Symmetric Hamiltonian Systems Research Articles

LIMIT CYCLE BIFURCATIONS FROM CENTERS OF SYMMETRIC HAMILTONIAN SYSTEMS PERTURBED BY CUBIC POLYNOMIALS

We study cubic near-Hamiltonian systems obtained by perturbing a symmetric cubic Hamiltonian system with two symmetric singular points. First, following [Han, 2012], we develop a method to study the analytical property of the Melnikov function near the origin for near-Hamiltonian system having the origin as its elementary center or nilpotent center. A computationally efficient algorithm based on the method is established to systematically compute the coefficients of the Melnikov function. Then, we consider the symmetric singular points and present the conditions for one of them to be an elementary center or a nilpotent center. Under the condition for the singular point to be a center, we obtain the standard form of the Hamiltonian system near the center. Moreover, perturbing the symmetric cubic Hamiltonian systems by cubic polynomials we study limit cycles bifurcating from the center. Finally, perturbing the symmetric Hamiltonian system by symmetric cubic polynomials, we consider the number of limit cycles near one of the symmetric centers of the symmetric near-Hamiltonian system, which is the same as that of another center.

Approximate integrals of motion

We determine approximate numerical integrals of motion of 2D symmetric Hamiltonian systems. We detail for a few gravitational potentials the conditions under which quasi-integrals are obtained as polynomial series. We show that each of these potentials has a wide range of regular orbits that are accurately modelled with a unique approximate integral of motion.

Dihedral Hamiltonian Cycle Systems of the Cocktail Party Graph

AbstractThe existence problem for a Hamiltonian cycle decomposition of (the so called cocktail party graph) with a dihedral automorphism group acting sharply transitively on the vertices is completely solved. Such Hamiltonian cycle decompositions exist for all even n while, for n odd, they exist if and only if the following conditions hold: (i) n is not a prime power; (ii) there is a suitable ℓ such that (mod 2ℓ) for all prime factors p of n and the number of the prime factors (counted with their respective multiplicities) such that (mod ) is even. Thus in particular one has a dihedral Hamiltonian cycle decomposition of the cocktail party graph on 8k vertices for all k, while it is known that no such cyclic Hamiltonian cycle decomposition exists. Finally, this paper touches on a recently introduced symmetry requirement by proving that there exists a dihedral and symmetric Hamiltonian cycle system of if and only if (mod 4).

Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems

In this paper, we consider the minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. We prove that if the Hamiltonian function $H\in C^2(\R^{2n}, \R)$ is super-quadratic and convex, for every number $\tau>0$, there exists at least one $\tau$-periodic brake orbit $(\tau,x)$ with minimal period $\tau$ or $\tau/2$ provided $H(Nx)=H(x)$.

Nonsensitive Nonlinear Homotopy Approach

Generally, natural scientific problems are so complicated that one has to establish some effective perturbation or nonperturbation theories with respect to some associated ideal models. We construct a new theory that combines perturbation and nonperturbation. An artificial nonlinear homotopy parameter plays the role of a perturbation parameter, while other artificial nonlinear parameters, which are independent of the original problems, introduced in the nonlinear homotopy models are nonperturbatively determined by means of the principle of minimal sensitivity. The method is demonstrated through several quantum anharmonic oscillators and a non-hermitian parity-time symmetric Hamiltonian system. In fact, the framework of the theory is rather general and can be applied to a broad range of natural phenomena. Possible applications to condensed matter physics, matter wave systems, and nonlinear optics are briefly discussed.

Periodic motion of a mass-spring system

The equations of planar motion of a mass attached to two anchored massless springs form a symmetric Hamiltonian system. The system has a single dimensionless parameter L, corresponding to the spacing between the anchors. For L > 1, there is a stable equilibrium at which the springs are in tension and lie on a line, but for L < 1, this equilibrium has both springs in compression and is unstable. However, there are then two stable equilibria at which both springs carry no force. Oscillations are studied in both regimes, but more systematically in the tension case, where techniques of bifurcation theory, numerical approximation and numerical simulation are used to explore the rich variety of periodic solutions.

Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand’s theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems. The demonstration relies on Perlick’s classification of Bertrand spacetimes and on the construction of a suitable, globally defined generalization of the Runge–Lenz vector.

Linear systems and determinantal random point fields

In random matrix theory, determinantal random point fields describe the distribution of eigenvalues of self-adjoint matrices from the generalized unitary ensemble. This paper considers symmetric Hamiltonian systems and determines the properties of kernels and associated determinantal random point fields that arise from them; this extends work of Tracy and Widom. The inverse spectral problem for self-adjoint Hankel operators gives sufficient conditions for a self-adjoint operator to be the Hankel operator on L 2 ( 0 , ∞ ) from a linear system in continuous time; thus this paper expresses certain kernels as squares of Hankel operators. For suitable linear systems ( − A , B , C ) with one-dimensional input and output spaces, there exists a Hankel operator Γ with kernel ϕ ( x ) ( s + t ) = C e − ( 2 x + s + t ) A B such that g x ( z ) = det ( I + ( z − 1 ) Γ Γ † ) is the generating function of a determinantal random point field on ( 0 , ∞ ) . The inverse scattering transform for the Zakharov–Shabat system involves a Gelfand–Levitan integral equation such that the trace of the diagonal of the solution gives ∂ ∂ x log g x ( z ) . When A ⩾ 0 is a finite matrix and B = C † , there exists a determinantal random point field such that the largest point has a generalised logistic distribution.

Numerical Bifurcation of Hamiltonian Relative Periodic Orbits

Relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur, for example, in celestial mechanics, molecular dynamics, and the motion of rigid bodies. RPOs are solutions which are periodic orbits of the symmetry-reduced system. In this paper we analyze certain symmetry-breaking bifurcations of RPOs in Hamiltonian systems with compact symmetry group and show how they can be detected and computed numerically. These are turning points of RPOs and relative period-doubling and relative period-halving bifurcations along branches of RPOs. In a comoving frame the latter correspond to symmetry-breaking/symmetry-increasing pitchfork bifurcations or to period-doubling/period-halving bifurcations. We apply our methods to the family of rotating choreographies which bifurcate from the famous figure eight solution of the three-body problem as angular momentum is varied. We find that the family of choreographies rotating around the $e^2$-axis bifurcates to the family of rotating choreographies that connects to the Lagrange relative equilibrium. Moreover, we compute several relative period-doubling bifurcations and a turning point of the family of planar rotating choreographies, which bifurcates from the figure eight solution when the third component of the angular momentum vector is varied.

Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation

A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria. In this paper the nonlinear problem near a degenerate relative equilibrium is considered. The degeneracy creates a saddle-center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. There is also an induced directional geometric phase in the homoclinic bifurcation. The backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension ( 2 n + 2 ) with an n-dimensional abelian symmetry group. Examples for n = 1 , 2 , 3 are presented to illustrate application of the theory.

Stability transitions for axisymmetric relative equilibria of Euclidean symmetric Hamiltonian systems

In the presence of noncompact symmetry, the stability of relative equilibria under momentum-preserving perturbations does not generally imply robust stability under momentum-changing perturbations. For axisymmetric relative equilibria of Hamiltonian systems with Euclidean symmetry, we investigate different mechanisms of stability: stability by energy–momentum confinement, KAM, and Nekhoroshev stability, and we explain the transitions between them. We apply our results to the Kirchhoff model for the motion of an axisymmetric underwater vehicle, and we numerically study dissipation induced instability of KAM stable relative equilibria for this system.

Relative periodic points of symplectic maps: persistence and bifurcations

In this paper, we study symplectic maps with a continuous symmetry group arising by periodic forcing of symmetric Hamiltonian systems. By Noether's theorem, for each continuous symmetry, the symplectic map has a conserved momentum. We study the persistence of relative periodic points of the symplectic map when momentum is varied and also treat subharmonic persistence and relative subharmonic bifurcations of relative periodic points.

