Periodic Orbits Of Hamiltonian Systems Research Articles

Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit

In this paper, using the Maslov index theory in symplectic geometry, we build up some stability criteria for symmetric periodic orbits in a Hamiltonian system, which is motivated by the recent discoveries in the n-body problem. The key ingredient is a generalized Bott-type iteration formula for periodic solution in the presence of finite group action on the orbit. For second order system, we prove, under general boundary conditions, the close formula for the relationship between the Morse index of an orbit in a Lagrangian system and the Maslov index of the fundamental solution for the corresponding orbit in its Hamiltonian system counterpart, and the boundary conditions cover the cases which appeared in the n-body problem. As an application we consider the stability problem of the celebrated figure-eight orbit due to Chenciner and Montgomery in the planar three-body problem with equal masses, and we clarify the relationship between linear stability and its variational nature on various loop spaces. The basic idea is as follows: the variational characterization of the figure-eight orbit provides information about its Morse index; based on its relation to the Maslov index, our stability criteria come into play.

Continuation of relative periodic orbits in a class of triatomic Hamiltonian systems

We study relative periodic orbits (i.e. time-periodic orbits in a frame rotating at constant velocity) in a class of triatomic Euclidean-invariant (planar) Hamiltonian systems. The system consists of two identical heavy atoms and a light one, and the atomic mass ratio is treated as a continuation parameter. Under some nondegeneracy conditions, we show that a given family of relative periodic orbits existing at infinite mass ratio (and parametrized by phase, rotational degree of freedom and period) persists for sufficiently large mass ratio and for nearby angular velocities (this result is valid for small angular velocities). The proof is based on a method initially introduced by Sepulchre and MacKay [J.-A. Sepulchre, R.S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997) 679–713] and further developed by Muñoz-Almaraz et al. [F.J. Muñoz-Almaraz, et al., Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D 181 (2003) 1–38] for the continuation of normal periodic orbits in Hamiltonian systems. Our results provide several types of relative periodic orbits, which extend from small amplitude relative normal modes [J.-P. Ortega, Relative normal modes for nonlinear Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 665–704] up to large amplitude solutions which are not restrained to a small neighborhood of a stable relative equilibrium. In particular, we show the existence of large amplitude motions of inversion, where the light atom periodically crosses the segment between heavy atoms. This analysis is completed by numerical results on the stability and bifurcations of some inversion orbits as their angular velocity is varied.

Systematic search of symmetric periodic orbits in 2DOF Hamiltonian systems

In this paper, we study in detail the grid search numerical method to locate symmetric periodic orbits in Hamiltonian systems of two degrees of freedom. The method is based on the classical search method but combining up-to-date numerical algorithms in the search and in the integration process. Instead of using Newton methods that requires to differentiate the Poincaré map we use the Brent’s method and in the integration process a Taylor series method that permits us to compute the orbits using extended precision, something highly interesting in the case of unstable periodic orbits. These facts have permitted us to obtain much more periodic orbits than other researchers. Once the families of periodic orbits have been found we study the bifurcations just by comparing with the stability index and the classical generic bifurcations for Hamiltonian systems with and without symmetries. We illustrate the method with four important classical Hamiltonian problems.

Maslov index for homoclinic orbits of Hamiltonian systems

A useful tool for studying nonlinear differential equations is index theory. For symplectic paths on bounded intervals, the index theory has been completely established, which revealed tremendous applications in the study of periodic orbits of Hamiltonian systems. Nevertheless, analogous questions concerning homoclinic orbits are still left open. In this paper we use a geometric approach to set up Maslov index for homoclinic orbits of Hamiltonian systems. On the other hand, a relative Morse index for homoclinic orbits will be derived through Fredholm index theory. It will be shown that these two indices coincide.

Periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces

This paper discusses the existence and multiplicity of periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces. We prove that each such energy hypersurface carries at least one symmetric periodic orbit. Under some suitable pinching conditions, we also get an existence result of multiple symmetric periodic orbits.

Periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces

This paper discusses the existence and multiplicity of periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces. We prove that each such energy hypersurface carries at least one symmetric periodic orbit. Under some suitable pinching conditions, we also get an existence result of multiple symmetric periodic orbits.

