Stability of equilibrium solutions of Hamiltonian systems with n-degrees of freedom and single resonance in the critical case

In this paper we give new results for the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom. Our Main Theorem generalizes several results existing in the literature and mainly we give information in the critical cases (i.e., the condition of stability and instability is not fulfilled). In particular, our Main Theorem provides necessary and sufficient conditions for stability of the equilibrium solutions under the existence of a single resonance. Using analogous tools used in the Main Theorem for the critical case, we study the stability or instability of degenerate equilibrium points in Hamiltonian systems with one degree of freedom. We apply our results to the stability of Hamiltonians of the type of cosmological models as in planar as in the spatial case.     

Stability of a Hamiltonian system in a limiting case

We give a fairly simple geometric proof that an equilibrium point of a Hamiltonian system of two degrees of freedom is Liapunov stable in a degenerate case. That is the 1: −1 resonance case where the linearized system has double pure imaginary eigenvalues ±iω, ω ≠ 0 and the Hamiltonian is indefinite. The linear system is weakly unstable, but if a particular coefficient in the normalized Hamiltonian is of the correct sign then Moser’s invariant curve theorem can be applied to show that the equilibrium point is encased in invariant tori and thus it is stable.     

The stability of the equilibrium position of Hamiltonian systems with two degrees of freedom

The stability of the equilibrium position at the origin of coordinates of a Hamiltonian system with two degrees of freedom with a Hamiltonian, the unperturbed part of which generates oscillators with a cubic restoring force, is considered. It is proved that the equilibrium position is Lyapunov conditionally stable for initial values which do not belong to a certain surface of the Hamiltonian level. A reduction of the system onto this surface shows that, in the generic case, unconditional Lyapunov stability also occurs.     

Control by Interconnection and Energy Shaping of the Timoshenko Beam

In this paper, the dynamical control of a mixed finite and infinite dimensional mechanical system is approached within the framework of port Hamiltonian systems. In particular, a flexible beam, modeled according to the Timoshenko theory and in distributed port Hamiltonian form, with a mass under gravity field connected at a free end, is considered. The control problem is approached by generalization of the concept of structural invariant (Casimir function) to the infinite dimensional case and the so-called control by interconnection technique is extended to the infinite dimensional case. In this way, finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy, i.e., by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipation effect is introduced.     

An Algorithm for Detecting Directional Quasi-Convexity

Directional Quasi-Convexity (DQC) is a sufficient condition for Nekhoroshev stability, that is, stability for finite but very long times, of elliptic equilibria of Hamiltonian systems. The numerical detection of DQC is elementary for system with three degrees of freedom. In this article, we propose a recursive algorithm to test DQC in any number n4 of degrees of freedom.     

On stability in Hamiltonian systems with two degrees of freedom

We consider the stability of the equilibrium position at the origin of coordinates of a Hamiltonian system with two degrees of freedom whose unperturbed part describes oscillators with restoring force of odd order greater than 1. It is proved that if the exponents of the restoring force of the oscillators are not equal, then the equilibrium position is Lyapunov stable. If the exponents are equal, then the equilibrium position is conditionally stable for trajectories not belonging to some level surface of the Hamiltonian. The reduction of the system to this surface shows that the equilibrium position is stable in the case of general position.     

On Hamiltonian Systems with a Homoclinic Orbit to a Saddle-Center

We consider a real analytic Hamiltonian system with two degrees of freedom having a homoclinic orbit to a saddle-center equilibrium (two nonzero real and two nonzero imaginary eigenvalues). We take a two-parameter unfolding for such a system and show that in the nonresonance case, there are countable sets of multi-round homoclinic orbits to a saddle-center. We also find families of periodic orbits accumulating at homoclinic orbits. Bibliography: 6 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 187–193.     

