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Tere M-Seara

Publications and source records attributed to Tere M-Seara.

8 recordsLinked to original sources
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.

math-ph↗
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 $\omega$ on a manifold M into a multiple of itself, $f^* \omega = \eta \omega$. We assume $\omega$ 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 $\eta$ 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 $\eta$) 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 $\omega$ is presymplectic. In dynamical systems with many rates (e.g., quasi-integrable systems near multiple resonances), pre-symplectic geometries appear naturally.

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arXiv2024

On the boundedness of solutions of a forced discontinuous oscillator

We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact symplectic map when written in suitable canonical coordinates. By an accurate study of the behaviour of the map for large amplitudes and by employing a parametrization KAM theorem, we show that the periodic solutions of the unperturbed oscillator persist as two-dimensional tori under conditions that depend on the Diophantine conditions of the frequency, but are independent on both the amplitude of the orbit and of the specific value of the frequency. This allows the construction of a sequence of nested invariant tori of increasing amplitude that confine the solutions within them, ensuring their boundedness. The same construction may be useful to address such persistence problem for a larger class of non-smooth forced oscillators.

math.DS↗
arXiv2023

Arnold diffusion in a model of dissipative system

For a mechanical system consisting of a rotator and a pendulum coupled via a small, time-periodic Hamiltonian perturbation, the Arnold diffusion problem asserts the existence of `diffusing orbits' along which the energy of the rotator grows by an amount independent of the size of the coupling parameter, for all sufficiently small values of the coupling parameter. There is a vast literature on establishing Arnold diffusion for such systems. In this work, we consider the case when an additional, dissipative perturbation is added to the rotator-pendulum system with coupling. Therefore, the system obtained is not symplectic but conformally symplectic. We provide explicit conditions on the dissipation parameter, so that the resulting system still exhibits energy growth. The fact that Arnold diffusion may play a role in systems with small dissipation was conjectured by Chirikov. In this work, the coupling is carefully chosen, however the mechanism we present can be adapted to general couplings and we will deal with the general case in future work.

math.DS↗
arXiv2022

Generalised analytical results on $n$-ejection-collision orbits in the RTBP. Analysis of bifurcations

In the planar circular restricted three-body problem and for any value of the mass parameter $μ\in (0,1)$ and $n\ge 1$, we prove the existence of four families of $n$-ejection-collision ($n$-EC) orbits, that is, orbits where the particle ejects from a primary, reaches $n$ maxima in the distance with respect to it and finally collides with the primary. Such EC orbits have a value of the Jacobi constant of the form $C=3μ+Ln^{2/3}(1-μ)^{2/3}$, where $L>0$ is big enough but independent of $μ$ and $n$. In order to prove this optimal result, we consider Levi-Civita's transformation to regularize the collision with one primary and a perturbative approach using an ad hoc small parameter once a suitable scale in the configuration plane and time has previously been applied. This result improves a previous work where the existence of the $n$-EC orbits was stated when the mass parameter $μ>0$ was small enough. In this paper, any possible value of $μ\in (0,1)$ and $n\ge 1$ is considered. Moreover, for decreasing values of $C$, there appear some bifurcations which are first numerically investigated and afterwards explicit expressions for the approximation of the bifurcation values of $C$ are discussed. Finally, a detailed analysis of the existence of $n$-EC orbits when $μ\to 1$ is also described. In a natural way Hill's problem shows up. For this problem, we prove an analytical result on the existence of four families of $n$-EC orbits and numerically we describe them as well as the appearing bifurcations.

math.DS↗
arXiv2021

Two regularizations of the grazing-sliding bifurcation giving non equivalent dynamics

We present two ways of regularizing a one parameter family of piece-wise smooth dynamical systems undergoing a codimension one grazing-sliding global bifurcation of periodic orbits. First we use the Sotomayor-Teixeira regularization and prove that the regularized family has a saddle-node bifurcation of periodic orbits. Then we perform a hysteretic regularization and show that the regularized family has chaotic dynamics. Our result shows that, in spite that the two regularizations will give the same dynamics in the sliding modes, when a tangency appears the hysteretic process generates chaotic dynamics.

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arXiv2018

Regularization around a generic codimension one fold-fold singularity

This paper is devoted to study the generic fold-fold singularity of Filippov systems on the plane, its unfoldings and its Sotomayor-Teixeira regularization. We work with general Filippov systems and provide the bifurcation diagrams of the fold-fold singularity and their unfoldings, proving that, under some generic conditions, is a codimension one embedded submanifold of the set of all Filippov systems. The regularization of this singularity is studied and its bifurcation diagram is shown. In the visible-invisible case, the use of geometric singular perturbation theory has been useful to give the complete diagram of the unfolding, specially the appearance and disappearance of periodic orbits that are not present in the Filippov vector field. In the case of a linear regularization, we prove that the regularized system is equivalent to a general slow-fast system studied by Krupa and Szmolyan [KrupaS01].

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