arXiv subjects

Joan Gimeno

Publications and source records attributed to Joan Gimeno.

15 recordsLinked to original sources

arXiv2026

Families of relative periodic orbits in the planar three-body problem via consecutive alignments

Relative periodic orbits (RPOs) are solutions of the three-body problem that are periodic in a uniformly rotating reference frame and, in general, quasi-periodic in inertial coordinates. We present a numerical procedure for computing and continuing one-parameter families of RPOs of the planar Newtonian three-body problem. The method exploits consecutive syzygies, understood here as configurations in which the three bodies are aligned and their velocities satisfy the corresponding symmetry conditions. Matching the positions and momenta at two consecutive alignments reduces the computation of RPOs to a low-dimensional nonlinear problem. Its solutions are then numerically continued, and linear stability is determined from the nontrivial eigenvalues of the rotated monodromy matrix after removing the neutral directions associated with conserved quantities and continuous symmetries. The procedure is applied to several mass distributions and initial configurations, producing families of Poincaré, Hill, and binary-type solutions. These families exhibit transitions from nearly circular to highly eccentric motion, changes of stability near resonances and turning points, and absolute periodic solutions when the rotation angle is a rational multiple of 2π. In the Hill families, the continuation connects satellite configurations with circumstellar motion as the smallest body loses its gravitational binding to the intermediate body. Circumbinary and circumstellar configurations are also obtained in the binary regime. The results illustrate the dynamical diversity of RPOs and provide coherent three-body motions that can be used as prescribed trajectories in restricted four-body models.

math.DS↗

arXiv2026

Numerical Computation of Quasiperiodic Reducible Saddle-Node Bifurcations: a Parameterization Method Approach

We present a method for computing reducible, normally hyperbolic, invariant tori with internal quasiperiodic dynamics in autonomous ordinary differential equation systems. The approach is based on the parameterization method of KAM theory; thus, it is a Newton scheme with small divisors. Since the inner dynamics of the torus is prescribed, the corresponding system parameters for which such a torus exists are simultaneously determined. The method is amenable to a form of pseudo-arclength continuation, enabling the traversal and computation of saddle-node bifurcations. We give explicit algorithms for the methods and demonstrate their applicability with two numerical examples.

math.DS↗

arXiv2026

Effective Stability of Near-Rectilinear Halo Orbits in the Earth-Moon System

Near-rectilinear halo orbits (NRHOs) around Earth-Moon L2 in the Circular Restricted 3-Body Problem (CR3BP) exhibit a complex dynamical landscape, featuring a band of normally elliptic orbits embedded within regions of strong instability. This coexistence of stable and unstable dynamics, amplified by the numerical sensitivity associated with close lunar passages, makes the long-term behavior of trajectories near NRHOs a delicate and intrinsically nonlinear problem. Understanding the effective stability of these elliptic orbits is therefore a critical challenge, lying at the intersection of local normal form theory and global instability mechanisms. To quantify finite-time confinement, we formulate a rigorous framework for effective stability using discrete Poincaré maps. By employing jet transport to compute high-order Taylor expansions, we construct explicit polynomial normal forms. We derive discrete Nekhoroshev-type estimates by identifying the normalization order, which balances the asymptotic convergence of the map's analyticity domain against the cumulative penalty of low-order small divisors. Applying this framework to the Earth-Moon system, we map the resulting geometric limits directly into physical spatial coordinates. Crucially, we demonstrate that for practical mission lifetimes (e.g., 10-50 years), the required stability is vastly shorter than the characteristic Nekhoroshev accumulation time. Consequently, the effective stability region is not constrained by the time-dependent exponential drift, but is instead governed entirely by the maximum analytical domain of the optimized normal form. These derived spatial envelopes establish explicit geometric boundaries for the intrinsic local stability of elliptic NRHOs, providing a rigorous mathematical characterization of their nonlinear confinement within the CR3BP.

math.DS↗

arXiv2026

Lag-Induced Critical Transitions to Extinction in Replicating Systems

Replicating systems sustained by error-prone enzymatic amplification can undergo critical transitions between persistence and extinction. In RNA viruses, such transitions are classically governed by mutation rates and fitness landscapes, giving rise to error thresholds and lethal mutagenesis. Motivated by experimental evidence that polymerase-targeting antivirals constrain replication, we analyze replicating systems with explicit delays in replication-enzyme availability. We identify a lag-induced (dynamical) critical transition driven by the loss of temporal coordination between genome translation and replication. At a fixed mutation rate and replicative fitness landscape, populations cross an extinction threshold solely due to time delays. Within the quasispecies framework, replication-translation timing emerges as an independent control parameter, defining a distinct dynamical route to extinction and suggesting new antiviral strategies based on modulating replicase availability. More generally, we propose that the pathway to collapse described in this article can be understood as lag-time-induced tipping (τ-tipping).

