arXiv subjects

Antonella Marchesiello

Publications and source records attributed to Antonella Marchesiello.

17 recordsLinked to original sources

arXiv2026

Perturbed isotropic oscillators with inversion symmetry

We consider the normal form of a 1:1 detuned resonance that can be related, via a double averaging process, to the three body problem of celestial mechanics. In view of the symmetry in the gravitational coupling between the planets, the obtained normal form exhibits inversion symmetry. We provide an analytical prediction of the possible bifurcations in terms of the detunings and the other physical parameters of the system. We show that the bifurcation scenario is rich. It includes bifurcation sequences where the unstable orbit appearing after a saddle-node bifurcation can have a reversed pitchfork bifurcation, where it turns into stability with the birth of two unstable orbits, which eventually reconnect with one of the two quasi-normal modes. Furthermore, another scenario shows a standard pitchfork in which one of the quasi-normal modes becomes unstable, giving rise to a pair of stable periodic orbits. By introducing coupling and asymmetry parameters, we provide an effective way to discern all the possible sequences of bifurcations.

math-ph↗

arXiv2026

Superintegrable 2D systems in magnetic fields with a parabolic type integral

We consider the problem on the existence of two dimensional superintegrable systems in the presence of a magnetic field in the two dimensional Euclidean space. We assume the existence of two integrals of motion, besides the Hamiltonian, that are quadratic polynomials in the momenta. This problem was already studied in the cases where one integral is of Cartesian or polar type [J. Bérubé, and P. Winternitz, J. Math. Phys., 45(5): 1959-1973, 2004]. We continue the investigation by assuming that one of the integrals is of parabolic type and the second integral is of elliptic or (''non-standard'') parabolic type, confirming so far that, on the Euclidean plane, the only two dimensional superintegrable system with quadratic integrals is the one with constant magnetic field and constant electrostatic potential.

math-ph↗

arXiv2025

Bifurcation sequences in the secular 3D planetary 3-Body problem: a geometric approach

We implement the geometric method proposed in ([9], [3], [16]) to analytically predict the sequence of bifurcations leading to a change of stability and/or the appearance of new periodic orbits in the secular 3D planetary three body problem. Stemming from the analysis in [17], we examine various normal form models as regards the extent to which they lead to a phase space dynamics qualitatively similar as that in the complete system. For fixed total angular momentum, the phase space in Hopf variables is the 3D sphere, and the complete sequence of bifurcations of new periodic orbits can be recovered through formulas yielding the tangencies or degenerate intersections between the sphere and the surfaces of a constant second integral of motion of the normal form flow. In particular, we find the critical values of the second integral giving rise to pitchfork and saddle-node bifurcations of new periodic orbits in the system. This analysis renders possible to predict the most important structural changes in the phase space, as well as the emergence of new possible stable periodic planetary orbital configurations which can take place as the mutual inclination between the two planets is allowed to increase.

math-ph↗

arXiv2024

Superintegrable families of magnetic monopoles with non-radial potential in curved background

We review some known results on the superintegrability of monopole systems in the three-dimensional (3D) Euclidean space and in the 3D generalized Taub-NUT spaces. We show that these results can be extended to certain curved backgrounds that, for suitable choice of the domain of the coordinates, can be related via conformal transformations to systems in Taub-NUT spaces. These include the multi-fold Kepler systems as special cases. The curvature of the space is not constant and depends on a rational parameter that is also related to the order of the integrals. New results on minimal superintegrability when the electrostatic potential depends on both radial and angular variables are also presented.

math-ph↗

arXiv2023

New classes of quadratically integrable systems with velocity dependent potentials: non-subgroup type cases