Relative equilibria near stable and unstable Hamiltonian relative equilibria

For a symmetric Hamiltonian system, lower bounds for the number of relative equilibria surrounding stable and formally unstable relative equilibria on nearby energy levels are given.

Bifurcation and forced symmetry breaking in Hamiltonian systems

We consider the phenomenon of forced symmetry breaking in a symmetric Hamiltonian system on a symplectic manifold. In particular we study the persistence of an initial relative equilibrium subjected to this forced symmetry breaking. We see that, under certain nondegeneracy conditions, an estimate can be made on the number of bifurcating relative equilibria. To cite this article: F. Grabsi et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Existence of relative periodic orbits near relative equilibria

We show existence of relative periodic orbits (a.k.a. relative nonlinear normal modes) near relative equilibria of a symmetric Hamiltonian system under an appropriate assumption on the Hessian of the Hamiltonian. This gives a relative version of the Moser-Weinstein theorem. The paper supersedes an earlier paper (this arxiv math.SG/9906007), which contains a mistake.

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.

Bifurcations and Control of Periodic Orbits in Symmetric Hamiltonian Systems; An Application to the Furuta Pendulum

The dynamical behavior of a simple pendulum hanging from a rotating arm (Furuta pendulum) has been investigated. The system is invariant under rotations around the axis and can be formulated as a two degrees of freedom integrable Hamiltonian system in the absence of external forcing. The bifurcation diagram is organized around the relative equilibria (solutions that are invariant under the symmetry) and bridges connecting different subharmonic bifurcations. Special attention has been given to those solutions that could shed some light into the stabilization of the upside down solution and the control problem

On the deficiency indices and self-adjointness of symmetric Hamiltonian systems

The main purpose of this paper is to investigate the formal deficiency indices N ± of a symmetric first-order system Jf′+Bf=λ Hf on an interval I, where I= R or I= R ± . Here J,B, H are n× n matrix-valued functions and the Hamiltonian H⩾0 may be singular even everywhere. We obtain two results for such a system to have minimal numbers ( N ±=0 if I= R resp. N ±=n if I= R + ) and a criterion for their maximality N ±=2n for I= R + (as well as the quasi-regularity). This covers the Kac–Krein and de Branges (Trans. Amer. Math. Soc. 99 (1961) 118) theorems on 2×2 canonical systems and some results from Kogan and Rofe–Beketov (Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75) 5). Some conditions for a canonical system to have intermediate formal deficiency indices are presented, too. We also obtain a generalization of the well known Titchmarsh–Sears theorem for second-order Sturm–Liouville-type equations. This contains results due to Lidskii and Krein as special cases. We present two approaches to the above problems: one dealing with formal deficiency indices and one dealing with (ordinary) deficiency indices. Our main (non-formal) approach is based on the investigation of a symmetric linear relation S min which is naturally associated to a first-order system. This approach works in the framework of extension theory and therefore we investigate in detail the domain D(S min ∗) of S min ∗ . In particular, we prove the so called regularity theorem for D(S min ∗) . As a byproduct of the regularity result we obtain very short proofs of (generalizations of) the main results of the paper by Kogan and Rofe–Beketov (1974/75).

Conformal Hamiltonian systems

Vector fields whose flow preserves a symplectic form up to a constant, such as simple mechanical systems with friction, are called “conformal”. We develop a reduction theory for symmetric conformal Hamiltonian systems, analogous to symplectic reduction theory. This entire theory extends naturally to Poisson systems: given a symmetric conformal Poisson vector field, we show that it induces two reduced conformal Poisson vector fields, again analogous to the dual pair construction for symplectic manifolds. Conformal Poisson systems form an interesting infinite-dimensional Lie algebra of foliate vector fields. Manifolds supporting such conformal vector fields include cotangent bundles, Lie–Poisson manifolds, and their natural quotients.

Stability of Relative Equilibria. Part I: Comparison of Four Methods

In this first part of the paper, we review methods for the investigation of stability of relative equilibria of symmetric Hamiltonian systems and explain them by means of the model problem of a rotating pendulum. For this example the modern approaches, known as energy momentum methods are compared with stability assessment by linearization and by the classical method of Routh.