74 research outputs found
Visualization and comparison of classical structures and quantum states of 4D maps
For generic 4D symplectic maps we propose the use of 3D phase-space slices
which allow for the global visualization of the geometrical organization and
coexistence of regular and chaotic motion. As an example we consider two
coupled standard maps. The advantages of the 3D phase-space slices are
presented in comparison to standard methods like 3D projections of orbits, the
frequency analysis, and a chaos indicator. Quantum mechanically, the 3D
phase-space slices allow for the first comparison of Husimi functions of
eigenstates of 4D maps with classical phase space structures. This confirms the
semi-classical eigenfunction hypothesis for 4D maps.Comment: For videos with rotated view of the 3D phase-space slices in high
resolution see http://www.comp-phys.tu-dresden.de/supp
The structure of invariant tori in a 3D galactic potential
We study in detail the structure of phase space in the neighborhood of stable
periodic orbits in a rotating 3D potential of galactic type. We have used the
color and rotation method to investigate the properties of the invariant tori
in the 4D spaces of section. We compare our results with those of previous
works and we describe the morphology of the rotational, as well as of the tube
tori in the 4D space. We find sticky chaotic orbits in the immediate
neighborhood of sets of invariant tori surrounding 3D stable periodic orbits.
Particularly useful for galactic dynamics is the behavior of chaotic orbits
trapped for long time between 4D invariant tori. We find that they support
during this time the same structure as the quasi-periodic orbits around the
stable periodic orbits, contributing however to a local increase of the
dispersion of velocities. Finally we find that the tube tori do not appear in
the 3D projections of the spaces of section in the axisymmetric Hamiltonian we
examined.Comment: 26 pages, 34 figures, accepted for publication in the International
Journal of Bifurcation and Chao
Puzzling out the coexistence of terrestrial planets and giant exoplanets. The 2/1 resonant periodic orbits
Hundreds of giant planets have been discovered so far and the quest of
exo-Earths in giant planet systems has become intriguing. In this work, we aim
to address the question of the possible long-term coexistence of a terrestrial
companion on an orbit interior to a giant planet, and explore the extent of the
stability regions for both non-resonant and resonant configurations. Our study
focuses on the restricted three-body problem, where an inner terrestrial planet
(massless body) moves under the gravitational attraction of a star and an outer
massive planet on a circular or elliptic orbit. Using the Detrended Fast
Lyapunov Indicator as a chaotic indicator, we constructed maps of dynamical
stability by varying both the eccentricity of the outer giant planet and the
semi-major axis of the inner terrestrial planet, and identify the boundaries of
the stability domains. Guided by the computation of families of periodic
orbits, the phase space is unravelled by meticulously chosen stable periodic
orbits, which buttress the stability domains. We provide all possible stability
domains for coplanar symmetric configurations and show that a terrestrial
planet, either in mean-motion resonance or not, can coexist with a giant
planet, when the latter moves on either a circular or an (even highly)
eccentric orbit. New families of symmetric and asymmetric periodic orbits are
presented for the 2/1 resonance. It is shown that an inner terrestrial planet
can survive long time spans with a giant eccentric outer planet on resonant
symmetric orbits, even when both orbits are highly eccentric. For 22 detected
single-planet systems consisting of a giant planet with high eccentricity, we
discuss the possible existence of a terrestrial planet. This study is
particularly suitable for the research of companions among the detected systems
with giant planets, and could assist with refining observational data.Comment: Accepted for publication in A&
Symbolic dynamics in a binary asteroid system
We highlight the existence of a topological horseshoe arising from a
a--priori stable model of the binary asteroid dynamics. The inspection is
numerical and uses correctly aligned windows, as described in a recent paper by
A. Gierzkiewicz and P. Zgliczy\'nski, combined with a recent analysis of an
associated secular problem.Comment: 20 pages, 10 figure
Time-frequency analysis of the restricted three-body problem: transport and resonance transitions
A method of time-frequency analysis based on wavelets is applied to the problem of transport between different regions of the solar system, using the model of the circular restricted three-body problem in both the planar and the spatial versions of the problem.. The method is based on the extraction of instantaneous frequencies from the wavelet transform of numerical solutions. Time-varying frequencies provide a good diagnostic tool to discern chaotic trajectories from regular ones, and we can identify resonance islands that greatly affect the dynamics. Good accuracy in the calculation of time-varying frequencies allows us to determine resonance trappings of chaotic trajectories and resonance transitions. We show the relation between resonance transitions and transport in different regions of the phase space
Fast numerical algorithms for the computation of invariant tori in Hamiltonian systems
In this paper, we develop numerical algorithms that use small requirements of storage and operations for the computation of invariant tori in Hamiltonian systems (exact symplectic maps and Hamiltonian vector fields). The algorithms are based on the parameterization method and follow closely the proof of the KAM theorem given in [LGJV05] and [FLS07]. They essentially consist in solving a functional equation satisfied by the invariant tori by using a Newton method. Using some geometric identities, it is possible to perform a Newton step using little storage and few operations. In this paper we focus on the numerical issues of the algorithms (speed, storage and stability) and we refer to the mentioned papers for the rigorous results. We show how to compute efficiently both maximal invariant tori and whiskered tori, together with the associated invariant stable and unstable manifolds of whiskered tori. Moreover, we present fast algorithms for the iteration of the quasi-periodic cocycles and the computation of the invariant bundles, which is a preliminary step for the computation of invariant whiskered tori. Since quasi-periodic cocycles appear in other contexts, this section may be of independent interest. The numerical methods presented here allow to compute in a unified way primary and secondary invariant KAM tori. Secondary tori are invariant tori which can be contracted to a periodic orbit. We present some preliminary results that ensure that the methods are indeed implementable and fast. We postpone to a future paper optimized implementations and results on the breakdown of invariant tori
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