arXiv subjects

Marcel Guardia

Publications and source records attributed to Marcel Guardia.

At least 19 recordsLinked to original sources

arXiv2026

Computation of a separatrix map and a normally hyperbolic invariant lamination for the RP3BP

In this paper we discuss the existence of a normally hyperbolic invariant lamination (NHIL) at the Kirkwood gap $3:1$ for the Restricted Planar Elliptic 3 Body Problem. This problem models the Sun-Jupiter-Asteroid dynamics. We also show that the induced dynamics on the NHIL is a partially hyperbolic skew-shift which is of the form \[ f:(\omega,I,\theta)\to (\sigma \omega, I+e_0 A_\omega(I)\cos(\theta+\psi_\omega)+\mathcal{O}(e^2_0), \theta+\Omega_\omega(I)+\mathcal{O}(e_0)),\] where $I\in [a,b], \theta\in \mathbb T, \omega\in\Sigma=\{0,1\}^\mathbb Z$, the space of sequences of $0,1$'s, $\sigma:\Sigma \to \Sigma$ is the shift in this space, $\Omega_\omega$ is the shear, $A_\omega$ is an amplitude, and $e_0$ is the eccentricity of Jupiter, which is taken as a small parameter. In the companion paper arXiv:2603.19894, relying on these skew-shift, we show the existence of stochastic diffusing behavior for Asteroids belonging to the Kirkwood gap provided the eccentricity of Jupiter is $e_0$ small enough. Key ingredients to construct the NHIL are the separatrix map associated to homoclinic channels to a normally hyperbolic invariant cylinder and an isolating block construction. Some of the necessary non-degeneracy conditions are verified numerically.

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arXiv2026

Stochastic behavior along mean motion resonances in the restricted planar 3-body problem

One of the most remarkable instability zones in the Solar system are Kirkwood gaps in the asteroid belt. In this paper we analyze instabilities in the famous Kirkwood gap $3:1$ in the regime of small eccentricity of Jupiter. Mathematically speaking, we study the evolution of asteroids under the influence of the Sun and Jupiter using the restricted planar elliptic 3-body problem (RPE3BP) for initial conditions near a mean motion resonance 3:1. The main result exhibits stochastic diffusing behavior of the eccentricity of the asteroid for a rich set of initial conditions. Roughly speaking, for small eccentricity $e_0$ of Jupiter, the evolution of the eccentricity of the asteroid $\mathbf{e}(t\cdot e_0^{-2})$ at the Kirkwood gap $3:1$ behaves like a diffusion process on the line, where the randomness comes from the initial conditions. Along with KAM theory, we have mixed behavior in the asteroid belt, that is coexistence of quasiperiodic (deterministic) and stochastic (diffusive) behavior. See also the companion paper arXiv:2603.19893

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arXiv2025

Second-species dynamics in the restricted planar circular three body problem: chaos, final motions and periodic orbits

Consider the Restricted Planar Circular Three Body Problem (RPC3BP), which models the motion of a massless particle (Asteroid) under the gravitational influence of two massive bodies (the primaries) moving on circular orbits. By considering the ratio between the masses of the primaries to be arbitrarily small, we construct orbits with close encounters with the smaller primary (Jupiter) that realize any combination of past and future final motions (in the sense of Chazy's), including oscillatory motions. We also obtain arbitrarily large ejection-collision orbits with Jupiter and ejection-collision orbits between the two primaries (Sun and Jupiter), as well as arbitrarily large periodic orbits that pass arbitrarily close to Jupiter. Our approach combines singular perturbation theory and Levi-Civita regularization near Jupiter, and McGehee regularization near infinity and near the Sun, together with a global analysis that leads to transverse intersections of invariant manifolds.

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arXiv2025

Partially hyperbolic dynamics in the 3-body problem

We construct symplectic blenders for two classical Hamiltonian systems: the 3-body problem and its restricted version. We use these objects to show that both models exhibit a robust, strong form of topological instability. We do not assume any smallness conditions on the masses but require only that at least two of them are distinct. Our construction is based on two abstract results which might be of independent interest. The first one gives an explicit condition under which a given pair of twist maps of the cylinder generates a locally transitive iterated function system. The second one extends this result to certain cylinder skew-products.

