Bifurcations Of Relative Equilibria Research Articles

Relative equilibria, stability and bifurcations in Hamiltonian galactic tidal models

ABSTRACT We study special solutions of Galactic Tidal models and the existence, stability and bifurcations of relative equilibria. Precisely, we found four equilibria and rectilinear, as well as circular solutions, previous to any manipulation of the original system. Moreover, by averaging of the Keplerian energy and making use of Reeb's theorem, we find four periodic orbits. The stability of these relative equilibria is analysed and determined in several cases that involve a relation between the Keplerian energy and the gravitational parameter. Further averaging on the angle giving the orbital plane node provides the Matese–Whitman model. This Hamiltonian is expressed as a function of the invariants of the twice-reduced space, which is given by three connected components. For the three connected components become attached to each other by two singular points. Although only the bounded component is endowed with physical meaning, we provide a full analysis of the stability and bifurcations for all the equilibria. By doing this, we get an insight into the bifurcation process. Precisely, we observe that for the singular leaf of the reduced space, some of the equilibria in the bounded and unbounded components switch their locations.

Bifurcation of Relative Equilibria in Infinite-Dimensional Hamiltonian Systems

Bifurcations of relative equilibria in perturbed infinite-dimensional Hamiltonian systems are studied. We assume that the unperturbed system has several symmetries and a family of relative equilibr...

Collision and annihilation of relative equilibrium points around asteroids with a changing parameter

In this work, we investigate the bifurcations of relative equilibria in the gravitational potential of asteroids. A theorem concerning a conserved quantity, which is about the eigenvalues and number of relative equilibria, is presented and proved. The conserved quantity can restrict the number of non-degenerate equilibria in the gravitational potential of an asteroid. It is concluded that the number of non-degenerate equilibria in the gravitational field of an asteroid varies in pairs and is an odd number. In addition, the conserved quantity can also restrict the kinds of bifurcations of relative equilibria in the gravitational potential of an asteroid when the parameter varies. Furthermore, studies have shown that there exist transcritical bifurcations, quasi-transcritical bifurcations, saddle-node bifurcations, saddle-saddle bifurcations, binary saddle-node bifurcations, supercritical pitchfork bifurcations, and subcritical pitchfork bifurcations for the relative equilibria in the gravitational potential of asteroids. It is found that for the asteroid 216 Kleopatra, when the rotation period varies as a parameter, the number of relative equilibria changes from 7 to 5 to 3 to 1, and the bifurcations for the relative equilibria are saddle-node bifurcations and saddle-saddle bifurcations.

Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry

For Hamiltonian systems with spherical symmetry there is a marked difference between zero and non-zero momentum values, and amongst all relative equilibria with zero momentum there is a marked difference between those of zero and those of non-zero angular velocity. We use techniques from singularity theory to study the family of relative equilibria that arise as a symmetric Hamiltonian which has a group orbit of equilibria with zero momentum is perturbed so that the zero-momentum relative equilibrium are no longer equilibria. We also analyze the stability of these perturbed relative equilibria, and consider an application to satellites controlled by means of rotors.

Stability and bifurcations of relative equilibria of a pendulum suspended on the equator

The problem of equilibria of a pendulum suspended at an equatorial point relative to the rotating Earth is considered. An altitude is determined at which the degree of instability of the inverted pendulum changes from two to unity. Relative equilibria are investigated that bifurcate from the radial one when its degree of instability changes. Their stability properties are studied.

Bifurcation of periodic solutions from a ring configuration of discrete nonlinear oscillators

This paper gives an analysis of the periodic solutions of a ring of $n$oscillators coupled to their neighbors. We prove the bifurcation of branchesof such solutions from a relative equilibrium, and we study theirsymmetries. We give complete results for a cubic Schr?dinger potential andfor a saturable potential and for intervals of the amplitude of theequilibrium. The tools for the analysis are the orthogonal degree andrepresentation of groups. The bifurcation of relative equilibria was givenin a previous paper.

Bifurcations of relative equilibria for one spheroidal and two spherical bodies

We discuss existence and bifurcations of non-collinear (Lagrangian) relative equilibria in a generalized three body problem. Specifically, one of the bodies is a spheroid (oblate or prolate) with its equatorial plane coincident with the plane of motion where only the “J2” term from its potential expansion is retained.

Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators

Given a regular polygonal arrangement of identical objects, turning around a central object (masses, vortices or dNLS oscillators), this paper studies the global bifurcation of relative equilibria in function of a natural parameter (central mass, central circulation or amplitude of the oscillation). The symmetries of the problem are used in order to find the irreducible representations, the linearization and, with the help of a degree theory, the symmetries of the bifurcated solutions.