We study quadratic integrability of systems with velocity dependent potentials in three-dimensional Euclidean space. Unlike in the case with only scalar potential, quadratic integrability with velocity dependent potentials does not imply separability in the configuration space. The leading order terms in the pairs of commuting integrals can either generalize or have no relation to the forms leading to separation in the absence of a vector potential. We call such pairs of integrals generalized, to distinguish them from the standard ones, which would correspond to separation. Here we focus on three cases of generalized non-subgroup type integrals, namely elliptic cylindrical, prolate / oblate spheroidal and circular parabolic integrals, together with one case not related to any coordinate system. We find two new integrable systems, non-separable in the configuration space, both with generalized elliptic cylindrical integrals. In the other cases, all systems found were already known and possess standard pairs of integrals. In the limit of vanishing vector potential, both systems reduce to free motion and therefore separate in every orthogonal coordinate system.

math-ph↗

arXiv2020

On the detuned 2:4 resonance

We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials this concerns the short axial orbits and in galactic dynamics the resulting stable periodic orbits are called "banana" orbits. Galactic potentials are symmetric with respect to the co-ordinate planes whence the potential -- and the normal form -- both have no cubic terms. This $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetry turns the 1:2 resonance into a higher order resonance and one therefore also speaks of the 2:4 resonance. In this paper we study the 2:4 resonance in its own right, not restricted to natural Hamiltonian systems where $H = T + V$ would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and "anti-banana" orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.

nlin.CD↗

arXiv2020

Bifurcations and Monodromy of the Axially Symmetric 1:1:-2 Resonance

We consider integrable Hamiltonian systems in three degrees of freedom near an elliptic equilibrium in 1:1:-2 resonance. The integrability originates from averaging along the periodic motion of the quadratic part and an imposed rotational symmetry about the vertical axis. Introducing a detuning parameter we find a rich bifurcation diagram, containing three parabolas of Hamiltonian Hopf bifurcations that join at the origin. We describe the monodromy of the resulting ramified 3-torus bundle as variation of the detuning parameter lets the system pass through 1:1:-2 resonance.

math.DS↗

arXiv2020

Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems which are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previous paper [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages], where one of the additional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems separable in Cartesian coordinates. We also describe two particular methods for constructing superintegrable systems with higher-order integrals.

math-ph↗

arXiv2018

An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals

We construct an additional independent integral of motion for a class of three dimensional minimally superintegrable systems with constant magnetic field. This class was introduced in [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages] and it is known to possess periodic closed orbits. In the present paper we demonstrate that it is maximally superintegrable. Depending on the values of the parameters of the system, the newly found integral can be of arbitrarily high polynomial order in momenta.

math-ph↗

arXiv2015

Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance

We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant under $Z_2 \times Z_2$ symmetry. The rich structure of these classical systems is investigated with geometric methods and the relation with the singularity theory approach is also highlighted. The geometric approach is the most straightforward way to obtain a general picture of the phase-space dynamics of the family as is defined by a complete subset in the space of control parameters complying with the symmetry constraint. It is shown how to find an energy-momentum map describing the phase space structure of each member of the family, a catastrophe map that captures its global features and formal expressions for action-angle variables. Several examples, mainly taken from astrodynamics, are used as applications.

math.DS↗

arXiv2015

Weak dependence for a class of local functionals of Markov chains on $\mathbb{Z}^d$

In some papers on infinite Markov chains in $\mathbb{Z}^d$, and notably in the work of R.A. Minlos and collaborators, one can prove the existence of a spectral gap for a suitable subspace of local functions. We consider functions of the type $f(\widehat η)$, where $\widehat η= \{η_{t}\}_{t=0}^{\infty}$ is the sequence of the states, and $f$ is local. In the case of a simple example of random walk in random environment with mutual interaction we show that there is a natural class of functions $f$, related to the Hölder continuos functions $\mathcal C^α$on the torus $T^{1}$, with $α\in (0,1)$ large enough, depending on the spectral gap, for which the Central Limit Theorem holds for the sequence $f(S^{k}\widehat η)$, $k=0,1,\ldots$, where $S$ is the time shift.

