Resonant Periodic Motions Research Articles

On Nonlinear Oscillations of a Near-Autonomous Hamiltonian System in One Case of Integer Nonequal Frequencies

This paper is concerned with the motions of a near-autonomous two-degree-of-freedom Hamiltonian system, $2\pi$-periodic in time, in a neighborhood of a trivial equilibrium. It is assumed that in the autonomous case, in the region where only necessary (which are not sufficient) conditions for the stability of this equilibrium are satisfied, for some parameter values of the system one of the frequencies of small linear oscillations is equal to two and the other is equal to one. An analysis is made of nonlinear oscillations of the system in a neighborhood of this equilibrium for the parameter values near a resonant point of parameter space. The boundaries of the parametric resonance regions are constructed which arise in the presence of secondary resonances in the transformed linear system (the cases of zero frequency and equal frequencies). The general case and both cases of secondary resonances are considered; in particular, the case of two zero frequencies is singled out. An analysis is made of resonant periodic motions of the system that are analytic in integer or fractional powers of the small parameter, and conditions for their linear stability are obtained. Using KAM theory, two- and three-frequency conditionally periodic motions (with frequencies of different orders in a small parameter) are described.

On Nonlinear Oscillations of a Time-Periodic Hamiltonian System at a 2:1:1 Resonance

We consider the motions of a near-autonomous Hamiltonian system $2\pi$-periodic in time, with two degrees of freedom, in a neighborhood of a trivial equilibrium. A multiple parametric resonance is assumed to occur for a certain set of system parameters in the autonomous case, for which the frequencies of small linear oscillations are equal to two and one, and the resonant point of the parameter space belongs to the region of sufficient stability conditions. Under certain restrictions on the structure of the Hamiltonian of perturbed motion, nonlinear oscillations of the system in the vicinity of the equilibrium are studied for parameter values from a small neighborhood of the resonant point. Analytical boundaries of parametric resonance regions are obtained, which arise in the presence of secondary resonances in the transformed linear system (the cases of zero frequency and equal frequencies). The general case, for which the parameter values do not belong to the parametric resonance regions and their small neighborhoods, and both cases of secondary resonances are considered. The question of the existence of resonant periodic motions of the system is solved, and their linear stability is studied. Two- and three-frequency conditionally periodic motions are described. As an application, nonlinear resonant oscillations of a dynamically symmetric satellite (rigid body) relative to the center of mass in the vicinity of its cylindrical precession in a weakly elliptical orbit are investigated.

On nonlinear oscillations of time-periodic Hamiltonian systems in the presence of double fourth-order resonances

The motion of a time-periodic Hamiltonian system with two degrees of freedom in a neighborhood of a linearly stable equilibrium is investigated. A double fourth-order resonance (of fundamental and combination types) is assumed to be realized in the system, or the parameters of the problem are close to the resonance values. The problem of the existence and number of resonant periodic motions in a small neighborhood of the equilibrium is investigated; the conditions for their stability in the linear approximation are analyzed. The results are applied to the problem of the motion of a dynamically symmetric satellite (a rigid body) near its stationary rotation (cylindrical precession) in the central Newtonian gravitational field in an elliptical orbit of arbitrary eccentricity.

On Nonlinear Oscillations of a Near-Autonomous Hamiltonian System in the Case of Two Identical Integer or Half-Integer Frequencies

This paper examines the motion of a time-periodic Hamiltonian system with two degrees of freedom in a neighborhood of trivial equilibrium. It is assumed that the system depends on three parameters, one of which is small; when it has zero value, the system is autonomous. Consideration is given to a set of values of the other two parameters for which, in the autonomous case, two frequencies of small oscillations of the linearized equations of perturbed motion are identical and are integer or half-integer numbers (the case of multiple parametric resonance). It is assumed that the normal form of the quadratic part of the Hamiltonian does not reduce to the sum of squares, i.e., the trivial equilibrium of the system is linearly unstable. Using a number of canonical transformations, the perturbed Hamiltonian of the system is reduced to normal form in terms through degree four in perturbations and up to various degrees in a small parameter (systems of first, second and third approximations). The structure of the regions of stability and instability of trivial equilibrium is investigated, and solutions are obtained to the problems of the existence, number, as well as (linear and nonlinear) stability of the system’s periodic motions analytic in fractional or integer powers of the small parameter. For some cases, conditionally periodic motions of the system are described. As an application, resonant periodic motions of a dynamically symmetric satellite modeled by a rigid body are constructed in a neighborhood of its steady rotation (cylindrical precession) on a weakly elliptic orbit and the problem of their stability is solved.

On orbital stability of resonant periodic motions originating from hyperboloidal precession of a dynamically symmetric satellite

Periodic motions arising from Hyperboloidal precession of a dynamically symmetric rigid-body satellite on a circular orbit are studied in this work. There exist two types of these motions categorized by frequencies ^ and co2 of the linearized system: short-periodic motions with period close to 2n/co2 and long-periodic ones with period close to 2?!/^ (co2 > co). These motions describe oscillations of the satellite’s principal axis in the neighbourhood Hyperboloidal precession and can be obtained as small-parameter power series of the oscillation amplitude. It has been shown before that in a resonant case there exist more than one family of long-periodic motions. In this work, families of long-periodic motions originating from Hyperboloidal precession in case of fourth order resonance are computed and analysed.

