Elements of Non-Linear Hamiltonian Dynamics

In this paper we present a set of generic results on Hamiltonian non-linear dynamics. We show the necessary conditions for a Hamiltonian system to present a non-twist scenario and from that we introduce the isochronous resonances. The generality of these resonances is shown from the Hamiltonian given by the Birkhof-Gustavson normal form, which can be considered a toy model, and from an optic system governed by the non-linear map of the annular billiard. We also define a special kind of transport barrier called robust torus. The meanders and shearless curves are also presented and we show the most robust shearless barrier associated with the rotation numbers.

Dynamical hierarchy in transition states: Why and how does a system climb over the mountain?

How a reacting system climbs through a transition state during the course of a reaction has been an intriguing subject for decades. Here we present and quantify a technique to identify and characterize local invariances about the transition state of an N-particle Hamiltonian system, using Lie canonical perturbation theory combined with microcanonical molecular dynamics simulation. We show that at least three distinct energy regimes of dynamical behavior occur in the region of the transition state, distinguished by the extent of their local dynamical invariance and regularity. Isomerization of a six-atom Lennard–Jones cluster illustrates this: up to energies high enough to make the system manifestly chaotic, approximate invariants of motion associated with a reaction coordinate in phase space imply a many-body dividing hypersurface in phase space that is free of recrossings even in a sea of chaos. The method makes it possible to visualize the stable and unstable invariant manifolds leading to and from the transition state, i.e., the reaction path in phase space, and how this regularity turns to chaos with increasing total energy of the system. This, in turn, illuminates a new type of phase space bottleneck in the region of a transition state that emerges as the total energy and mode coupling increase, which keeps a reacting system increasingly trapped in that region.

Energy-based motion control of marine vehicles using interconnection and damping assignment passivity-based control : a survey

This paper reviews some recent results in motion control of marine vehicles using a technique called Interconnection and Damping Assignment Passivity-based Control (IDA-PBC). This approach to motion control exploits the fact that vehicle dynamics can be described in terms of energy storage, distribution, and dissipation, and that the stable equilibrium points of mechanical systems are those at which the potential energy attains a minima. The control forces are used to transform the closed-loop dynamics into a port-controlled Hamiltonian system with dissipation. This is achieved by shaping the energy-storing characteristics of the system, modifying its interconnection structure (how the energy is distributed), and injecting damping. The end result is that the closed-loop system presents a stable equilibrium (hopefully global) at the desired operating point. By forcing the closed-loop dynamics into a Hamiltonian form, the resulting total energy function of the system serves as a Lyapunov function that can be used to demonstrate stability. We consider the tracking and regulation of fully actuated unmanned underwater vehicles, its extension to under-actuated slender vehicles, and also manifold regulation of under-actuated surface vessels. The paper is concluded with an outlook on future research.

Renormalization methods in KAM theory

It is well known that an integrable (in the sense of Arnold-Jost) Hamiltonian system gives rise to quasi-periodic motion with trajectories running on invariant tori. These tori foliate the whole phase space. If we perturb an integrable system, the Kolmogorow-Arnold-Moser (KAM) theorem states that, provided some non-degeneracy condition and that the perturbation is sufficiently small, most of the invariant tori carrying quasi-periodic motion persist, getting only slightly deformed. The measure of the persisting invariant tori is large together with the inverse of the size of the perturbation. In the first part of the thesis we shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non analytic perturbation (the latter will only be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which theperturbations are analytic approximations of the original one. We will finally show that the approximate solutions will converge to a differentiable solution of our original problem. In the second part we will use an RG scheme using continuous scales, so that instead of solving an iterative equation as in the classical RG KAM, we will end up solving a partial differential equation. This will allow us to reduce the complications of treating a sequence of iterative equations to the use of the Banach fixed point theorem in a suitable Banach space.

Undamped Oscillations of Collisionless Stellar Systems: Spheres,Spheroids and Discs.

The collisionless Boltzmann equation governing self-gravitating systems such as galaxies has recently been shown to admit exact oscillating solutions with planar and spherical symmetry. The relation of the spherically symmetric solutions to the Virial theorem, as well as generalizations to non-uniform spheres, uniform spheroids and discs form the subject of this paper. These models generalize known families of static solutions. The case of the spheroid is worked out in some detail. Quasiperiodic as well as chaotic time variation of the two axes is demonstrated by studying the surface of section for the associated Hamiltonian system with two degrees of freedom. The relation to earlier work and possible implications for the general problem of collisionless relaxation in self gravitating systems are also discussed.

Long term stability studies of particle storage rings using polynomial Maps

Long-term stability studies of particle storage rings can not be carried out using conventional numerical integration algorithms. We require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the sym-plectic map representing the Hamiltonian system is refactorized using polynomial symplectic maps. This method is used to perform long term integration on a particle storage ring.