Relative equilibria and bifurcations in the generalized van der Waals 4D oscillator

A uniparametric 4-DOF family of perturbed Hamiltonian oscillators in 1:1:1:1 resonance is studied as a generalization for several models for perturbed Keplerian systems. Normalization by Lie transforms (only first order is considered here) as well as geometric reduction related to the invariants associated to the symmetries is used based on the previous work of the authors. A description is given for the lower dimensional relative equilibria in such normalized systems for which we introduce, in this context, the new concept of moment polytope. In addition bifurcations of relative equilibria corresponding to 3D tori are studied in some particular cases where we focus on Hamiltonian–Hopf bifurcations and bifurcations in the 3D van der Waals and Zeeman systems.

Regularization of the Amended Potential and the Bifurcation of Relative Equilibria

In the context of simple mechanical systems with symmetry, we give a method based on blowing up the amended potential for obtaining symmetry-breaking branches of relative equilibria bifurcating from a given set of symmetric relative equilibria. The general method is illustrated with two concrete mechanical examples, the double spherical pendulum and the symmetric coupled rigid bodies.

Perturbations of the 1 : 1 : 1 resonance with tetrahedral symmetry: a three degree of freedom analogue of the two degree of freedom Hénon–Heiles Hamiltonian

We study a class of three degree of freedom (3-DOF) Hamiltonian systems that share certain characteristics with the 2-DOF Hénon–Heiles Hamiltonian. Our systems represent a 1 : 1 : 1 resonant three-oscillator whose principal nonlinear perturbation is the cubic potential term xyz with tetrahedral symmetry. After normalizing and reducing the 1 : 1 : 1 oscillator symmetry, we show that near the limit of linearization all our systems can be described as a one-parametric family. Such reduced systems have been suggested earlier by Hecht (1960 J. Mol. Spectrosc. 5 355) and later by Patterson (1985 J. Chem. Phys. 83 4618) to model triply degenerate vibrations of tetrahedral molecules. We describe relative equilibria (RE) of these systems, classify all qualitatively different family members, and discuss bifurcations of RE involved in the transitions from one region of regular parameter values to the other.

Bifurcation of relative equilibria in mechanical systems with symmetry

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describing the set of relative equilibria in a neighborhood of a given relative equilibrium. The structure of the reduced equations is studied in a few relevant situations. In particular, a persistence result of Lerman and Singer [Nonlinearity 11 (1998) 1637–1649] is generalized to the framework of Abelian proper actions. Also, a Hamiltonian version of the Equivariant Branching Lemma and a study of bifurcations with maximal isotropy are presented. An elementary example illustrates the use of this approach.

Bifurcation of Relative Equilibria in the Main Problem of Artificial Satellite Theory for a Prolate Body

In this paper, we study circular orbits of the J2 problem that are confined to constant-z planes. They correspond to fixed points of the dynamics in a meridian plane. It turns out that, in the case of a prolate body, such orbits can exist that are not equatorial and branch from the equatorial one through a saddle-center bifurcation. A closed-form parametrization of these branching solutions is given and the bifurcation is studied in detail. We show both theoretically and numerically that, close to the bifurcation point, quasi-periodic orbits are created, along with two families of reversible orbits that are homoclinic to each one of them.

On the geometry and dynamics of floating four-bar linkages

In this paper, we investigate the kinematics and dynamics of floating, planar four-bar linkages. The geometry of configuration space is analyzed in part through the classical theory of mechanisms due to Grashof. The techniques of symplectic and Poisson reduction are used to understand the dynamics of the system. Bifurcations of relative equilibria for linkages admitting symmetric shapes are studied using the techniques of singularity theory. The problem of reconstruction of the full dynamics and its relation to geometric phases is discussed through some examples. This research reveals that a coupled mechanical system with kinematic loops possesses richer and more complicated dynamical aspects in comparison with systems which have the same number of degrees of freedom, but no kinematic loops

Bifurcations of relative equilibria in the 4- and 5-body problem

AbstractThe equilateral triangle family of relative equilibria of the 4-body problem consists of three particles of mass 1 at the vertices of an equilateral triangle and the fourth particle of arbitrary massmat the centroid. For one value of the massmthis relative equilibrium is degenerate. We show that families of isosceles triangle relative equilibria bifurcate from the equilateral triangle family asmpasses through the degenerate value.The square family of relative equilibria of the 5-body problem consists of four particles of mass 1 at the vertices of a square and the fifth particle of arbitrary massmat the centroid. For one value of the massmthis relative equilibrium is degenerate. We show that families of kite and isosceles trapezoidal relative equilibria bifurcate from the square family asmpasses through the degenerate value.

Relative equilibria of the four-body problem

AbstractBy employing a regularizing transformation, the problem of bifurcation of relative equilibria in the Newtonian 4-body problem is reduced to a study of an algebraic correspondence between real algebraic varieties. The finiteness theorems of algebraic geometry are used to find an upper bound for the number of affine equivalence classes of relative equilibria which holds for all masses in the complement of a proper, algebraic subset of the space of all masses.