On the number of periodic orbits of Hamiltonian systems on positive-type hypersurfaces in [formula omitted

In this paper we prove some existence theorems of multiple periodic orbits of Hamiltonian systems on a class of hypersurfaces in R 2n that are more general than star-shaped ones. Our results extend some earlier works on this subject.

Calculations of Periodic Orbits for Hamiltonian Systems with Regularizable Singularities

We describe a numerical method for calculating periodic orbits, which is a generalization of the monodromy method by Baranger et al. to the case of an arbitrary autonomous dynamical system with a constant of motion C. By specifying the constant C to be the pseudoenergy \(\), i.e. a value of the regularized Hamilton function, and applying the general method to the corresponding Hamilton’s equations, we show that this method can be used successfully for calculating periodic orbits of Hamiltonian systems having regularizable singularities in their original Hamilton functions.

Existence of multiple periodic orbits of Hamiltonian systems on positive-type hypersurfaces in [formula omitted

This paper deals with the fixed energy problem of Hamiltonian systems in R2n. The main result is an existence theorem of multiple periodic orbits for Hamiltonian vector fields on a class of contact type but not necessarily star-shaped hypersurfaces. This is the first multiplicity result for hypersurfaces that are more general than star-shaped ones. The main result is achieved by using critical point theory for a special Hamiltonian function, and central gravity of the proof relies on a delicate construction of such Hamiltonian function.

Stability of traveling water waves on the sphere

We study the stability of a class of traveling waves in a model of weakly nonlinear water waves on the sphere. The model describes free surface potential flow of a fluid layer surrounding a gravitating sphere, and the evolution equations are Hamiltonian. For small amplitude oscillations the Hamiltonian can be expanded in powers of the wave amplitude, yielding simpler model equations. We integrate numerically Galerkin truncations of such a model, focusing on a class of traveling and standing waves that are “near-monochromatic” in space, i.e. have amplitude consisting of one spherical harmonic plus small corrections. We observe that such motions are stable for long times. To explain the observed behavior we use methods of Hamiltonian dynamics, first showing that decay to all but a small number of modes must be very slow. To understand the interaction between these modes we obtain general conditions for the long time nonlinear stability of a certain class of periodic orbits in Hamiltonian systems of resonantly coupled harmonic oscillators.

Spectral Flow and Bifurcation of Critical Points of Strongly Indefinite Functionals: Part II. Bifurcation of Periodic Orbits of Hamiltonian Systems

Our main results here are as follows: Let Xλ be a family of 2π-periodic Hamiltonian vectorfields that depend smoothly on a real parameter λ in [a, b] and has a known, trivial, branch sλ of 2π-periodic solutions. Let Pλ be the Poincaré map of the linearization of Xλ at sλ. If the Conley–Zehnder index of the path Pλ does not vanish, then any neighborhood of the trivial branch of periodic solutions contains 2π-periodic solutions not on the branch. Moreover, if each solution sλ is constant and each linearization Aλ of Xλ at sλ is time independent, then bifurcation of 2π-periodic orbits from the branch of equilibria arises whenever i(Ab)≠i(Ab), where i(A) is the index of the linear Hamiltonian system Ju=Au.

ERROR GROWTH AND A POSTERIORI ERROR ESTIMATES FOR CONSERVATIVE GALERKIN APPROXIMATIONS OF PERIODIC ORBITS IN HAMILTONIAN SYSTEMS

We prove an a posteriori error estimate for approximations of periodic orbits in Hamiltonian systems using Galerkin methods which conserve the Hamiltonian. The error is estimated in terms of the local time step, the residual obtained by inserting the approximate solution into the differential equation, and a stability factor describing relevant stability properties of the adjoint linearized problem. The quantitative growth of the stability factor as a function of time is of particular interest. We show that the stability factor grows linearly with time for a certain class of problems when the conservative scheme is used, in contrast to the quadratic growth of the stability factor, expected for non-conservative schemes in general.

Non-linear stability of singular relative periodic orbits in Hamiltonian systems with symmetry

We generalize the sufficient condition for the stability of relative periodic orbits in symmetric Hamiltonian systems presented in [J.-P. Ortega, T.S. Ratiu, J. Geom. Phys. 32 (1999) 131–159] to the case in which these orbits have non-trivial symmetry. We also describe a block diagonalization, similar in philosophy to the one presented in [J.-P. Ortega, T.S. Ratiu, Nonlinearity 12 (1999) 693–720] for relative equilibria, that facilitates the use of this result in particular examples and shows the relation between the stability of the relative periodic orbit and the orbital stability of the associated singular reduced periodic orbit.

