855 research outputs found
Effective power-law dependence of Lyapunov exponents on the central mass in galaxies
Using both numerical and analytical approaches, we demonstrate the existence
of an effective power-law relation between the mean Lyapunov
exponent of stellar orbits chaotically scattered by a supermassive black
hole in the center of a galaxy and the mass parameter , i.e. ratio of the
mass of the black hole over the mass of the galaxy. The exponent is found
numerically to obtain values in the range --. We propose a
theoretical interpretation of these exponents, based on estimates of local
`stretching numbers', i.e. local Lyapunov exponents at successive transits of
the orbits through the black hole's sphere of influence. We thus predict
with --. Our basic model refers to elliptical
galaxy models with a central core. However, we find numerically that an
effective power law scaling of with holds also in models with central
cusp, beyond a mass scale up to which chaos is dominated by the influence of
the cusp itself. We finally show numerically that an analogous law exists also
in disc galaxies with rotating bars. In the latter case, chaotic scattering by
the black hole affects mainly populations of thick tube-like orbits surrounding
some low-order branches of the family of periodic orbits, as well as its
bifurcations at low-order resonances, mainly the Inner Lindbland resonance and
the 4/1 resonance. Implications of the correlations between and to
determining the rate of secular evolution of galaxies are discussed.Comment: 27 pages, 19 figure
Analytical description of the structure of chaos
We consider analytical formulae that describe the chaotic regions around the
main periodic orbit of the H\'{e}non map. Following our previous
paper (Efthymiopoulos, Contopoulos, Katsanikas ) we introduce new
variables in which the product (constant) gives
hyperbolic invariant curves. These hyperbolae are mapped by a canonical
transformation to the plane , giving "Moser invariant curves". We
find that the series are convergent up to a maximum value of
. We give estimates of the errors due to the finite truncation of
the series and discuss how these errors affect the applicability of analytical
computations. For values of the basic parameter of the H\'{e}non map
smaller than a critical value, there is an island of stability, around a stable
periodic orbit , containing KAM invariant curves. The Moser curves for are completely outside the last KAM curve around , the curves
with intersect the last KAM curve and the curves with are completely inside the last KAM curve. All orbits in
the chaotic region around the periodic orbit , although they seem
random, belong to Moser invariant curves, which, therefore define a "structure
of chaos". Orbits starting close and outside the last KAM curve remain close to
it for a stickiness time that is estimated analytically using the series
. We finally calculate the periodic orbits that accumulate close to the
homoclinic points, i.e. the points of intersection of the asymptotic curves
from , exploiting a method based on the self-intersections of the
invariant Moser curves. We find that all the computed periodic orbits are
generated from the stable orbit for smaller values of the H\'{e}non
parameter , i.e. they are all regular periodic orbits.Comment: 22 pages, 9 figure
Resonant normal form and asymptotic normal form behavior in magnetic bottle Hamiltonians
We consider normal forms in `magnetic bottle' type Hamiltonians of the form
(second
frequency equal to zero in the lowest order). Our main results are:
i) a novel method to construct the normal form in cases of resonance, and ii) a
study of the asymptotic behavior of both the non-resonant and the resonant
series. We find that, if we truncate the normal form series at order , the
series remainder in both constructions decreases with increasing down to a
minimum, and then it increases with . The computed minimum remainder turns
to be exponentially small in , where is the
mirror oscillation energy, while the optimal order scales as an inverse power
of . We estimate numerically the exponents associated with the
optimal order and the remainder's exponential asymptotic behavior. In the
resonant case, our novel method allows to compute a `quasi-integral' (i.e.
truncated formal integral) valid both for each particular resonance as well as
away from all resonances. We applied these results to a specific magnetic
bottle Hamiltonian. The non resonant normal form yields theorerical invariant
curves on a surface of section which fit well the empirical curves away from
resonances. On the other hand the resonant normal form fits very well both the
invariant curves inside the islands of a particular resonance as well as the
non-resonant invariant curves. Finally, we discuss how normal forms allow to
compute a critical threshold for the onset of global chaos in the magnetic
bottle.Comment: 20 pages, 7 figure
Partial Integrability of 3-d Bohmian Trajectories
In this paper we study the integrability of 3-d Bohmian trajectories of a
system of quantum harmonic oscillators. We show that the initial choice of
quantum numbers is responsible for the existence (or not) of an integral of
motion which confines the trajectories on certain invariant surfaces. We give a
few examples of orbits in cases where there is or there is not an integral and
make some comments on the impact of partial integrability in Bohmian Mechanics.
