86,765 research outputs found
From order to chaos in Earth satellite orbits
We consider Earth satellite orbits in the range of semi-major axes where the
perturbing effects of Earth's oblateness and lunisolar gravity are of
comparable order. This range covers the medium-Earth orbits (MEO) of the Global
Navigation Satellite Systems and the geosynchronous orbits (GEO) of the
communication satellites. We recall a secular and quadrupolar model, based on
the Milankovitch vector formulation of perturbation theory, which governs the
long-term orbital evolution subject to the predominant gravitational
interactions. We study the global dynamics of this two-and-a-half
degrees-of-freedom Hamiltonian system by means of the fast Lyapunov indicator
(FLI), used in a statistical sense. Specifically, we characterize the degree of
chaoticity of the action space using angle-averaged normalized FLI maps,
thereby overcoming the angle dependencies of the conventional stability maps.
Emphasis is placed upon the phase-space structures near secular resonances,
which are of first importance to the space debris community. We confirm and
quantify the transition from order to chaos in MEO, stemming from the critical
inclinations, and find that highly inclined GEO orbits are particularly
unstable. Despite their reputed normality, Earth satellite orbits can possess
an extraordinarily rich spectrum of dynamical behaviors, and, from a
mathematical perspective, have all the complications that make them very
interesting candidates for testing the modern tools of chaos theory.Comment: 30 pages, 9 figures. Accepted for publication in the Astronomical
Journa
The Control of Dynamical Systems - Recovering Order from Chaos -
Following a brief historical introduction of the notions of chaos in
dynamical systems, we will present recent developments that attempt to profit
from the rich structure and complexity of the chaotic dynamics. In particular,
we will demonstrate the ability to control chaos in realistic complex
environments. Several applications will serve to illustrate the theory and to
highlight its advantages and weaknesses. The presentation will end with a
survey of possible generalizations and extensions of the basic formalism as
well as a discussion of applications outside the field of the physical
sciences. Future research avenues in this rapidly growing field will also be
addressed.Comment: 18 pages, 9 figures. Invited Talk at the XXIth International
Conference on the Physics of Electronic and Atomic Collisions (ICPEAC), July
22-27, 1999 (Sendai, Japan
Resurvey of order and chaos in spinning compact binaries
This paper is mainly devoted to applying the invariant, fast, Lyapunov
indicator to clarify some doubt regarding the apparently conflicting results of
chaos in spinning compact binaries at the second-order post-Newtonian
approximation of general relativity from previous literatures. It is shown with
a number of examples that no single physical parameter or initial condition can
be described as responsible for causing chaos, but a complicated combination of
all parameters and initial conditions is responsible. In other words, a
universal rule for the dependence of chaos on each parameter or initial
condition cannot be found in general. Chaos does not depend only on the mass
ratio, and the maximal spins do not necessarily bring the strongest effect of
chaos. Additionally, chaos does not always become drastic when the initial spin
vectors are nearly perpendicular to the orbital plane, and the alignment of
spins cannot trigger chaos by itself.Comment: 16 pages, 7 figure
Out-of-time-order correlators in quantum mechanics
The out-of-time-order correlator (OTOC) is considered as a measure of quantum
chaos. We formulate how to calculate the OTOC for quantum mechanics with a
general Hamiltonian. We demonstrate explicit calculations of OTOCs for a
harmonic oscillator, a particle in a one-dimensional box, a circle billiard and
stadium billiards. For the first two cases, OTOCs are periodic in time because
of their commensurable energy spectra. For the circle and stadium billiards,
they are not recursive but saturate to constant values which are linear in
temperature. Although the stadium billiard is a typical example of the
classical chaos, an expected exponential growth of the OTOC is not found. We
also discuss the classical limit of the OTOC. Analysis of a time evolution of a
wavepacket in a box shows that the OTOC can deviate from its classical value at
a time much earlier than the Ehrenfest time.Comment: 30 pages, 13 figure
Transition from Regular to Chaotic Circulation in Magnetized Coronae near Compact Objects
Accretion onto black holes and compact stars brings material in a zone of
strong gravitational and electromagnetic fields. We study dynamical properties
of motion of electrically charged particles forming a highly diluted medium (a
corona) in the regime of strong gravity and large-scale (ordered) magnetic
field. We start our work from a system that allows regular motion, then we
focus on the onset of chaos. To this end, we investigate the case of a rotating
black hole immersed in a weak, asymptotically uniform magnetic field. We also
consider a magnetic star, approximated by the Schwarzschild metric and a test
magnetic field of a rotating dipole. These are two model examples of systems
permitting energetically bound, off-equatorial motion of matter confined to the
halo lobes that encircle the central body. Our approach allows us to address
the question of whether the spin parameter of the black hole plays any major
role in determining the degree of the chaoticness. To characterize the motion,
we construct the Recurrence Plots (RP) and we compare them with Poincar\'e
surfaces of section. We describe the Recurrence Plots in terms of the
Recurrence Quantification Analysis (RQA), which allows us to identify the
transition between different dynamical regimes. We demonstrate that this new
technique is able to detect the chaos onset very efficiently, and to provide
its quantitative measure. The chaos typically occurs when the conserved energy
is raised to a sufficiently high level that allows the particles to traverse
the equatorial plane. We find that the role of the black-hole spin in setting
the chaos is more complicated than initially thought.Comment: 21 pages, 20 figures, accepted to Ap
Entanglement Across a Transition to Quantum Chaos
We study the relation between entanglement and quantum chaos in one- and
two-dimensional spin-1/2 lattice models, which exhibit mixing of the
noninteracting eigenfunctions and transition from integrability to quantum
chaos. Contrary to what occurs in a quantum phase transition, the onset of
quantum chaos is not a property of the ground state but take place for any
typical many-spin quantum state. We study bipartite and pairwise entanglement
measures, namely the reduced Von Neumann entropy and the concurrence, and
discuss quantum entanglement sharing. Our results suggest that the behavior of
the entanglement is related to the mixing of the eigenfunctions rather than to
the transition to chaos.Comment: 14 pages, 14 figure
A phenomenological approach to normal form modeling: a case study in laser induced nematodynamics
An experimental setting for the polarimetric study of optically induced
dynamical behavior in nematic liquid crystal films has allowed to identify most
notably some behavior which was recognized as gluing bifurcations leading to
chaos. This analysis of the data used a comparison with a model for the
transition to chaos via gluing bifurcations in optically excited nematic liquid
crystals previously proposed by G. Demeter and L. Kramer. The model of these
last authors, proposed about twenty years before, does not have the central
symmetry which one would expect for minimal dimensional models for chaos in
nematics in view of the time series. What we show here is that the simplest
truncated normal forms for gluing, with the appropriate symmetry and minimal
dimension, do exhibit time signals that are embarrassingly similar to the ones
found using the above mentioned experimental settings. The gluing bifurcation
scenario itself is only visible in limited parameter ranges and substantial
aspect of the chaos that can be observed is due to other factors. First, out of
the immediate neighborhood of the homoclinic curve, nonlinearity can produce
expansion leading to chaos when combined with the recurrence induced by the
homoclinic behavior. Also, pairs of symmetric homoclinic orbits create extreme
sensitivity to noise, so that when the noiseless approach contains a rich
behavior, minute noise can transform the complex damping into sustained chaos.
Leonid Shil'nikov taught us that combining global considerations and local
spectral analysis near critical points is crucial to understand the
phenomenology associated to homoclinic bifurcations. Here this helps us
construct a phenomenological approach to modeling experiments in nonlinear
dissipative contexts.Comment: 25 pages, 9 figure
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