arXiv subjects

Marian Gidea

Publications and source records attributed to Marian Gidea.

At least 19 recordsLinked to original sources

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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arXiv2025

Arnold diffusion in the elliptic Hill four-body problem: geometric method and numerical verification

We present a mechanism for Arnold diffusion in energy in a model of the elliptic Hill four-body problem. Our model is expressed as a small perturbation of the circular Hill four-body problem, with the small parameter being the eccentricity of the orbits of the primaries. The mechanism relies on the existence of two normally hyperbolic invariant manifolds (NHIMs), and on the corresponding homoclinic and heteroclinic connections. The dynamics along homoclinic/heteroclinic orbits is encoded via scattering maps, which we compute numerically. We provide two arguments for diffusion. In the first argument, we successively apply a single scattering map and use Birkhoff's Ergodic Theorem to obtain pseudo-orbits that, on average, gain energy. In the second argument, we use two scattering maps and, at each step, select one that increases energy. Either argument yields pseudo-orbits of scattering maps along which the energy grows by an amount independent of the small parameter. A shadowing lemma concludes the existence of diffusing orbits.

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arXiv2025

Arnold Diffusion in the Full Three-Body Problem

The full three-body problem, on the motion of three celestial bodies under their mutual gravitational attraction, is one of the oldest unsolved problems in classical mechanics. The main difficulty comes from the presence of unstable and chaotic motions, which make long-term prediction impossible. In this paper, we show that the full three-body problem exhibits a strong form of instability known as Arnold diffusion. We consider the planar full three-body problem, formulated as a perturbation of both the Kepler problem and the planar circular restricted three-body problem. We show that the system exhibits Arnold diffusion, in the sense that there is a transfer of energy -- of an amount independent of the perturbation parameter -- between the Kepler problem and the restricted three-body problem. Our argument is based on the topological method of correctly aligned windows, which is implemented into a computer assisted proof. We demonstrate that the approach can be applied to physically relevant masses of the bodies, choosing a Neptune-Triton-asteroid system as an example. In this case, we obtain explicit estimates for the range of the perturbation parameter and for the diffusion time.

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arXiv2025

Semi-analytic construction of global transfers between quasi-periodic orbits in the spatial R3BP

Consider the spatial restricted three-body problem, as a model for the motion of a spacecraft relative to the Sun-Earth system. We focus on the dynamics near the equilibrium point $L_1$, located between the Sun and the Earth. We show that we can transfer the spacecraft from a quasi-periodic orbit that is nearly planar relative to the ecliptic to a quasi-periodic orbit that has large vertical amplitude, at zero energy cost. (In fact, the final orbit has the maximum vertical amplitude that can be obtained through the particular mechanism that we consider. Moreover, the transfer can be made through any prescribed sequence of quasi-periodic orbits in between). Our transfer mechanism is based on selecting trajectories homoclinic to a normally hyperbolic invariant manifold (NHIM) near $L_1$, and then gluing them together. We present a theoretical result establishing the existence of such transfer orbits, and we verify numerically its applicability to our model. We provide several explicit constructions of such transfers, and also develop an algorithm to design trajectories that achieve the shortest transfer time for this particular mechanism. The change in the vertical amplitude along a homoclinic trajectory can be described via the scattering map. We develop a new tool, the `Standard Scattering Map' (SSM), which is a series representation of the exact scattering map. We use the SSM to obtain a complete description of the dynamics along homoclinic trajectories. The SSM can be used in many other situations, from Arnold diffusion problems to transport phenomena in applications.

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arXiv2024

Energy Transport in Random Perturbations of Mechanical Systems

We describe a mechanism for transport of energy in a mechanical system consisting of a pendulum and a rotator subject to a random perturbation. The perturbation that we consider is the product of a Hamiltonian vector field and a scalar, continuous, stationary Gaussian process with Hölder continuous realizations, scaled by a smallness parameter. We show that for almost every realization of the stochastic process, there is a distinguished set of times for which there exists a random normally hyperbolic invariant manifold with associated stable and unstable manifolds that intersect transversally, for all sufficiently small values of the smallness parameter. We derive the existence of orbits along which the energy changes over time by an amount proportional to the smallness parameter. This result is related to the Arnold diffusion problem for Hamiltonian systems, which we treat here in the random setting.

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arXiv2023

Why Topological Data Analysis Detects Financial Bubbles?

We present a heuristic argument for the propensity of Topological Data Analysis (TDA) to detect early warning signals of critical transitions in financial time series. Our argument is based on the Log-Periodic Power Law Singularity (LPPLS) model, which characterizes financial bubbles as super-exponential growth (or decay) of an asset price superimposed with oscillations increasing in frequency and decreasing in amplitude when approaching a critical transition (tipping point). We show that whenever the LPPLS model is fitting with the data, TDA generates early warning signals. As an application, we illustrate this approach on a sample of positive and negative bubbles in the Bitcoin historical price.

q-fin.ST↗

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.

