observables of the tori used in the renormalization group of infinite dimensional spaces. In contrast with previously studied simple models, the behavior at breakdown of the spin-orbit problem does not satisfy standard scaling relations which implies that the spin-orbit problem is not described by a hyperbolic fixed point of a renormalization operator.

math.DS↗

arXiv2023

A simple proof of Gevrey estimates for expansions of quasi-periodic orbits: dissipative models and lower dimensional tori

We consider standard-like/Froeschlé maps with a dissipation and nonlinear perturbation. That is \[ T_\varepsilon(p,q) = \left( (1 - γ\varepsilon^3) p + μ+ \varepsilon V'(q), q + (1 - γ\varepsilon^3) p + μ+ \varepsilon V'(q) \mod 2 π\right) \] where $p \in \mathbb{R}^D$, $q \in 2 π\mathbb{T}^D$ are the dynamical variables. The $μ\in \mathbb{R}^D, γ\in \mathbb{R}$ are parameters of the model. We assume that the potential $V$ is a trigonometric polynomial. Note that when $γ\ne 0$, the perturbation parameter $\varepsilon$ creates dissipation, which has a drastic effect on the existence of quasi-periodic orbits, hence it is a singular perturbation. We fix a frequency $ω\in \mathbb{R}^D$ and study the existence of quasiperiodic orbits. When there is dissipation, having a quasiperiodic orbit of frequency $ω$ requires adjusting the parameter $μ$, called \textit{the drift}. We first study the Lindstedt series (formal power series in $\varepsilon$) for quasiperidic orbits with $D$ independent frequencies and the drift when $γ\ne 0$. We show that, when $ω$ is irrational, the series exist to all orders, and when $ω$ is Diophantine, we show that the formal Lindstedt series are Gevrey. We also study the case when $D = 2$, but the quasi-periodic orbits have only one independent frequency (Lower dimensional tori). Both when $γ= 0$ and when $γ\ne 0$, we show that, under some mild non-degeneracy conditions on $V$, there are (at least two) formal Lindstedt series defined to all orders and that they are Gevrey. Furthermore, we can take $μ$ along a $1$ dimensional space.

math.DS↗

arXiv2022

Nonconmutative coboundary equations over integrable systems

\def\G{\mathcal G} \def\M{\mathcal M} \def\cE{\mathcal E} We prove an analog of Livšic theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra $\G$ or a Lie group. Namely, we consider an integrable dynamical system $f:\M \equiv\torus^d \times [-1,1]^d\to \M$, $f(θ, I)=(θ+ I, I)$, and a real-analytic family of cocycles $η_\eps : \M \to \G$, indexed by a complex parameter $\eps$ in an open ball $\cE_ρ\in\CC$. We show that if $η_\eps$ has trivial periodic data, i.e., $$ η_\eps(f^{n-1}(p))\dots η_{\eps} (f(p))\cdot η_{\eps} (p)=Id $$ for each periodic point $p=f^n p$ and each $\eps \in \cE_ρ$, then there exists a real-analytic family of maps $ϕ_\eps: \M \to \G$ satisfying the coboundary equation $$ η_\eps(θ, I)=ϕ_\eps^{-1}\circ f(θ, I)\cdot ϕ_\eps (θ, I) $$ for all $(θ, I)\in \M$ and $\eps \in \cE_{ρ/2}$. We also show that if the coboundary equation above with an analytic left-hand side $η_\eps$ has a solution in the sense of formal power series in $\eps$, then it has an analytic solution.

math.DS↗

arXiv2022

Resonances and Phase Locking Phenomena for Foliation Preserving Torus Maps

It is well known for experts that resonances in nonlinear systems lead to new invariant objects that lead to new behaviors. The goal of this paper is to study the invariant sets generated by resonances under foliation preserving torus maps. That is torus which preserve a foliation of irrational lines $L_{θ_{0}}=\{θ_{0}+Ωt | t\in\mathbb{R}\}\subset\mathbb{T}^{d}$. Foliation preserving maps appear naturally as reparametrization of linear flows in the torus and also play an important role in several applications involving coupled oscillators, delay equations, resonators with moving walls, etc. The invariant objects we find here, lead to predictions on the behavior of these models. Since the results of this paper are meant to be applied for other problems, we have developed very quantitative results giving very explicit descriptions of the phenomena and the invariant objects that control them. The structure of the phase locking regions for foliation preserving maps is very different than for generic maps of the torus. Indeed, for the sake of completeness, we have developed similar analysis for the case of generic maps of the torus and shown that the objects that appear in foliation preserving maps are quantitatively and qualitatively different from those of generic torus maps. This has consequences in applications.