math-ph↗

arXiv2015

Three-dimensional superintegrable systems in a static electromagnetic field

We consider a charged particle moving in a static electromagnetic field described by the vector potential $\vec A(\vec x)$ and the electrostatic potential $V(\vec x)$. We study the conditions on the structure of the integrals of motion of the first and second order in momenta, in particular how they are influenced by the gauge invariance of the problem. Next, we concentrate on the three possibilities for integrability arising from the first order integrals corresponding to three nonequivalent subalgebras of the Euclidean algebra, namely $(P_1,P_2)$, $(L_3,P_3)$ and $(L_1,L_2,L_3)$. For these cases we look for additional independent integrals of first or second order in the momenta. These would make the system superintegrable (minimally or maximally). We study their quantum spectra and classical equations of motion. In some cases nonpolynomial integrals of motion occur and ensure maximal superintegrability.

math-ph↗

arXiv2014

An energy-momentum map for the time-reversal symmetric 1:1 resonance with Z_2 X Z_2 symmetry

We present a general analysis of the bifurcation sequences of periodic orbits in general position of a family of reversible 1:1 resonant Hamiltonian normal forms invariant under $\Z_2\times\Z_2$ symmetry. The rich structure of these classical systems is investigated both with a singularity theory approach and geometric methods. The geometric approach readily allows to find an energy-momentum map describing the phase space structure of each member of the family and a catastrophe map that captures its global features. Quadrature formulas for the actions, periods and rotation number are also provided.

nlin.CD↗

arXiv2013

Equivariant singularity analysis of the 2:2 resonance

We present a general analysis of the bifurcation sequences of 2:2 resonant reversible Hamiltonian systems invariant under spatial $\Z_2\times\Z_2$ symmetry. The rich structure of these systems is investigated by a singularity theory approach based on the construction of a universal deformation of the detuned Birkhoff normal form. The thresholds for the bifurcations are computed as asymptotic series also in terms of physical quantities for the original system.

nlin.CD↗

arXiv2013

The symmetric 1:2 resonance

This paper illustrates the application of Lie transform normal-form theory to the construction of the 1:2 resonant normal form corresponding to a wide class of natural Hamiltonian systems. We show how to compute the bifurcations of the main periodic orbits in a potential with double reflection symmetries. The stability analysis of the normal modes and of the periodic orbits in general position allows us to get overall informations on the phase-space structure of systems in which this resonance is dominating. As an example we apply these results to a class of models useful as galactic potentials.

nlin.CD↗

arXiv2012

Resonances and bifurcations in systems with elliptical equipotentials

We present a general analysis of the orbit structure of 2D potentials with self-similar elliptical equipotentials by applying the method of Lie transform normalization. We study the most relevant resonances and related bifurcations. We find that the 1:1 resonance is associated only to the appearance of the loops and leads to the destabilization of either one or the other normal modes, depending on the ellipticity of equipotentials. Inclined orbits are never present and may appear only when the equipotentials are heavily deformed. The 1:2 resonance determines the appearance of bananas and anti-banana orbits: the first family is stable and always appears at a lower energy than the second, which is unstable. The bifurcation sequence also produces the variations in the stability character of the major axis orbit and is modified only by very large deformations of the equipotentials. Higher-order resonances appear at intermediate or higher energies and can be described with good accuracy.

astro-ph.GA↗

arXiv2011

Relevance of the 1:1 resonance in galactic dynamics

This paper aims to illustrate the applications of resonant Hamiltonian normal forms to some problems of galactic dynamics. We detail the construction of the 1:1 resonant normal form corresponding to a wide class of potentials with self-similar elliptical equi-potentials and apply it to investigate relevant features of the orbit structure of the system. We show how to compute the bifurcation of the main periodic orbits in the symmetry planes of a triaxial ellipsoid and in the meridional plane of an axi-symmetric spheroid and briefly discuss how to refine these results with higher-order approaches.

astro-ph.GA↗