A Dirichlet criterion for the stability of periodic and relative periodic orbits in Hamiltonian systems

The classical Dirichlet criterion on the stability of equilibria in Hamiltonian systems is generalized to the orbital stability of periodic orbits. The result obtained is adapted to the study of the stability of relative periodic orbits in Hamiltonian systems with symmetry that lie at regular level sets of the corresponding momentum map.

Calculations of periodic orbits: The monodromy method and application to regularized systems

We describe a numerical method for calculating periodic orbits, which is a generalization of the monodromy method by Baranger et al. to the case of an arbitrary autonomous dynamical system. Two variants of the method are developed, using the midpoint and the Runge-Kutta discretization of equations of motion, respectively. Particularly, we adapt the first variant for calculating periodic orbits of Hamiltonian systems when the period or the energy is given a priori. Finally, we consider the application of the monodromy method to the case of regularized mechanical systems and demonstrate the use by two examples. (c) 1999 American Institute of Physics.

Holomorphic cylinders with lagrangian boundaries and periodic orbits of hamiltonian systems

This paper outlines how to use holomorphic cylinders with Lagrangian boundaries to prove existence results for periodic orbits of Hamiltonian systems. We describe the case of the cotangent bundle of the Klein bottle, where these results lead to a new obstruction to the existence of Lagrangian embeddings of the Klein bottle into C 2.

Periodic orbits near bifurcations of codimension two: Classical mechanics, semiclassics and Stokes transitions

We investigate classical and semiclassical aspects of codimension--two bifurcations of periodic orbits in Hamiltonian systems. A classification of these bifurcations in autonomous systems with two degrees of freedom or time-periodic systems with one degree of freedom is presented. We derive uniform approximations to be used in semiclassical trace formulas and determine also certain global bifurcations in conjunction with Stokes transitions that become important in the ensuing diffraction catastrophe integrals.

POMULT: A program for computing periodic orbits in hamiltonian systems based on multiple shooting algorithms

POMULT is a FORTRAN code for locating Periodic Orbits and Equilibrium Points in Hamiltonian systems based on 2-point boundary value solvers which use multiple shooting algorithms. The code has mainly been developed for locating periodic orbits in molecular Hamiltonian systems with many degrees of freedom and it utilizes a damped Newton—Raphson method and a secant method. The Graphical User Interface has also been written in the tcl-tk script language for interactively manipulating the input and output data. POMULT provides routines for a general analysis of a dynamical system such as fast Fourier transform of the trajectories, Poincaré surfaces of sections, maximum Lyapunov exponents and evaluation of the classical autocorrelation functions and power spectra.

Persistance d’orbites périodiques relatives dans les systèmes hamiltoniens symétriques

It was known to Poincaré that a non-degenerate periodic orbit in a Hamiltonian system persists to nearby energy-levels. In this Note, we consider the analogous problem for relative periodic orbits in symmetric Hamiltonian systems. We show that non-degenerate relative periodic orbits also persist when shifting to nearby values of the energy-momentum map, under the hypothesis that the group of symmetries acts freely.

Bifurcation of the periodic orbits of Hamiltonian systems: An analysis using normal form theory.

We develop an analytic technique to study the dynamics in the neighborhood of a periodic trajectory of a Hamiltonian system. The theory begins with Poincar\'e and Birkhoff; major modern contributions are due to Meyer, Arnol'd, and Deprit. The realization of the method relies on local Fourier-Taylor series expansions with numerically obtained coefficients. The procedure and machinery are presented in detail on the example of the ``perpendicular'' (z=0) periodic trajectory of the diamagnetic Kepler problem. This simple one-parameter problem well exhibits the power of our technique. Thus, we obtain a precise analytic description of bifurcations observed by J.-M. Mao and J. B. Delos [Phys. Rev. A 45, 1746 (1992)] and explain the underlying dynamics and symmetries. \textcopyright{} 1996 The American Physical Society.