arXiv subjects

Pau Martín

Publications and source records attributed to Pau Martín.

17 recordsLinked to original sources

arXiv2026

Degenerate fixed points of maps in Banach spaces and lattices with decay and their invariant manifolds

Degenerate fixed points of maps appear in many interesting problems in Celestial Mechanics, Economics and Chemistry. Lattice systems, that is, dynamical systems consisting in an infinite array of finite dimensional subsystems interacting locally among them appear in many models of Biology, Physics and Mathematics. In this work, we extend the results concerning the existence of stable and unstable invariant manifolds of degenerate fixed points to lattice systems with decay properties. As an example, we find such manifolds in perturbations of the Toda lattice.

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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 \[ f:(ω,I,θ)\to (σω, I+e_0 A_ω(I)\cos(θ+ψ_ω)+\mathcal{O}(e^2_0), θ+Ω_ω(I)+\mathcal{O}(e_0)),\] where $I\in [a,b], θ\in \mathbb T, ω\inΣ=\{0,1\}^\mathbb Z$, the space of sequences of $0,1$'s, $σ:Σ\to Σ$ is the shift in this space, $Ω_ω$ is the shear, $A_ω$ 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

Chaotic phenomena in generic unfoldings of the Hamilton Hopf bifurcation with emphasis on the restricted planar circular 3-body problem beyond the Gascheau-Routh mass ratio

In this work, we prove that a generic unfolding of an analytic Hamiltonian Hopf singularity (in an open set with codimension 1 boundary) possesses transverse homoclinic orbits for subcritical values of the parameter close to the bifurcation parameter. As a consequence, these systems display chaotic dynamics with arbitrarily large topological entropy. We verify that the Hamiltonian of the restricted planar circular three-body problem (RPC3BP) close to the Lagrangian point $L_4$ falls within this open set. The generic condition ensuring the presence of transversal homoclinic intersections is subtle and involves the so-called Stokes constant. Thus, in the case of the RPC3BP close to $L_4$, our result holds conditionally on the value of this constant.

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arXiv2024

Parabolic saddles and Newhouse domains in Celestial Mechanics

In the 70s McGehee introduced a compactification of the phase space of the restricted 3-body problem by gluing a manifold of periodic orbits "at infinity". Although from the dynamical point of view these periodic orbits are parabolic (the linearization of the Poincaré map is the identity matrix), one of them, denoted here by $O$, possesses stable and unstable manifolds which, moreover, separate the regions of bounded and unbounded motion. This observation prompted the investigation of the homoclinic picture associated to $O$, starting with the work of Alekseev and Moser. We continue this research and extend, to this degenerate setting, some classical results in the theory of homoclinic bifurcations. More concretely, we prove that there exist Newhouse domains $\mathcal N$ in parameter space (the ratio of masses of the bodies) and residual subsets $\mathcal R\subset \mathcal N$ for which the homoclinic class of $O$ has maximal Hausdorff dimension and is accumulated by generic elliptic periodic orbits. One of the main consequences of our work is the fact that, for a (locally) topologically large set of parameters of the restricted 3-body problem the union of its elliptic islands forms an unbounded subset of the phase space and, moreover, the closure of the set of generic elliptic periodic orbits contains hyperbolic sets with Hausdorff dimension arbitrarily close to maximal. Other instances of the restricted $n$-body problem such as the Sitnikov problem and the case $n=4$ are also considered.

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arXiv2024

Invariant manifolds of degenerate tori and double parabolic orbits to infinity in the (n+2)-body problem

There are many interesting dynamical systems in which degenerate invariant tori appear. We give conditions under which these degenerate tori have stable and unstable invariant manifolds, with stable and unstable directions having arbitrary finite dimension. The setting in which the dimension is larger than one was not previously considered and is technically more involved because in such case the invariant manifolds do not have, in general, polynomial approximations. As an example, we apply our theorem to prove that there are motions in the $(n+2)$-body problem in which the distances among the first $n$ bodies remain bounded for all time, while the relative distances between the first $n$-bodies and the last two and the distances between the last bodies tend to infinity, when time goes to infinity. Moreover, we prove that the final motion of the first $n$ bodies corresponds to a KAM torus of the $n$-body problem.

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arXiv2024

Chaotic scattering of He atoms off a Cu surface with corrugated Morse potential

We consider a Hamiltonian system that models the scattering of helium atoms off a copper surface. The interaction between the He and the Cu atoms is described by a corrugated Morse potential. Using corrugation coefficients values in the potential obtained by fitting to experimental values, we prove that, provided some coefficient of an auxiliary function is different from 0, there are regions of the phase space, corresponding to sufficiently large energy of the incident atom, where the scattering is chaotic. Furthermore, we prove that the system has oscillatory motions.

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arXiv2022

Novel slow-fast behaviour in an oscillator driven by a frequency-switching force

When an oscillator switches abruptly between different frequencies, there is some ambiguity in deciding how the system should be modelled at the switch. Here we describe two seemingly natural models of a switch in a simple periodically-forced harmonic oscillator, which disagree starkly in their predictions of its long time behaviour. Attempting to resolve the disagreement by `regularizing' the switch not only preserves the disagreement, but shows it increases with time. One of the models corresponds to a conventional `Filippov' description of a nonsmooth system, while the second exhibits a structure that irreversibly ages, developing a number of novel multi-scale behaviours that we believe have not been reported before. These include slow-fast staircases, novel mixed-mode oscillations, and a synchronized canard explosion. These features are proven to exist using asymptotic analysis, but as they involve a slow-fast time-scale separation that increases with time, they lie beyond the reach of numerical methods.