The structure of semiclassical asymptotic expansions of antisymmetric solutions of the stationary Schrödinger equation

We consider the semiclassical asymptotics of eigenfunctions for the Hamiltonian of a quantum-mechanical system ofN identical fermions withd degrees of freedom without spin interaction. In the one-dimensional case (d=1), examples are known in which the ground antisymmetric state in the semiclassical limit is the product ofN(N−1)/2 two-particle wave functions. We construct a nontrivial generalization of this property ford>1. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 257–269, February, 2000.     

Nonlinear Stability in Resonant Cases: A Geometrical Approach

Summary. In systems with two degrees of freedom, Arnold's theorem is used for studying nonlinear stability of the origin when the quadratic part of the Hamiltonian is a nondefinite form. In that case, a previous normalization of the higher orders is needed, which reduces the Hamiltonian to homogeneous polynomials in the actions. However, in the case of resonances, it could not be possible to bring the Hamiltonian to the normal form required by Arnold's theorem. In these cases, we determine the stability from analysis of the normalized phase flow. Normalization up to an arbitrary order by Lie-Deprit transformation is carried out using a generalization of the Lissajous variables. Received November 8, 2000; accepted January 6, 2001 Online publication March 23, 2001     

Twistless Tori Near Low-Order Resonances

In this paper, we investigate the behavior of the twist near low-order resonances of a periodic orbit or an equilibrium of a Hamiltonian system with two degrees of freedom. Namely, we analyze the case where the Hamiltonian has multiple eigenvalues (the Hamiltonian Hopf bifurcation) or a zero eigenvalue near the equilibrium and the case where the system has a periodic orbit whose multipliers are equal to 1 (the saddle-center bifurcation) or −1 (the period-doubling bifurcation). We show that the twist does not vanish at least in a small neighborhood of the period-doubling bifurcation. For the saddle-center bifurcation and the resonances of the equilibrium under consideration, we prove the existence of a “twistless” torus for sufficiently small values of the bifurcation parameter. An explicit dependence of the energy corresponding to the twistless torus on the bifurcation parameter is derived. Bibliography: 6 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 135–144.     

A note on degenerate corank-one singularities of integrable Hamiltonian systems

We prove that, in a neighborhood of a corank-1 singularity of an analytic integrable Hamiltonian system with n degrees of freedom, there is a locally-free analytic symplectic \Bbb T^n-1 {\Bbb T}^{n-1} -action which preserves the moment map, under some mild conditions. This result allows one to classify generic degenerate corank-one singularities of integrable Hamiltonian systems. It can also be applied to the study of (non)integrability of perturbations of integrable systems.     

Stability sets of multiparameter Hamiltonian systems

A real linear Hamiltonian system with constant coefficients that depend on several real parameters is considered. A method is proposed for calculating the sets of all values of the parameters for which the stationary solution of this system is stable for fixed values of the parameters (that is, the stability sets). The application of the method is demonstrated for a gyroscopic problem described by a Hamiltonian system with four degrees of freedom and three parameters. Computer algebra, in particular, a Gröbner basis and a Power Geometry are used. It is shown that the four-parameter generalization of this problem does not contain fundamentally new difficulties.     

On nonlinear motions of Hamiltonian system in case of fourth order resonance

We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin, which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3: 1. We study nonlinear conditionally periodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the so-called truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionally periodic. By using the KAM theory methods we show that most of the conditionally periodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that are not conditionally periodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession.     

On the Constructive Algorithm for Stability Analysis of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Second-order Resonance Case

This paper is concerned with a nonautonomous Hamiltonian system with two degrees of freedom whose Hamiltonian is a 2π-periodic function of time and analytic in a neighborhood of an equilibrium point. It is assumed that the system exhibits a secondorder resonance, i. e., the system linearized in a neighborhood of the equilibrium point has a double multiplier equal to ?1. The case of general position is considered when the monodromy matrix is not reduced to diagonal form and the equilibrium point is linearly unstable. In this case, a nonlinear analysis is required to draw conclusions on the stability (or instability) of the equilibrium point in the complete system.In this paper, a constructive algorithm for a rigorous stability analysis of the equilibrium point of the above-mentioned system is presented. This algorithm has been developed on the basis of a method proposed in [1]. The main idea of this method is to construct and normalize a symplectic map generated by the phase flow of a Hamiltonian system.It is shown that the normal form of the Hamiltonian function and the generating function of the corresponding symplectic map contain no third-degree terms. Explicit formulae are obtained which allow one to calculate the coefficients of the normal form of the Hamiltonian in terms of the coefficients of the generating function of a symplectic map.The developed algorithm is applied to solve the problem of stability of resonant rotations of a symmetric satellite.     