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arXiv2024

On a countable sequence of homoclinic orbits arising near a saddle-center point

Exponential small splitting of separatrices in the singular perturbation theory leads generally to nonvanishing oscillations near a saddle--center point and to nonexistence of a true homoclinic orbit. It was conjectured long ago that the oscillations may vanish at a countable set of small parameter values if there exist a quadruplet of singularities in the complex analytic extension of the limiting homoclinic orbit. The present paper gives a rigorous proof of this conjecture for a particular fourth-order equation relevant to the traveling wave reduction of the modified Korteweg--de Vries equation with the fifth-order dispersion term.

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arXiv2024

Oscillatory motions, parabolic orbits and collision orbits in the planar circular restricted three-body problem

In this paper we consider the planar circular restricted three body problem (PCRTBP), which models the motion of a massless body under the attraction of other two bodies, the primaries, which describe circular orbits around their common center of mass. In a suitable system of coordinates, this is a two degrees of freedom Hamiltonian system. The orbits of this system are either defined for all (future or past) time or eventually go to collision with one of the primaries. For orbits defined for all time, Chazy provided a classification of all possible asymptotic behaviors, usually called final motions. By considering a sufficiently small mass ratio between the primaries, we analyze the interplay between collision orbits and various final motions and construct several types of dynamics. In particular, we show that orbits corresponding to any combination of past and future final motions can be created to pass arbitrarily close to the massive primary. Additionally, we construct arbitrarily large ejection-collision orbits (orbits which experience collision in both past and future times) and periodic orbits that are arbitrarily large and get arbitrarily close to the massive primary. Furthermore, we also establish oscillatory motions in both position and velocity, meaning that as time tends to infinity, the superior limit of the position or velocity is infinity while the inferior limit remains a real number.

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arXiv2024

On the role of the fast oscillations in the secular dynamics of the lunar coplanar perturbation on Galileo satellites

Motivated by the practical interest in the third-body perturbation as a natural cleaning mechanism for high-altitude Earth orbits, we investigate the dynamics stemming from the secular Hamiltonian associated with the lunar perturbation, assuming that the Moon lies on the ecliptic plane. The secular Hamiltonian defined in that way is characterized by two timescales. We compare the location and stability of the fixed points associated with the secular Hamiltonian averaged with respect to the fast variable with the corresponding periodic orbits of the full system. Focusing on the orbit of the Galileo satellites, it turns out that the two dynamics cannot be confused, as the relative difference depends on the ratio between the semi-major axis of Galileo and the one of the Moon, that is not negligible. The result is relevant to construct rigorously the Arnold diffusion mechanism that can drive a natural growth in eccentricity that allows a satellite initially on a circular orbit in Medium Earth Orbit to reenter into the Earth's atmosphere.

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arXiv2024

On the Arnold diffusion mechanism in Medium Earth Orbit

Space debris mitigation guidelines represent the most effective method to preserve the circumterrestrial environment. Among them, end-of-life disposal solutions play a key role. A growing effort is devoted to exploit natural perturbations to lead the satellites towards an atmospheric reentry, reducing the disposal cost, also if departing from high-altitude regions. In the case of the Medium Earth Orbit region, home of the navigation satellites (like Galileo), the main driver is the gravitational perturbation due to the Moon, that can increase the eccentricity in the long term. In this way, the pericenter altitude can get into the atmospheric drag domain and the satellite can eventually reenter. In this work, we show how an Arnold diffusion mechanism can trigger the eccentricity growth. Focusing on the case of Galileo, we consider a hierarchy of Hamiltonian models, assuming that the main perturbations on the motion of the spacecraft are the oblateness of the Earth and the gravitational attraction of the Moon. First, the Moon is assumed to lay on the ecliptic plane and periodic orbits and associated stable and unstable invariant manifolds are computed for various energy levels, in the neighborhood of a given resonance. Along each invariant manifold, the eccentricity increases naturally, achieving its maximum at the first intersection between them. This growth is, however, not sufficient to achieve reentry. By moving to a model where the inclination of the Moon is taken into account, the problem becomes non-autonomous and the satellite is able to move along different energy levels. Under the ansatz of transversality of the manifolds in the autonomous case, checked numerically, Poincar\'e-Melnikov techniques are applied to show how diffusion can be attained, by constructing a sequence of homoclinic orbits that connect invariant tori at different energy levels.