On resonant periodic motions close to conical precession of a dynamically symmetric satellite in a weakly elliptic orbit

The motion of a dynamically symmetric rigid body (satellite) about its centre of mass in an elliptical orbit is considered. It is assumed that the eccentricity of the orbit is small and an external resonance caused by gravitational torque takes place. Periodic motions emanating from conical precession of the satellite are studied. Such periodic motions can be constructed in a form of convergent series of fractional powers of the eccentricity. The first terms of these series expansion have been calculated in an explicit form. The bifurcation phenomenon of the above periodic motions is investigated.

A Galilean dance 1:2:4 resonant periodic motions and their librations of Jupiter and his Galilean moons

The four Galilean moons of Jupiter were discovered by Galileo in the early 17th century, and their motion was first seen as a miniature solar system. Around 1800 Laplace discovered that the Galilean motion is subjected to an orbital \begin{document}$ 1{:}2{:}4 $\end{document} -resonance of the inner three moons Io, Europa and Ganymedes. In the early 20th century De Sitter gave a mathematical explanation for this in a Newtonian framework. In fact, he found a family of stable periodic solutions by using the seminal work of Poincare, which at the time was quite new. In this paper we review and summarize recent results of Broer, Hansmann and Zhao on the motion of the entire Galilean system, so including the fourth moon Callisto. To this purpose we use a version of parametrised Kolmogorov-Arnol'd-Moser theory where a family of multi-periodic isotropic invariant three-dimensional tori is found that combines the periodic motions of De Sitter and Callisto. The \begin{document}$ 3 $\end{document} -tori are normally elliptic and excite a family of invariant Lagrangean \begin{document}$ 8 $\end{document} -tori that project down to librational motions. Both the \begin{document}$ 3 $\end{document} - and the \begin{document}$ 8 $\end{document} -tori occur for an almost full Hausdorff measure set in the product of corresponding dimension in phase space and a parameter space, where the external parameters are given by the masses of the moons.

The 1/1 resonance in extrasolar systems

We present families of symmetric and asymmetric periodic orbits at the 1/1 resonance, for a planetary system consisting of a star and two small bodies, in comparison to the star, moving in the same plane under their mutual gravitational attraction. The stable 1/1 resonant periodic orbits belong to a family which has a planetary branch, with the two planets moving in nearly Keplerian orbits with non zero eccentricities and a satellite branch, where the gravitational interaction between the two planets dominates the attraction from the star and the two planets form a close binary which revolves around the star. The stability regions around periodic orbits along the family are studied. Next, we study the dynamical evolution in time of a planetary system with two planets which is initially trapped in a stable 1/1 resonant periodic motion, when a drag force is included in the system. We prove that if we start with a 1/1 resonant planetary system with large eccentricities, the system migrates, due to the drag force, along the family of periodic orbits and is finally trapped in a satellite orbit. This, in principle, provides a mechanism for the generation of a satellite system: we start with a planetary system and the final stage is a system where the two small bodies form a close binary whose center of mass revolves around the star.

The 1/1 resonance in extrasolar planetary systems

We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits.

On periodic orbits and resonance in extrasolar planetary systems

We study the evolution of an extrasolar planetary system with two planets, for planar motion, starting from an exact resonant periodic motion and increasing the deviation from the equilibrium solution. We keep the semimajor axes and the eccentricities of the two planets fixed and we change the initial conditions by rotating the orbit of the outer planet by Δω. In this way the resonance is preserved, but we deviate from the exact periodicity and there is a transition from order to chaos as the deviation increases. There are three different routes to chaos, as far as the evolution of (ω2 − ω1) is concerned: (a) Libration → rotation → chaos, with intermittent transition from libration to rotation in between, (b) libration → chaos and (c) libration → intermittent interchange between libration and rotation → chaos. This indicates that resonant planetary systems where the angle (ω2 − ω1) librates or rotates are not different, but are closely connected to the exact periodic motion.

Resonance transitions associated to weak capture in the restricted three-body problem

An interesting dynamics is studied in the restricted three-body problem where a particle abruptly transitions between resonance states, called a resonance hop. It occurs in a region about the secondary mass point which supports weak capture. This region, called a weak stability boundary, was recently proven to give rise to chaotic dynamics. Although it was numerically known that the resonance hop was associated with this boundary, this process was not well understood. In addition, the dynamical structure of the weak stability boundary has not been well understood. In this paper, we give a way to reveal the global structure of the weak stability boundary associated to resonance motions. This structure is shown to be surprisingly rich in resonant periodic motions interconnected by invariant manifolds. In this case, nearly all the motions are approximately resonant in nature where resonance hops can occur. The correlation dimension of orbits undergoing resonant motions, associated to the weak stability boundary, is also examined. The dynamics analyzed in the present paper is related to that studied by J. Marsden et al. under the perspective of Lyapunov orbits and the associated invariant manifolds. Applications are discussed.