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arXiv2023

Regularization of the Hill four-body problem with oblate bodies

We consider the Hill four-body problem where three oblate, massive bodies form a relative equilibrium triangular configuration, and the fourth, infinitesimal body orbits in a neighborhood of the smallest of the three massive bodies. We regularize collisions between the infinitesimal body and the smallest massive body, via McGehee coordinate transformation. We describe the corresponding collision manifold and show that it undergoes a bifurcation when the oblateness coefficient of the small massive body passes through the zero value.

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arXiv2022

Diffusion along chains of normally hyperbolic cylinders

We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on $\mathbb{A}^3$. Our approach can also be applied to more general Hamiltonians that are not necessarily convex. The main geometric objects in our framework are $3$-dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define chains of cylinders, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections. Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders -- i.e., the existence of Poincaré sections with the return maps satisfying a tilt condition -- and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders -- i.e., certain compatibility conditions between the tilt map and the homoclinic maps attached to its essential invariant circles. We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems.

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arXiv2021

Melnikov method for non-conservative perturbations of the three-body problem

We consider the planar circular restricted three-body problem (PCRTBP), as a model for the motion of a spacecraft relative to the Earth-Moon system. We focus on the Lagrange equilibrium points $L_1$ and $L_2$. There are families of Lyapunov periodic orbits around either $L_1$ or $L_2$, forming Lyapunov manifolds. There also exist homoclinic orbits to the Lyapunov manifolds around either $L_1$ or $L_2$, as well as heteroclinic orbits between the Lyapunov manifold around $L_1$ and the one around $L_2$. The motion along the homoclinic/heteroclinic orbits can be described via the scattering map, which gives the future asymptotic of a homoclinic orbit as a function of the past asymptotic. In contrast with the more customary Melnikov theory, we do not need to assume that the asymptotic orbits have a special nature (periodic, quasi-periodic, etc.). We add a non-conservative, time-dependent perturbation, as a model for a thrust applied to the spacecraft for some duration of time, or for some other effect, such as solar radiation pressure. We compute the first order approximation of the perturbed scattering map, in terms of fast convergent integrals of the perturbation along homoclinic/heteroclinic orbits of the unperturbed system. As a possible application, this result can be used to determine the trajectory of the spacecraft upon using the thrust.

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arXiv2021

Surface Gravity of Rotating Dumbbell Shapes

We investigate the problem of determining the shape of a rotating celestial object - e.g., a comet or an asteroid - under its own gravitational field. More specifically, we consider an object symmetric with respect to one axis - such as a dumbbell - that rotates around a second axis perpendicular to the symmetry axis. We assume that the object can be modeled as an incompressible fluid of constant mass density, which is regarded as a first approximation of an aggregate of particles. In the literature, the gravitational field of a body is often described as a multipolar expansion involving spherical coordinates (Kaula, 1966). In this work we describe the shape in terms of cylindrical coordinates, which are most naturally adapted to the symmetry of the body, and we express the gravitational potential generated by the rotating body as a simple formula in terms of elliptic integrals. An equilibrium shape occurs when the gravitational potential energy and the rotational kinetic energy at the surface of the body balance each other out. Such an equilibrium shape can be derived as a solution of an optimization problem, which can be found via the variational method. We give an example where we apply this method to a two-parameter family of dumbbell shapes, and find approximate numerical solutions to the corresponding optimization problem.

astro-ph.EP↗

arXiv2020

Arnold Diffusion, Quantitative Estimates and Stochastic Behavior in the Three-Body Problem

We consider a class of autonomous Hamiltonian systems subject to small, time-periodic perturbations. When the perturbation parameter is set to zero, the energy of the system is preserved. This is no longer the case when the perturbation parameter is non-zero. We describe a topological method to establish orbits which diffuse in energy for every suitably small perturbation parameter $\varepsilon>0$. The method yields quantitative estimates: (i) The existence of orbits along which the energy drifts by an amount independent of $\varepsilon$. The time required by such orbits to drift is $O(1/\varepsilon)$; (ii) The existence of orbits along which the energy makes chaotic excursions; (iii) Explicit estimates for the Hausdorff dimension of the set of such chaotic orbits; (iv) The existence of orbits along which the time evolution of energy approaches a stopped diffusion process (Brownian motion with drift), as $\varepsilon$ tends to $0$. For each $\varepsilon$ fixed, the set of initial conditions of the orbits that yield the diffusion process has positive Lebesgue measure, and in the limit the measure of these sets approaches zero. Moreover, we can obtain any desired values of the drift and variance for the limiting Brownian motion, for appropriate sets of initial conditions. A key feature of our topological method is that it can be implemented in computer assisted proofs. We give an application to a concrete model of the planar elliptic restricted three-body problem, on the motion of an infinitesimal body relative to the Neptune-Triton system.

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arXiv2019

Global effect of non-conservative perturbations on homoclinic orbits

We study the effect of time-dependent, non-conservative perturbations on the dynamics along homoclinic orbits to a normally hyperbolic invariant manifold. We assume that the unperturbed system is Hamiltonian, and the normally hyperbolic invariant manifold is parametrized via action-angle coordinates. The homoclinic excursions can be described via the scattering map, which gives the future asymptotic of an orbit as a function of the past asymptotic. We provide explicit formulas, in terms of convergent integrals, for the perturbed scattering map expressed in action-angle coordinates. We illustrate these formulas in the case of a perturbed rotator-pendulum system.