Finally, we make a connection between our present results for the integrability
in the 3-d case and analogous results found in the 2-d and 4-d cases.Comment: 18 pages, 3 figure
Origin of chaos in 3-d Bohmian trajectories
We study the 3-d Bohmian trajectories of a quantum system of three harmonic
oscillators. We focus on the mechanism responsible for the generation of
chaotic trajectories. We demonstrate the existence of a 3-d analogue of the
mechanism found in earlier studies of 2-d systems, based on moving 2-d `nodal
point - X-point complexes'. In the 3-d case, we observe a foliation of nodal
point - X-point complexes, forming a `3-d structure of nodal and X-points'.
Chaos is generated when the Bohmian trajectories are scattered at one or more
close encounters with such a structure.Comment: 7 pages, 8 figure
Structures induced by companions in galactic discs
Using N-body simulations we study the structures induced on a galactic disc
by repeated flybys of a companion in decaying eccentric orbit around the disc.
Our system is composed by a stellar disc, bulge and live dark matter halo, and
we study the system's dynamical response to a sequence of a companion's flybys,
when we vary i) the disc's temperature (parameterized by Toomre's Q-parameter)
and ii) the companion's mass and initial orbit. We use a new 3D Cartesian grid
code: MAIN (Mesh-adaptive Approximate Inverse N-body solver). The main features
of MAIN are reviewed, with emphasis on the use of a new Symmetric Factored
Approximate Sparse Inverse (SFASI) matrix in conjunction with the multigrid
method that allows the efficient solution of Poisson's equation in three space
variables. We find that: i) companions need to be assigned initial masses in a
rather narrow window of values in order to produce significant and more
long-standing non-axisymmetric structures (bars and spirals) in the main
galaxy's disc by the repeated flyby mechanism. ii) a crucial phenomenon is the
antagonism between companion-excited and self-excited modes on the disc. Values
of are needed in order to allow for the growth of the
companion-excited modes to prevail over the the growth of the disc's
self-excited modes. iii) We give evidence that the companion-induced spiral
structure is best represented by a density wave with pattern speed nearly
constant in a region extending from the ILR to a radius close to, but inside,
corotation.Comment: Published in MNRA
Quantum vortices and trajectories in particle diffraction
We investigate the phenomenon of the diffraction of charged particles by thin
material targets using the method of the de Broglie-Bohm quantum trajectories.
The particle wave function can be modeled as a sum of two terms
. A thin separator exists between the
domains of prevalence of the ingoing and outgoing wavefunction terms. The
structure of the quantum-mechanical currents in the neighborhood of the
separator implies the formation of an array of \emph{quantum vortices}. The
flow structure around each vortex displays a characteristic pattern called
`nodal point - X point complex'. The X point gives rise to stable and unstable
manifolds. We find the scaling laws characterizing a nodal point-X point
complex by a local perturbation theory around the nodal point. We then analyze
the dynamical role of vortices in the emergence of the diffraction pattern. In
particular, we demonstrate the abrupt deflections, along the direction of the
unstable manifold, of the quantum trajectories approaching an X-point along its
stable manifold. Theoretical results are compared to numerical simulations of
quantum trajectories. We finally calculate the {\it times of flight} of
particles following quantum trajectories from the source to detectors placed at
various scattering angles , and thereby propose an experimental test of
the de Broglie - Bohm formalism.Comment: 17 pages, 7 figures, accepted by IJB
Order in de Broglie - Bohm quantum mechanics
A usual assumption in the so-called {\it de Broglie - Bohm} approach to
quantum dynamics is that the quantum trajectories subject to typical `guiding'
wavefunctions turn to be quite irregular, i.e. {\it chaotic} (in the dynamical
systems' sense). In the present paper, we consider mainly cases in which the
quantum trajectories are {\it ordered}, i.e. they have zero Lyapunov
characteristic numbers. We use perturbative methods to establish the existence
of such trajectories from a theoretical point of view, while we analyze their
properties via numerical experiments. Using a 2D harmonic oscillator system, we
first establish conditions under which a trajectory can be shown to avoid close
encounters with a moving nodal point, thus avoiding the source of chaos in this
system. We then consider series expansions for trajectories both in the
interior and the exterior of the domain covered by nodal lines, probing the
domain of convergence as well as how successful the series are in comparison
with numerical computations or regular trajectories. We then examine a
H\'{e}non - Heiles system possessing regular trajectories, thus generalizing
previous results. Finally, we explore a key issue of physical interest in the
context of the de Broglie - Bohm formalism, namely the influence of order in
the so-called {\it quantum relaxation} effect. We show that the existence of
regular trajectories poses restrictions to the quantum relaxation process, and
we give examples in which the relaxation is suppressed even when we consider
initial ensembles of only chaotic trajectories, provided, however, that the
system as a whole is characterized by a certain degree of order.Comment: 25 pages, 12 figure
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