math.DS↗

arXiv2022

Lyapunov exponents everywhere and rigidity

In the present work we obtain rigidity results analysing the set of regular points, in the sense of Oseledec's Theorem. It is presented a study on the possibility of an Anosov diffeomorphisms having all Lyapunov exponents defined everywhere. We prove that this condition implies local rigidity of an Anosov automorphism of the torus $\mathbb{T}^d, d \geq 3,$ $C^1-$close to a linear automorphism diagonalizable over $\mathbb{R}$ and such that its characteristic polynomial is irreducible over $\mathbb{Q}.$

math.DS↗

arXiv2021

Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the following a priori unstable Hamiltonian system with a time-periodic perturbation \[\mathcal{H}_\varepsilon(p,q,I,φ,t)=h(I)+\sum_{i=1}^n\pm \left(\frac{1}{2}p_i^2+V_i(q_i)\right)+\varepsilon H_1(p,q,I,φ, t), \] where $(p,q)\in \mathbb{R}^n\times\mathbb{T}^n$, $(I,φ)\in\mathbb{R}^d\times\mathbb{T}^d$ with $n, d\geq 1$, $V_i$ are Morse potentials, and $\varepsilon$ is a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations $H_1$. Indeed, the set of admissible $H_1$ is $C^ω$ dense and $C^3$ open (a fortiori, $C^ω$ open). Our perturbative technique for the genericity is valid in the $C^k$ topology for all $k\in [3,\infty)\cup\{\infty, ω\}$.

math.DS↗

arXiv2021

Rapid and Accurate Methods for Computing Whiskered Tori and their Manifolds in Periodically Perturbed Planar Circular Restricted 3-Body Problems

When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation as well as rotation numbers; 3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds; 4) regularize the equations of motion; and 5) globalize these manifolds. Our methodology improves on efficiency and accuracy compared to prior studies, and applies to a variety of periodic perturbations. We demonstrate the tools near resonances in the planar elliptic RTBP.

math.DS↗

arXiv2021

Computation and Analysis of Jupiter-Europa and Jupiter-Ganymede Resonant Orbits in the Planar Concentric Circular Restricted 4-Body Problem

Many unstable periodic orbits of the planar circular restricted 3-body problem (PCRTBP) persist as invariant tori when a periodic forcing is added to the equations of motion. In this study, we compute tori corresponding to exterior Jupiter-Europa and interior Jupiter-Ganymede PCRTBP resonant periodic orbits in a concentric circular restricted 4-body problem (CCR4BP). Motivated by the 2:1 Laplace resonance between Europa and Ganymede's orbits, we then attempt the continuation of a Jupiter-Europa 3:4 resonant orbit from the CCR4BP into the Jupiter-Ganymede PCRTBP. We strongly believe that the resulting dynamical object is a KAM torus lying near but not on the 3:2 Jupiter-Ganymede resonance.

math.DS↗

arXiv2021

High-Order Resonant Orbit Manifold Expansions For Mission Design In the Planar Circular Restricted 3-Body Problem

In recent years, stable and unstable manifolds of invariant objects (such as libration points and periodic orbits) have been increasingly recognized as an efficient tool for designing transfer trajectories in space missions. However, most methods currently used in mission design rely on using eigenvectors of the linearized dynamics as local approximations of the manifolds. Since such approximations are not accurate except very close to the base invariant object, this requires large amounts of numerical integration to globalize the manifolds and locate intersections. In this paper, we study hyperbolic resonant periodic orbits in the planar circular restricted 3-body problem, and transfer trajectories between them, by: 1) determining where to search for resonant periodic orbits; 2) developing and implementing a parameterization method for accurate computation of their invariant manifolds as Taylor series; and 3) developing a procedure to compute intersections of the computed stable and unstable manifolds. We develop and implement algorithms that accomplish these three goals, and demonstrate their application to the problem of transferring between resonances in the Jupiter-Europa system.

math.DS↗

arXiv2021

The existence of solutions for nonlinear elliptic equations: Simple proofs and extensions of a paper by Y. Shi

The paper [Shi19] uses the Craig-Wayne-Bourgain method to construct solutions of an elliptic problem involving parameters. The results of [Shi19] include regularity assumptions on the perturbation and involve excluding parameters. The paper [Shi19] also constructs response solutions to a quasi-periodically perturbed (ill-posed evolution) problem. In this paper, we use several classical methods (freezing of coefficients, alternative methods for nonlinear elliptic equations) to extend the results of [Shi19]. We weaken the regularity assumptions on the perturbation and we describe the phenomena that happens for all parameters. In the ill-posed problem, we use a recently developed time-dependent center manifold theorem which allows to reduce the problem to a finite-dimensional ODE with quasi-periodic dependence on time. The bounded and sufficiently small solutions of these ODE give solutions of the ill-posed PDE.

math.AP↗

arXiv2021

KAM quasi-periodic tori for the dissipative spin-orbit problem

We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold. We consider the spin-orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity. Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter. The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in [CCGdlL20a]. The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing that standard numerics. We hope that our work could stimulate a computer-assisted proof.

math.DS↗