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arXiv2023

Coorbital homoclinic and chaotic dynamics in the Restricted 3-Body Problem

The description of unstable motions in the Restricted Planar Circular 3-Body Problem, modeling the dynamics of a Sun-Planet-Asteriod system, is one of the fundamental problems in Celestial Mechanics. The goal of this paper is to analyze homoclinic and instability phenomena at coorbital motions, that is when the negligible mass Asteroid is at 1:1 mean motion resonance with the Planet (i.e. nearly equal periods) and performs close to circular motions. Several bodies in our Solar system belong to such regimes. In this paper, we obtain the following results. First, we prove that, for a sequence of ratios between the masses of the Planet and the Sun going to 0, there exist a 2-round homoclinic orbit to the Lagrange point L3, i.e. homoclinic orbits that approach the critical point twice. Second, we construct chaotic motions (hyperbolic sets with symbolic dynamics) as a consequence of the existence of transverse homoclinic orbits to Lyapunov periodic orbits associated to L3. Finally, we prove that the RPC3BP possesses Newhouse domains by proving that the energy level unfolds generically a quadratic homoclinic tangency to a periodic orbit.

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arXiv2023

Breakdown of homoclinic orbits to L3: Nonvanishing of the Stokes constant

The Restricted Planar Circular 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries, which perform circular orbits coplanar with that of the massless body. In rotating coordinates, it can be modelled by a two degrees of freedom Hamiltonian system, which has five critical points called the Lagrange points. Among them, the point L3 is a saddle-center which is collinear with the primaries and beyond the largest of the two. The papers arXiv:2107.09942 and arXiv:2107.09941 provide an asymptotic formula for the distance between the one dimensional stable and unstable manifolds of L3 in a transverse section for small values of the mass ratio $\mu$. This distance is exponentially small with respect to $\mu$ and its first order depends on what is usually called a Stokes constant. The non-vanishing of this constant implies that the distance between the invariant manifolds at the section is not zero. In this paper, we prove that the Stokes constant is non-zero. The proof is computer assisted.

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arXiv2023

A remark on the onset of resonance overlap

Chirikov's celebrated criterion of resonance overlap has been widely used in celestial mechanics and Hamiltonian dynamics to detect global instability, but is rarely rigourous. We introduce two simple Hamiltonian systems, each depending on two parameters measuring respectively the distance to resonance overlap and non-integrability. Within some thin region of the parameter plane, classical perturbation theory shows the existence of global instability and symbolic dynamics, thus illustrating Chirikov's criterion.

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arXiv2023

A counterexample to the theorem of Laplace-Lagrange on the stability of semimajor axes

A longstanding belief has been that the semimajor axes, in the Newtonian planetary problem, are stable. In the course of the XIX century, Laplace, Lagrange and others gave stronger and stronger arguments in this direction, thus culminating in what has commonly been referred to as the first Laplace-Lagrange stability theorem. In the problem with 3 planets, we prove the existence of orbits along which the semimajor axis of the outer planet undergoes large random variations thus disproving the theorem of Laplace-Lagrange. The time of instability varies as a negative power of the masses of the planets. The orbits we have found fall outside the scope of the theory of Nekhoroshev-Niederman because they are not confined by the conservation of angular momentum and because the Hamiltonian is not (uniformly) convex with respect to the Keplerian actions.

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arXiv2023

Effective chaos for the Kirchhoff equation on tori

We consider the Kirchhoff equation on tori of any dimension and we construct solutions whose Sobolev norms oscillates in a chaotic way on certain long time scales. The chaoticity is encoded in the time between oscillations of the norm, which can be chosen in any prescribed way. This phenomenon, that we name as effective chaos (it occurs over a long, but finite, time scale), is consequence of the existence of symbolic dynamics for an effective system. Since the first order resonant dynamics has been proved to be essentially stable, we need to perform a second order analysis to find an effective model displaying chaotic dynamics. More precisely, after some reductions, this model behaves as two weakly coupled pendulums.

math.AP↗

arXiv2023

A degenerate Arnold diffusion mechanism in the Restricted 3 Body Problem

A major question in dynamical systems is to understand the mechanisms driving global instability in the 3 Body Problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0,m_1$. We consider the Restricted Planar Elliptic 3 Body Problem (RPE3BP) where the primaries revolve in Keplerian ellipses. We prove that the RPE3BP exhibits topological instability: for any values of the masses $m_0,m_1$ (except $m_0=m_1$), we build orbits along which the angular momentum of the massless body experiences an arbitrarily large variation provided the eccentricity of the orbit of the primaries is positive but small enough. In order to prove this result we show that a degenerate Arnold Diffusion Mechanism, which moreover involves exponentially small phenomena, takes place in the RPE3BP. Our work extends the one by Delshams, Kaloshin, De la Rosa, Seara (2019) for the a priori unstable case $m_1/m_0\ll1$, to the case of arbitrary masses $m_0,m_1>0$, where the model displays features of the so-called a priori stable setting.