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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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arXiv2021

Oscillatory Motions and Parabolic Manifolds at Infinity in the Planar Circular Restricted Three Body Problem

Consider the Restricted Planar Circular 3 Body Problem with both realistic mass ratio and Jacobi constant for the Sun-Jupiter pair. We prove the existence of all possible combinations of past and future final motions. In particular, we obtain the existence of oscillatory motions. All the constructed trajectories cross the orbit of Jupiter but avoid close encounters with it. The proof relies on the method of correctly aligned windows and is computer assisted.

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arXiv2017

Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points

We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to $1$. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem.

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arXiv2016

Invariant manifolds of parabolic fixed points (I). Existence and dependence on parameters

In this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases, using the parametrization method. The parametrization method relies on a suitable approximate solution of a functional equation. In the case of parabolic points, if the manifolds have dimension two or higher, in general this approximation cannot be obtained in the ring of polynomials but as a sum of homogeneous functions and it is given in~\cite{BFM2015b}. Assuming a sufficiently good approximation is found, here we provide an "a posteriori" result which gives a true invariant manifold close to the approximated one. In the differentiable case, in some cases, there is a loss of regularity. We also consider the case of parabolic periodic orbits of periodic vector fields and the dependence of the manifolds on parameters. Examples are provided. We apply our method to prove that in several situations, namely, related to the parabolic infinity in the elliptic spatial three body problem, these invariant manifolds exist and do have polynomial approximations.

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arXiv2016

Invariant manifolds of parabolic fixed points (II). Approximations by sums of homogeneous functions

We study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields. If the dimension of these manifolds is two or greater, in general, it is not possible to obtain polynomial approximations. Here we develop an algorithm to obtain them as sums of homogeneous functions by solving suitable cohomological equations. We deal with both the differentiable and analytic cases. We also study the dependence on parameters. In the companion paper, Invariant manifolds of parabolic fixed points (I), these approximations are used to obtain the existence of true invariant manifolds close by. Examples are provided.

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arXiv2015

Oscillatory orbits in the restricted elliptic planar three body problem

The restricted planar elliptic three body problem models the motion of a massless body under the Newtonian gravitational force of the two other bodies, the primaries, which evolve in Keplerian ellipses. A trajectory is called oscillatory if it leaves every bounded region but returns infinitely often to some fixed bounded region. We prove the existence of such type of trajectories for any values for the masses of the primaries provided they make almost circular orbits.

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arXiv2015

Exponentially small asymptotic formulas for the length spectrum in some billiard tables

Let $q \ge 3$ be a period. There are at least two $(1,q)$-periodic trajectories inside any smooth strictly convex billiard table, and all of them have the same length when the table is an ellipse or a circle. We quantify the chaotic dynamics of axisymmetric billiard tables close to their borders by studying the asymptotic behavior of the differences of the lengths of their axisymmetric $(1,q)$-periodic trajectories as $q \to +\infty$. Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor $q^{-3} e^{-r q}$ times either a constant or an oscillating function, and the exponent $r$ is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the $(1,q)$-periodic trajectories. Our experiments are restricted to some perturbed ellipses and circles, which allows us to compare the numerical results with some analytical predictions obtained by Melnikov methods and also to detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.

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arXiv2014

On the length and area spectrum of analytic convex domains

Area-preserving twist maps have at least two different $(p,q)$-periodic orbits and every $(p,q)$-periodic orbit has its $(p,q)$-periodic action for suitable couples $(p,q)$. We establish an exponentially small upper bound for the differences of $(p,q)$-periodic actions when the map is analytic on a $(m,n)$-resonant rotational invariant curve (resonant RIC) and $p/q$ is "sufficiently close" to $m/n$. The exponent in this upper bound is closely related to the analyticity strip width of a suitable angular variable. The result is obtained in two steps. First, we prove a Neishtadt-like theorem, in which the $n$-th power of the twist map is written as an integrable twist map plus an exponentially small remainder on the distance to the RIC. Second, we apply the MacKay-Meiss-Percival action principle. We apply our exponentially small upper bound to several billiard problems. The resonant RIC is a boundary of the phase space in almost all of them. For instance, we show that the lengths (respectively, areas) of all the $(1,q)$-periodic billiard (respectively, dual billiard) trajectories inside (respectively, outside) analytic strictly convex domains are exponentially close in the period $q$. This improves some classical results of Marvizi, Melrose, Colin de Verdière, Tabachnikov, and others about the smooth case.

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arXiv2012

The inner equation for generalized standard maps

We study particular solutions of the inner equation associated to the splitting of separatrices on generalized standard maps. An exponentially small complete expression for their difference is obtained. We also provide numerical evidence that the inner equation provides quantitative information of the splitting of separatrices even in the case when the limit flow does not.

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