A search for invariant relative satellite motion

This report presents the canonical Hamiltonian formulation of relative satellite motion. The unperturbed Hamiltonian model is shown to be equivalent to the well known Hill-Clohessy-Wilshire (HCW) linear formulation. The in°uence of perturbations of the nonlinear Gravitational potential and the oblateness of the Earth; J2 perturbations are also modelled within the Hamiltonian formulation. The modelling incorporates eccentricity of the reference orbit. The corresponding Hamiltonian vector ¯elds are computed and implemented in Simulink. A numerical method is presented aimed at locating periodic or quasi-periodic relative satellite motion. The numerical method outlined in this paper is applied to the Hamiltonian system. Although the orbits considered here are weakly unstable at best, in the case of eccentricity only, the method ¯nds exact periodic orbits. When other perturbations such as nonlinear gravitational terms are added, drift is signicantly reduced and in the case of the J2 perturbation with and without the nonlinear gravitational potential term, bounded quasi-periodic solutions are found. Advantages of using Newton's method to search for periodic or quasi-periodic relative satellite motion include simplicity of implementation, repeatability of solutions due to its non-random nature, and fast convergence. Given that the use of bounded or drifting trajectories as control references carries practical di±culties over long-term missions, Principal Component Analysis (PCA) is applied to the quasi-periodic or slowly drifting trajectories to help provide a closed reference trajectory for the implementation of closed loop control. In order to evaluate the e®ect of the quality of the model used to generate the periodic reference trajectory, a study involving closed loop control of a simulated master/follower formation was performed. 2 The results of the closed loop control study indicate that the quality of the model employed for generating the reference trajectory used for control purposes has an important in°uence on the resulting amount of fuel required to track the reference trajectory. The model used to generate LQR controller gains also has an e®ect on the e±ciency of the controller.

Chirikov diffusion in the asteroidal three-body resonance (5,-2,-2)

The theory of diffusion in many-dimensional Hamiltonian system is applied to asteroidal dynamics. The general formulation developed by Chirikov is applied to the NesvornA1/2-Morbidelli analytic model of three-body (three-orbit) mean-motion resonances (Jupiter-Saturn-asteroid). In particular, we investigate the diffusion along and across the separatrices of the (5, -2, -2) resonance of the (490) Veritas asteroidal family and their relationship to diffusion in semi-major axis and eccentricity. The estimations of diffusion were obtained using the Melnikov integral, a Hadjidemetriou-type sympletic map and numerical integrations for times up to 10(8) years.

Plasma confinement in tokamaks with robust torus

We present a non-linear symplectic map that describes the alterations of the magnetic field lines inside the tokamak plasma due to the presence of a robust torus (RT) at the plasma edge. This RT prevents the magnetic field lines from reaching the tokamak wall and reduces, in its vicinity, the islands and invariant curve destruction due to resonant perturbations. The map describes the equilibrium magnetic field lines perturbed by resonances created by ergodic magnetic limiters (EMLs). We present the results obtained for twist and non-twist mappings derived for monotonic and non-monotonic plasma current density radial profiles, respectively. Our results indicate that the RT implementation would decrease the field line transport at the tokamak plasma edge. (C) 2010 Elsevier B.V. All rights reserved.

Tunneling among rotation tori

We consider an integrable Hamiltonian system generated by the resonant normal form in order to study a particular mechanism of tunneling. We isolated near doublets of energy corresponding to rotation tori of the classical dynamics counterpart and the degeneracies breakdown is attributed to rotation-rotation tunneling. (C) 2008 Elsevier B.V. All rights reserved.

Tunneling among rotation tori

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

The defocusing mechanism of isochronous resonances

We present a numerical study concerning the defocusing mechanism of isochronous resonance island chains in the presence of two permanent robust tori. The process is initialized and concluded through bifurcations of fixed points located on the robust tori. Our approach is based on a Hamiltonian system derived from the resonant normal form. Choosing a convenient parameter in this system, we are able to depict a comprehensive analysis of the dynamics of the problem. (c) 2004 Elsevier B.V. All rights reserved.

Plasma confinement in tokamaks with robust torus

We present a non-linear symplectic map that describes the alterations of the magnetic field lines inside the tokamak plasma due to the presence of a robust torus (RT) at the plasma edge. This RT prevents the magnetic field lines from reaching the tokamak wall and reduces, in its vicinity, the islands and invariant curve destruction due to resonant perturbations. The map describes the equilibrium magnetic field lines perturbed by resonances created by ergodic magnetic limiters (EMLs). We present the results obtained for twist and non-twist mappings derived for monotonic and non-monotonic plasma current density radial profiles, respectively. Our results indicate that the RT implementation would decrease the field line transport at the tokamak plasma edge. © 2010 Elsevier B.V. All rights reserved.

Periodic orbits and non-integrability of Armbruster-Guckenheimer-Kim potential

In this paper we study the periodic orbits of the Hamiltonian system with the Armburster-Guckenheimer Kim potential and its C1 non-integrability in the sense of Liouville-Arnold.

Solution of optimal continuous low-thrust transfer using Lie transforms

This paper addresses the problem of optimal constant continuous low-thrust transfer in the context of the restricted two-body problem (R2BP). Using the Pontryagin’s principle, the problem is formulated as a two point boundary value problem (TPBVP) for a Hamiltonian system. Lie transforms obtained through the Deprit method allow us to obtain the canonical mapping of the phase flow as a series in terms of the order of magnitude of the thrust applied. The reachable set of states starting from a given initial condition using optimal control policy is obtained analytically. In addition, a particular optimal transfer can be computed as the solution of a non-linear algebraic equation. Se investiga el uso de series y transformadas de Lie en problemas de optimización de trayectorias de satélites impulsados por motores de bajo empuje