Symmetric Orbits Research Articles

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

We consider the planar restricted three-body problem and the collinear equilibrium point L 3, as an example of a center × saddle equilibrium point in a Hamiltonian with two degrees of freedom. We explore numerically the existence of symmetric and non-symmetric homoclinic orbits to L 3, when varying the mass parameter μ. Concerning the symmetric homoclinic orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely, given a transversal value of μ for which there is a 1-round SHO, say μ 1, we show that for any m ≥ 2, there are countable sets of values of μ, tending to μ 1, corresponding to m-round SHO. Some comments on related analytical results are also made.

An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem

The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically obtained solutions.

Spin axis evolution of two interacting bodies

We consider the solid–solid interactions in the two body problem. The relative equilibria have been previously studied analytically and general motions were numerically analyzed using some expansion of the gravitational potential up to the second order, but only when there are no direct interactions between the orientation of the bodies. Here we expand the potential up to the fourth order and we show that the secular problem obtained after averaging over fast angles, as for the precession model of Boué and Laskar [Boué, G., Laskar, J., 2006. Icarus 185, 312–330], is integrable, but not trivially. We describe the general features of the motions and we provide explicit analytical approximations for the solutions. We demonstrate that the general solution of the secular system can be decomposed as a uniform precession around the total angular momentum and a periodic symmetric orbit in the precessing frame. More generally, we show that for a general n-body system of rigid bodies in gravitational interaction, the regular quasiperiodic solutions can be decomposed into a uniform precession around the total angular momentum, and a quasiperiodic motion with one frequency less in the precessing frame.

Invariant Manifold of Symmetric Orbits for Globally Optimal Biped Locomotion

This paper proposes novel walking gait generation for a double-pendulum-like biped walking model based on the family of the symmetric orbits. By introducing an involution R associated with the leg switching map, the flow of the saddle-center dynamics of the uncontrolled pendulum possesses time-reversal symmetry with respect to R, and the conjunction of the flow and R forms a family of symmetric orbits parameterized by the orbital energy and the stride. The invariant manifold of the family of symmetric orbits is obtained numerically or approximately using a perturbation method. The proposed methods are evaluated on numerical simulations. By constraining the solution onto the invariant manifold using the control inputs, stable walking gaits are generated in a semi-global manner in simulations. Based on the passivity of the closed-loop system, a robust speed-controlled walking is achieved in a very simple way.

Symmetric and non-symmetric periodic orbits for the digital filter map

We exhibit instances of non-symmetric periodic orbits for the digital filter map, resolving a question posed in the literature as to whether such orbits can exist. This piecewise irrational rotation, depending on a parameter a = 2cos θ, is an isometry of [−1, 1) × [−1, 1) and reflections in the two diagonals are time-reversing symmetries for the map. Symmetric orbits are plentiful and have been much investigated. Each periodic orbit is paired with a symbolic string, from the alphabet {−, 0, +}, arising under iteration of the map because of the presence of a line of discontinuity. We prove the existence of an infinite family of non-symmetric orbits where the period N starts at 29 and increases in steps of 5; they correspond to the strings (+00)5(+−)2 0 N−19. We describe several computer algorithms to find non-symmetric periodic orbits and their symbolic strings and list non-symmetric strings both for a = 0.5, and for N ≤ 100 across the parameter range. Our evidence suggests that non-symmetric orbits, though not plentiful, are characteristic of the dynamics of the map for all parameter values.

Bifurcation of degenerate homoclinic orbits to saddle-center in reversible systems

The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoclinic orbits near the primary homoclinic orbits is developed. Some known results are extended.

Totally geodesic singular orbits and symmetric singular orbits in cohomogeneity one Riemannian manifolds

We classify cohomogeneity one G-manifolds M of a simple Lie group G such that one of the singular orbits is a simply connected symmetric space. We give a simple necessary and sufficient condition that a singular orbit P of a cohomogeneity one Riemannian G-manifold is totally geodesic. Using it, we obtain a list of cohomogeneity one G-manifolds M as above such that the singular orbit P is totally geodesic with respect to any G-invariant Riemannian metric of M. Such G-manifolds have no G-invariant metric of positive or negative curvature if the rank of the symmetric space P is greater than one.