Resonant periodic motions of Hamiltonian systems with one degree of freedom when the Hamiltonian is degenerate

The motions of a non-autonomous Hamiltonian system with one degree of freedom which is periodic in time and where the Hamiltonian contains a small parameter is considered. The origin of coordinates of the phase space is the equilibrium position of the unperturbed or complete system, which is stable in the linear approximation. It is assumed that there is degeneracy in the unperturbed Hamiltonian when account is taken of terms no higher than the fourth degree (the frequency of the small linear oscillations depends on the amplitude) and, in this case, one of the resonances of up to the fourth order inclusive is realized in the system. Model Hamiltonians are constructed for each case of resonance and a qualitative investigation of the motions of the model system is carried out. Using Poincaré's theory of periodic motions and KAM-theory, a rigorous solution is given of the problem of the existence, bifurcations and stability of the periodic motions of the initial system, which are analytic with respect to fractional powers of the small parameter. The resonant periodic motions (in the case of the degeneracy being considered) of a spherical pendulum with an oscillating suspension point are investigated as an application.

Stable Manifolds and Homoclinic Points Near Resonances in the Restricted Three-Body Problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$ that circle each other with period equal to $2\pi$. For small $\mu$, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers $p$ and $q$, if its period around the heavier primary is approximately $2\pi p/q$, and by its approximate eccentricity $e$. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with $q/p$ equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any $\mu$ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for $\mu$ small enough in the unaveraged restricted three-body problem.

Chaos and Quasi-Periodic Motions on the Homoclinic Surface of Nonlinear Hamiltonian Systems With Two Degrees of Freedom

Abstract The numerical prediction of chaos and quasi-periodic motion on the homoclinic surface of a two-degree-of-freedom (2-DOF) nonlinear Hamiltonian system is presented through the energy spectrum method. For weak interactions, the analytical conditions for chaotic motion in such a Hamiltonian system are presented through the incremental energy approach. The Poincaré mapping surfaces of chaotic motions for this specific nonlinear Hamiltonian system are illustrated. The chaotic and quasi-periodic motions on the phase planes, displacement subspace (or potential domains), and the velocity subspace (or kinetic energy domains) are illustrated for a better understanding of motion behaviors on the homoclinic surface. Through this investigation, it is observed that the chaotic and quasi-periodic motions almost fill on the homoclinic surface of the 2-DOF nonlinear Hamiltonian system. The resonant-periodic motions for such a system are theoretically countable but numerically inaccessible. Such conclusions are similar to the ones in the KAM theorem even though the KAM theorem is based on the small perturbation.

Linear Stability Analysis of Resonant Periodic Motions in the Restricted Three-Body Problem

The equations of the restricted three-body problem describe the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$, $0\leq \mu \leq 1/2$, that circle each other with period equal to $2\pi$. When $\mu=0$, the problem admits orbits for the massless particle that are ellipses of eccentricity $e$ with the primary of mass 1 located at one of the focii. If the period is a rational multiple of $2\pi$, denoted $2\pi p/q$, some of these orbits perturb to periodic motions for $\mu > 0$. For typical values of $e$ and $p/q$, two resonant periodic motions are obtained for $\mu > 0$. We show that the characteristic multipliers of both these motions are given by expressions of the form $1\pm\sqrt{C(e,p,q)\mu}+O(\mu)$ in the limit $\mu\to 0$. The coefficient $C(e,p,q)$ is analytic in $e$ at $e=0$ and $C(e,p,q)=O(e^{\abs{p-q}})$. The coefficients in front of $e^{\abs{p-q}}$, obtained when $C(e,p,q)$ is expanded in powers of $e$ for the two resonant periodic motions, sum to zero. Typically, if one of the two resonant periodic motions is of elliptic type the other is of hyperbolic type. We give similar results for retrograde periodic motions and discuss periodic motions that nearly collide with the primary of mass $1-\mu$.

Resonant forced motions of a weakly coupled linear and strictly non-linear oscillator

This paper is concerned with the investigation of the resonant periodic motions, and their stability, for a weakly coupled linear and strictly non-linear oscillator, subject to weak periodic forcing. The non-linear oscillator may be performing either (large) oscillations or rotations. The natural frequencies of the linear oscillator and of the non-linear oscillator (with appropriate energy) are assumed to be close to rational multiples of the forcing frequency. The equations of motion are transformed into an appropriate standard form, and previously obtained results are applied in order to investigate the resonant periodic motions, and their stability, in the general case. The investigation is carried out analytically for a particular example, for which the resonance for the linear oscillator must be fundamental. The resonance for the non-linear oscillator may be fundamental, or a subharmonic one in the oscillatory case, and an ultrasubharmonic one in the rotary case.