The Conley-Zehnder index and the saddle-center equilibrium

We consider Hamiltonian systems with two degrees of freedom. We suppose the existence of a saddle-center equilibrium in a strictly convex component S of its energy level. Moser's normal form for such equilibriums and a theorem of Hofer, Wysocki and Zehnder are used to establish the existence of a periodic orbit in S with several topological properties. We also prove the explosion of the Conley-Zehnder index of any periodic orbit that passes close to the saddle-center equilibrium.     

On the Stability of a Periodic Hamiltonian System with One Degree of Freedom in a Transcendental Case

The stability of an equilibrium of a nonautonomous Hamiltonian system with one degree of freedom whose Hamiltonian function depends 2π-periodically on time and is analytic near the equilibrium is considered. The multipliers of the system linearized around the equilibrium are assumed to be multiple and equal to 1 or–1. Sufficient conditions are found under which a transcendental case occurs, i.e., stability cannot be determined by analyzing the finite-power terms in the series expansion of the Hamiltonian about the equilibrium. The equilibrium is proved to be unstable in the transcendental case.     

On the Stability of Equilibria in Two-Degrees-of- Freedom Hamiltonian Systems Under Resonances

We consider the problem of stability of equilibrium points in Hamiltonian systems of two degrees of freedom under resonances. Determining the stability or instability is based on a geometrical criterion based on how two surfaces, related with the normal form, intersect one another. The equivalence of this criterion with a result of Cabral and Meyer is proved. With this geometrical procedure, the hypothesis may be extended to more general cases.     

Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case

This paper concerns with the study of the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a neighborhood of the equilibrium point with 1-degree of freedom in the degenerate case H = q⁴ + H₅ + H₆ +.... Our main results complement the study initiated by Markeev in [9].     

Bifurcations in a family of Hamiltonian systems and associated nontwist cubic maps

The aim of the paper is to study systems with one-and-a-half degrees of freedom generated by a Hamiltonian with a quartic unperturbed part and broad perturbation spectrum. To this end, an approximate interpolating Hamiltonian system is firstly studied. Behaviour of the Poincaré–Birkhoff or dimerised chains in their routes to reconnection when the perturbation parameter varies is particularly presented. In the second step, a discrete system associated to the full Hamiltonian system is constructed and studied. We point out interesting properties of the dynamics of the Poincaré–Birkhoff or dimerised chains, such as pairs of homoclinic orbits to the same equilibrium point (sandglass) and triple reconnection. Then we use the scenario of reconnections to explain the destruction of transport barriers in the non-autonomous system.     

Non-linear oscillations of a Hamiltonian system with 2:1 resonance

Non-linear oscillations of an autonomous Hamiltonian system with two degrees of freedom in the neighbourhood of a stable equilibrium are considered. It is assumed that the frequency ratio of the linear oscillations is close to or equal to two, and that the Hamiltonian is sign-definite in the neighbourhood of the equilibrium. A solution is presented to the problem of the orbital stability of periodic motions emanating from the equilibrium position. Conditionally periodic motions of an approximate system are analysed taking into account terms of order up to and including three in the normalized Hamiltonian. The KAM theory is used to consider the problem of maintaining these motions taking into account fourth- and higher-order terms in the series expansion of the Hamiltonian in a sufficiently small neighbourhood of the equilibrium. The results are used to investigate non-linear oscillations of an elastic pendulum.