A topological existence proof for the Schubart orbits in the collinear three-body problem

A topological existence proof is presented for certain symmetrical periodic orbits of the collinear three-body problem with two equal masses, called Schubart orbits. The proof is based on the construction of a Wazewski set $W$ in the phase space. The periodic orbits are found by a shooting argument in which symmetrical initial conditions entering $W$ are followed under the flow until they exit $W$. Topological considerations show that the image of the symmetrical entrance states under this flow map must intersect an appropriate set of symmetrical exit states.

Chaotic and regular motion around generalized Kalnajs discs

The motion of test particles in the gravitational fields generated by the first four members of the infinite family of generalized Kalnajs discs is studied. In first instance, we analyse the stability of circular orbits under radial and vertical perturbations and describe the behaviour of general equatorial orbits and so we find that radial stability and vertical instability dominate such disc models. Then we study bounded axially symmetric orbits by using the Poincaré surfaces of section and Lyapunov characteristic numbers and find chaos in the case of disc-crossing orbits and completely regular motion in other cases.

Resonance bifurcation of a three-degree-of-freedom vibratory system with symmetrical rigid stops

Abstract A three-degree-of-freedom vibratory system impacting symmetrical rigid stops is considered. Dynamics of the vibroimpact system is studied by use of a mapping derived from the equations of motion, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact system, associated with 1 : 3 strong resonance, are analyzed. Neimark–Sacker bifurcations of period-1 double-impact symmetrical orbits, near 1 : 3 resonance point, are found, and the transition of quasi-periodic impact motions to unstable period-3 six-impact motions, are obtained numerically. The results conform to 1 : 3 resonance bifurcation theory of maps available. More complicated “tire-like” quasi-periodic attractor and torus bifurcation associated with the transition of the closed invariant curve to torus, near 1 : 3 resonance point, are found to exist in the nonlinear system. The results imply that there exist possibly more complicated bifurcation sequences near 1 : 3 resonance points of nonlinear dynamical systems.

Higher-order, asymmetric orbit multipactors

Multipactors involving asymmetric back-and-forth orbits with different flight times and launch phases are studied. Resonant orbits with the rf involve periodic sequences of two or more different impact phases. Such orbits appear when the rf field and emission velocity reach values that make the usual symmetric orbits unstable, through a stability bifurcation process. The number of resonant impact phases contained in each orbit doubles at each bifurcation, leading to higher-order multipactor. Existence of second- and third-order multipactors is demonstrated, extending the theoretical upper-field boundary for multipactor discharge. The study focuses on orbital properties, without addressing secondary yield in detail.

Isotropy subalgebras of elliptic orbits in semisimple Lie algebras, and the canonical representatives of pseudo-Hermitian symmetric elliptic orbits

We give a method of determining the centralizer of an elliptic element in a real semisimple Lie algebra g , in relation with the maximal compact subalgebra of g and the compact dual of g . Moreover, we determine a special central element (called the H -element) of the isotropy subalgebra of each simple irreducible pseudo-Hermitian symmetric Lie algebra.

Circular solution of two unequal mass particles in post-Minkowski approximation

A Fokker action for a post-Minkowski approximation with the first post-Newtonian correction is introduced in our previous paper, and a solution for the helically symmetric circular orbit is obtained. We present supplemental results for the circular solution of two unequal mass point particles. Circular solutions for selected mass ratios are found numerically, and analytic formulas in the extreme mass ratio limit are derived. The leading terms of the analytic formulas agree with the first post-Newtonian formulas in this limit.