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arXiv2018

Hill four-body problem with oblate tertiary: an application to the Sun-Jupiter-Hektor-Skamandrios system

We consider a restricted four-body problem with a precise hierarchy between the bodies: two point-mass bigger bodies, a smaller one with oblate shape, and an infinitesimal body in the neighborhood of the oblate body. The three heavy bodies are assumed to move in a plane under their mutual gravity, and the fourth body moves under the gravitational influence of the three heavy bodies, but without affecting them. We start by finding the triangular central configurations of the three heavy bodies; since one body is oblate, the triangle is isosceles, rather than equilateral as in the point mass case. We assume that the three heavy bodies are in such a central configuration and we perform a Hill's approximation of the equations of motion describing the dynamics of the infinitesimal body in a neighborhood of the oblate body. Through the use of Hill's variables and a limiting procedure, this approximation amounts to sending the two other bodies to infinity. Finally, for the Hill approximation, we find the equilibrium points of the infinitesimal body and determine their stability. As a motivating example, we consider the dynamics of the moonlet Skamandrios of Jupiter's Trojan asteroid Hektor.

math.DS↗

arXiv2018

Topological recognition of critical transitions in time series of cryptocurrencies

We analyze the time series of four major cryptocurrencies (Bitcoin, Ethereum, Litecoin, and Ripple) before the digital market crash at the end of 2017 - beginning 2018. We introduce a methodology that combines topological data analysis with a machine learning technique -- $k$-means clustering -- in order to automatically recognize the emerging chaotic regime in a complex system approaching a critical transition. We first test our methodology on the complex system dynamics of a Lorenz-type attractor, and then we apply it to the four major cryptocurrencies. We find early warning signals for critical transitions in the cryptocurrency markets, even though the relevant time series exhibit a highly erratic behavior.

q-fin.MF↗

arXiv2017

Global Melnikov Theory in Hamiltonian Systems with General Time-dependent Perturbations

We consider a mechanical system consisting of $n$ penduli and a $d$-dimensional generalized rotator subject to a time-dependent perturbation. The perturbation is not assumed to be either Hamiltonian, or periodic or quasi-periodic. The strength of the perturbation is given by a parameter $ε\in\mathbb{R}$. For all $|ε|$ sufficiently small, the augmented flow has a $(2d + 1)$-dimensional normally hyperbolic locally invariant manifold $\tildeΛ_ε$. We define a Melnikov vector, which gives the first order expansion of the displacement of the stable and unstable manifolds of $\tildeΛ_0$ under the perturbation. We provide an explicit formula for the Melnikov vector in terms of convergent improper integrals of the perturbation along homoclinic orbits of the unperturbed system. We show that if the perturbation satisfies some explicit non-degeneracy conditions, then the stable and unstable manifolds of $\tildeΛ_ε$, $W^s(\tildeΛ_ε)$ and $W^u(\tildeΛ_ε)$, respectively, intersect along a transverse homoclinic manifold, and, moreover, the splitting of $W^s(\tildeΛ_ε)$ and $W^u(\tildeΛ_ε)$ can be explicitly computed, up to the first order, in terms of the Melnikov vector. This implies that the excursions along some homoclinic trajectories yield a non-trivial increase of order $O(ε)$ in the action variables of the rotator, for all sufficiently small perturbations. The formulas that we obtain are independent of the unperturbed motions and give, at the same time, the effects on periodic, quasi-periodic, or general orbits. When the perturbation is Hamiltonian, we express the effects of the perturbation, up to the first order, in terms of a Melnikov potential. In addition, if the perturbation is periodic, we obtain that the non-degeneracy conditions on the Melnikov potential are generic.

math.DS↗

arXiv2017

Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics

We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach relies on the method of `correctly aligned windows'. We subdivide the `windows' into cubical complexes, and we assign to the vertices of the cubes labels determined by the dynamics. In this way we encode the dynamics information into a combinatorial structure. We use a version of the Sperner Lemma saying that if the labeling satisfies certain conditions, then there exist fixed points/periodic orbits/orbits with prescribed itineraries. Our arguments are elementary.

math.DS↗

arXiv2017

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the `outer dynamics' along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined map, referred to as the `scattering map'. We find pseudo-orbits of the scattering map that keep advancing in some privileged direction. Then we use the recurrence property of the `inner dynamics', restricted to the normally hyperbolic invariant manifold, to return to those pseudo-orbits. Finally, we apply topological methods to show the existence of true orbits that follow the successive applications of the two dynamics. This method differs, in several crucial aspects, from earlier works. Unlike the well known `two-dynamics' approach, the method we present relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, as its invariant objects (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets) are not used at all. The method applies to unperturbed Hamiltonians of arbitrary degrees of freedom that are not necessarily convex. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, as well as to establish diffusion in generic systems. We include several applications, such as bridging large gaps in a priori unstable models in any dimension, and establishing diffusion in cases when the inner dynamics is a non-twist map.

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