Carles Simó

Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits

In this paper we deal with an alternative technique to study global dynamics in Hamiltonian systems, the mean exponential growth factor of nearby orbits (MEGNO), that proves to be efficient to investigate both regular and stochastic components of phase space. It provides a clear picture of resonance structures, location of stable and unstable periodic orbits as well as a measure of hyperbolicity in chaotic domains which coincides with that given by the Lyapunov characteristic number. Here the MEGNO is applied to a rather simple model, the 3D perturbed quartic oscillator, in order to visualize the structure of its phase space and obtain a quite clear picture of its resonance structure. Examples of application to multi-dimensional canonical maps are also included.

Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem

Abstract We consider an n -degrees of freedom Hamiltonian system near an elliptic equilibrium point. The system is put in normal form (up to an arbitrary order and with respect to some resonance module) and estimates are obtained for both the size of the remainder and for the domain of convergence of the transformation leading to normal form. A bound to the rate of diffusion is thus found, and by optimizing the order of normalization exponential estimates of Nekhoroshevs type are obtained. This provides explicit estimates for the stability properties of the elliptic point, and leads in some cases to “effective stability,” i.e., stability up to finite but long times. An application to the stability of the triangular libration points in the spatial restricted three body is also given.

Study of the transfer from the Earth to a halo orbit around the equilibrium pointL 1

The purpose of this paper is to study a transfer strategy from the vicinity of the Earth to a halo orbit around the equilibrium pointL1 of the Earth-Sun system. The study is done in the real solar system (we use the DE-118 JPL ephemeris in the simulations of motion) although some simplified models, such as the restricted three body problem (RTBP) and the bicircular problem, have been also used in order to clarify the geometrical aspects of the problem. The approach used in the paper makes use of the hyperbolic character of the halo orbits under consideration. The invariant stable manifold of these orbits enables the transfer to be achieved with, theoretically, only one manoeuvre: the one of insertion into the stable manifold. For the total Δv required, the figures obtained are similar to the ones given by the standard procedures of optimization.

Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

Abstract The restricted three-body problem is considered for values of the Jacobi constant C near the value C 2 associated to the Euler critical point L 2 . A Lyapunov family of periodic orbits near L 2 , the so-called family ( c ), is born for C = C 2 and exists for values of C less than C 2 . These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ≲ C 2 and μ ≳ 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.