q-bio.PE↗

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

Numerical computation of high-order expansions of invariant manifolds of high-dimensional tori

In this paper we present a procedure to compute reducible invariant tori and their stable and unstable manifolds in stroboscopic Poincaré maps. The method has two steps. In the first step we compute, by means of a quadratically convergent scheme, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. If the torus has stable and/or unstable directions, in the second step we compute the Taylor-Fourier expansions of the corresponding invariant manifolds up to a given order. The paper also discusses the case in which the torus is highly unstable so that a multiple shooting strategy is needed to compute the torus. If the order of the Taylor expansion of the manifolds is fixed and N is the number of Fourier modes, the whole computational effort (torus and manifolds) increases as O(N log N) q and the memory required behaves as O(N). This makes the algorithm very suitable to compute high-dimensional tori for which a huge number of Fourier modes are needed. Besides, the algorithm has a very high degree of parallelism. The paper includes examples where we compute invariant tori (of dimensions up to 5) of quasi-periodically forced ODEs. The computations are run in a parallel computer and its efficiency with respect to the number of processors is also discussed.

math.DS↗

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↗

arXiv2022

Deep Non-Crossing Quantiles through the Partial Derivative

Quantile Regression (QR) provides a way to approximate a single conditional quantile. To have a more informative description of the conditional distribution, QR can be merged with deep learning techniques to simultaneously estimate multiple quantiles. However, the minimisation of the QR-loss function does not guarantee non-crossing quantiles, which affects the validity of such predictions and introduces a critical issue in certain scenarios. In this article, we propose a generic deep learning algorithm for predicting an arbitrary number of quantiles that ensures the quantile monotonicity constraint up to the machine precision and maintains its modelling performance with respect to alternative models. The presented method is evaluated over several real-world datasets obtaining state-of-the-art results as well as showing that it scales to large-size data sets.

cs.LG↗

arXiv2021

Persistence of Periodic Orbits under State-dependent Delayed Perturbations: Computer-assisted Proofs

A computer-assisted argument is given, which provides existence proofs for periodic orbits in state-dependent delayed perturbations of ordinary differential equations (ODEs). Assuming that the unperturbed ODE has an isolated periodic orbit, we introduce a set of polynomial inequalities whose successful verification leads to the existence of periodic orbits in the perturbed delay equation. We present a general algorithm, which describes a way of computing the coefficients of the polynomials and optimizing their variables so that the polynomial inequalities are satisfied. The algorithm uses the tools of validated numerics together with Chebyshev series expansion to obtain the periodic orbit of the ODE as well as the solution of the variational equations, which are both used to compute rigorously the coefficients of the polynomials. We apply our algorithm to prove the existence of periodic orbits in a state-dependent delayed perturbation of the van der Pol equation.

math.DS↗

arXiv2021

The spin-spin problem in Celestial Mechanics

We study the dynamics of two homogeneous rigid ellipsoids subject to their mutual gravitational influence. We assume that the spin axis of each ellipsoid coincides with its shortest physical axis and is perpendicular to the orbital plane. Due to such assumptions, the problem is planar and depends on particular parameters of the ellipsoids, most notably, the equatorial oblateness and the flattening with respect to the shortest physical axes. We consider two models for such configuration: while in the full model, there is a coupling between the orbital and rotational motions, in the Keplerian model, the centers of mass of the bodies are constrained to move on coplanar Keplerian ellipses. The Keplerian case, in the approximation that includes the coupling between the spins of the two ellipsoids, is what we call spin-spin problem, that is a generalization of the classical spin-orbit problem. In this paper we continue the investigations of [Mis21] on the spin-spin problem by comparing it with the spin-orbit problem and also with the full model. Beside detailing the models associated to the spin-orbit and spin-spin problems, we introduce the notions of standard and balanced resonances, which lead us to investigate the existence of periodic and quasi-periodic solutions. We also give a qualitative description of the phase space and provide results on the linear stability of solutions for the spin-orbit and spin-spin problems. We conclude by providing a comparison between the full and the Keplerian models with particular reference to the interaction between the rotational and orbital motions.

math.DS↗

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↗

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.

math.NA↗

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.

math.DS↗

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.

math.DS↗

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 $C^1$).

math.DS↗