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arXiv2022

Why are inner planets not inclined?

Poincar\'e's work more than one century ago, or Laskar's numerical simulations from the 1990's on, have irrevocably impaired the long-held belief that the Solar System should be stable. But mathematical mechanisms explaining this instability have remained mysterious. In 1968, Arnold conjectured the existence of "Arnold diffusion" in celestial mechanics. We prove Arnold's conjecture in the planetary spatial $4$-body problem as well as in the corresponding hierarchical problem (where the bodies are increasingly separated), and show that this diffusion leads, on a long time interval, to some large-scale instability. Along the diffusive orbits, the mutual inclination of the two inner planets is close to $\pi/2$, which hints at why even marginal stability in planetary systems may exist only when inner planets are not inclined. More precisely, consider the normalised angular momentum of the second planet, obtained by rescaling the angular momentum by the square root of its semimajor axis and by an adequate mass factor (its direction and norm give the plane of revolution and the eccentricity of the second planet). It is a vector of the unit $3$-ball. We show that any finite sequence in this ball may be realised, up to an arbitrary precision, as a sequence of values of the normalised angular momentum in the $4$-body problem. For example, the second planet may flip from prograde nearly horizontal revolutions to retrograde ones. As a consequence of the proof, the non-recurrent set of any finite-order secular normal form accumulates on circular motions -- a weak form of a celebrated conjecture of Herman.

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arXiv2022

Hyperbolic dynamics and oscillatory motions in the 3 Body Problem

Consider the planar 3 Body Problem with masses $m_0,m_1,m_2>0$. In this paper we address two fundamental questions: the existence of oscillatory motions and of chaotic hyperbolic sets. In 1922, Chazy classified the possible final motions of the three bodies, that is the behaviors the bodies may have when time tends to infinity. One of the possible behaviors are oscillatory motions, that is, solutions of the 3 Body Problem such that the positions of the bodies $q_0, q_1, q_2$ satisfy \[ \liminf_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|<+\infty \quad \text{ and }\quad \limsup_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|=+\infty. \] Assume that all three masses $m_0,m_1,m_2>0$ are not equal. Then, we prove that such motions exists. We also prove that one can construct solutions of the three body problem whose forward and backward final motions are of different type. This result relies on constructing invariant sets whose dynamics is conjugated to the (infinite symbols) Bernouilli shift. These sets are hyperbolic for the symplectically reduced planar 3 Body Problem. As a consequence, we obtain the existence of chaotic motions, an infinite number of periodic orbits and positive topological entropy for the 3 Body Problem.

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arXiv2022

Breakdown of heteroclinic connections in the analytic Hopf-Zero singularity: Rigorous computation of the Stokes constant

Consider analytic generic unfoldings of the three dimensional conservative Hopf-Zero singularity. Under open conditions on the parameters determining the singularity, the unfolding possesses two saddle-foci when the unfolding parameter is small enough. One of them has one dimensional stable manifold and two dimensional unstable manifold whereas the other one has one dimensional unstable manifold and two dimensional stable manifold. Baldom\'a, Castej\'on and Seara [BCS13] gave an asymptotic formula for the distance between the one dimensional invariant manifolds in a suitable transverse section. This distance is exponentially small with respect to the perturbative parameter, and it depends on what is usually called a Stokes constant. The non-vanishing of this constant implies that the distance between the invariant manifolds at the section is not zero. However, up to now there do not exist analytic techniques to check that condition. In this paper we provide a method for obtaining accurate rigorous computer assisted bounds for the Stokes constant. We apply it to two concrete unfoldings of the Hopf-Zero singularity, obtaining a computer assisted proof that the constant is non-zero.

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arXiv2022

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

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

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