Hamiltonian systems with three or more degrees of freedom

Preface. List of Participants. Lectures. Contributions. List of Authors. Subject Index. Lectures. Inflection points, extatic points and curve shortening S. Angenent. Topologically necessary singularities on moving wavefronts and caustics V.I. Arnold. Heteroclinic chains of skew product Hamiltonian systems S.V. Bolotin. Order and chaos in 3-D systems G. Contopoulos, et al. Splitting of separatrices in Hamiltonian systems and symplectic maps A. Delshams, et al. On the discrete one-dimensional quasi-periodic Schrodinger equation and other smooth quasi-periodic skew products L.H. Eliasson. Lindstedt series and Kolmogorov theorem G. Gallavotti. A classical self-contained proof of Kolmogorovs theorem on invariant tori A. Giorgilli, U. Locatelli. Dynamical Stability in Lagrangian Systems P. Boyland, C. Gole. The origin of chaotic behaviour in the Kirkwood gaps J. Henrard. Examples of compact hypersurfaces in R2p, 2p >= 6, with no periodic orbits M. Herman. Hamiltonian systems with three degrees of freedom and Hydrodynamics V.V. Kozlov. Introduction to frequency map analysis J. Laskar. Lindstedt series for lower dimensional tori A. Jorba, et al. Arnold diffusion a compendium of remarks and questions P. Lochak. Old and new applications of KAM theory J. Moser. On adiabatic invariance in two-frequency systems A. Neishtadt. The method of rational approximations: theory and applications J. Seimenis. Dynamical systems methods for space missions on a vicinity of collinear libration points C. Simo. A mechanism of ergodicity in standard-like maps Ya. G. Sinai. Continuous averaging in Hamiltonian systems D.V. Treschev. Phase Space Geometry and Dynamics Associatedwith the 1:2:2 Resonance S. Wiggins. From singular point analysis to rigorous results on integrability: A dream of S. Kowalevskaya H. Yoshida. Contributions. Time singularities for polynomial Hamiltonians with analytic time dependence S. Abenda. Numerical study of turbulence in N-body Hamilton-ian systems with long range force I. Relaxation M. Antoni, et al. Phase space structures in 3 and 4 degrees of freedom: Application to chemical reactions K.M. Atkins, M. Hutson. Modulated diffusion for symplectic maps A. Bazzani, F. Brini. On the Jeans-Landau-Teller approximation for adiabatic invariants G. Benettin. Periodic orbits and quantum mechanics of molecular Hamiltonian systems R.M. Benito, F. Borondo. Chaos in atom-surface collisions F. Borondo, et al. On the non-integrability of the Mixmaster Universe model T. Bountis, L. Drossos. Branching of solutions as obstruction to the existence of a meromorphic integral in many-dimensional systems S.A. Dovbysh. The energy surfaces of the Kovalevskaya-top H.R. Dullin. Filling rates for linear flow on the torus: recent progress and applications H.S. Dumas. Invariant spectra of orbits in multidimensional symplectic maps N. Voglis, et al. On the convergence of formal series containing small divisors L. Chierchia, C. Falcolini. Systems with an invariant measure on Lie groups Y.N. Fedorov. Stochasticity of the 2/1 asteroidal resonance: a symplectic mapping approach S. Ferraz-Mello. On perturbed oscillators in 1-1-1 resonance: critical inclination in the 3D Henon-Heiles potential S. Ferrer, et al. Splitting of separatrices for (fast) quasiperiodic forcing A. Delshams, et al. A possible mechanism for the KAM tori breakdown G. Gentile, V.

Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem

Abstract The classical Hill’s problem is a simplified version of the restricted three body problem (RTBP) where the distance of the two massive bodies (say, primary for the largest one and secondary for the smallest) is made infinity through the use of Hill’s variables and a limiting procedure so that a neighborhood of the secondary can be studied in detail. In this way it is the zeroth-order approximation in powers of μ 1/3 . The Levi-Civita regularization takes the Hamiltonian into the form of two uncoupled harmonic oscillators perturbed by the Coriolis force and the Sun action, polynomials of degree 4 and 6, respectively. The goal of this paper is multiple. It presents a detailed description of the main features, including a global description of the dynamics, when the zero velocity curve (zvc) confines the motion. Then it focuses on the collinear equilibrium points and its nearby periodic orbits. Several homoclinic and heteroclinic connections are displayed. Persistence of confined motion when the zvc opens is one of the major concerns. The geometrical behavior of the center-stable/unstable manifolds of the libration points L 1 and L 2 is studied. Suitable Poincare sections make apparent the relation between these manifolds and the destruction of the invariant KAM tori surrounding the secondary. These results extend immediately to the RTBP. Some practical applications to astronomy and space missions are mentioned. The methodology presented here can be useful on a more general framework for many readers in other areas and not only in celestial mechanics.

On quasi-periodic perturbations of elliptic equilibrium points

This work focuses on quasi-periodic time-dependent perturbations of ordinary differential equations near elliptic equilibrium points. This means studying \[ \dot x = (A + \varepsilon Q(t,\varepsilon ))x + \varepsilon g(t,\varepsilon ) + h(x,t,\varepsilon ), \] where A is elliptic and h is

On the reducibility of linear differential equations with quasiperiodic coefficients

\mathcal{O}(x^2 )

The splitting of separatrices for analytic diffeomorphisms

. It is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to

New Families of Solutions in N-Body Problems

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