Modelling gap junctions in a neural field model

We study a nonlinear, one-dimensional neural field model based upon a partial integro-differential equation, that is used to model spatial patterns in working memory. Through the application of Fourier transforms to the PIDE, steady states of spatially localised areas of high activity can be represented by solutions of a fourth order ODE. Recent research has shown a high density of gap junctions in areas of the brain that experience epileptic events. We extend the model by including a diffusion-like term to model gap junctions and derive a sixth order ODE which we use to investigate changes in the dynamics of spatially localised solutions. We find that symmetric homoclinic orbits to a zero steady state exist for a wide area of parameter space. Numerical work shows families of solutions are destroyed as the strength of the term modelling gap junctions increases.

Asymptotic orbits in the restricted four-body problem

This paper studies the asymptotic solutions of the restricted planar problem of four bodies, three of which are finite, moving in circular orbits around their center of masses, while the fourth is infinitesimal. Two of the primaries have equal mass and the most-massive primary is located at the origin of the system. We found the invariant unstable and stable manifolds around the hyperbolic Lyapunov periodic orbits which emanate from the collinear equilibrium points L i , i = 1 , … , 4 , as well as the invariant manifolds from the Lagrangian critical points L 5 and L 6 . We construct numerically, applying forward and backward integration from the intersection points of the appropriate Poincaré cuts, homo- and hetero-clinic, symmetric and non-symmetric asymptotic orbits. We present the characteristic curves of the 24 families which consist of symmetric simple-periodic orbits of the problem for a fixed value of the mass parameter b. The stability of the families is computed and also presented. Sixteen families contain as terminal points asymptotic periodic orbits which intersect the x-axis perpendicularly and tend asymptotically to L 5 for t → + ∞ and to L 6 for t → - ∞ , spiralling into (and out of) these points. The corresponding 16 terminating heteroclinic asymptotic orbits, for b = 2 , are illustrated.

Continuation of normal doubly symmetric orbits in conservative reversible systems

In this paper we introduce the concept of a quasi-submersive mapping between two finite-dimensional spaces, we obtain the main properties of such mappings, and we introduce “normality conditions” under which a particular class of so-called “constrained mappings” are quasi-submersive at their zeros. Our main application is concerned with the continuation properties of normal doubly symmetric orbits in time-reversible systems with one or more first integrals. As examples we study the continuation of the figure-eight and the supereight choreographies in the N-body problem.

GLUING BIFURCATIONS IN CHUA OSCILLATOR

Gluing bifurcation in a modified Chua's oscillator is reported. Keeping other parameters fixed when a control parameter is varied in the modified oscillator model, two symmetric homoclinic orbits to saddle focus at origin, which are mirror images of each other, are glued together for a particular value of the control parameter. In experiments, two asymmetric limit cycles are homoclinic to the saddle focus origin for different values of the control parameter. However, imperfect gluing bifurcation has been observed, in experiments, when one stable and unstable limit cycles merge to the saddle focus origin via saddle-node bifurcation.

Symmetric and asymmetric 3:1 resonant periodic orbits with an application to the 55Cnc extra-solar system

We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit.

Dynamics of the Lorenz–Robbins system with control

In this paper, the existence of periodic orbits and homoclinic orbits in the Lorenz equations with high Rayleigh number r , i.e., the Lorenz–Robbins system, is rigorously proved by the generalized Melnikov method for the three-dimensional slowly varying systems. We analyze stability of these periodic orbits and show that the existence of these nontransverse but symmetrical homoclinic orbits implies the existence of chaos in the Lorenz–Robbins system. The results obtained analytically show the existence of chaotic dynamics in the Lorenz–Robbins system for the first time, but also solve a disagreement on the conditions of existence of periodic orbits in the system. In addition, a simple adaptive algorithm, which was recently developed by the author [D. Huang, Stabilizing near-nonhyperbolic chaotic systems with applications, Phys. Rev. Lett. 93 (2004) 214101] for stabilizing the near-nonhyperbolic chaotic systems, is used to successfully control the chaotic mixing of the Lorenz flows with high Rayleigh number found.

Orbit trap rendering method for generating artistic images with cyclic or dihedral symmetry

This paper explores combining chaotic functions having cyclic ( Z n ) or dihedral ( D n ) symmetry with the orbit trap rendering method. By setting symmetric orbit traps, visually fascinating images with Z n or D n symmetry can be created.