33 research outputs found
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
Asymptotic Orbits in Barred Spiral Galaxies
We study the formation of the spiral structure of barred spiral galaxies,
using an -body model. The evolution of this -body model in the adiabatic
approximation maintains a strong spiral pattern for more than 10 bar rotations.
We find that this longevity of the spiral arms is mainly due to the phenomenon
of stickiness of chaotic orbits close to the unstable asymptotic manifolds
originated from the main unstable periodic orbits, both inside and outside
corotation. The stickiness along the manifolds corresponding to different
energy levels supports parts of the spiral structure. The loci of the disc
velocity minima (where the particles spend most of their time, in the
configuration space) reveal the density maxima and therefore the main
morphological structures of the system. We study the relation of these loci
with those of the apocentres and pericentres at different energy levels. The
diffusion of the sticky chaotic orbits outwards is slow and depends on the
initial conditions and the corresponding Jacobi constant.Comment: 17 pages, 24 figure
Perturbed precessing ellipses as the building blocks of spiral arms in a barred galaxy with two pattern speeds
Observations and simulations of barred spiral galaxies have shown that, in
general, the spiral arms rotate at a different pattern speed to that of the
bar. The main conclusion from the bibliography is that the bar rotates faster
than the spiral arms with a double or even a triple value of angular velocity.
The theory that prevails in explaining the formation of the spiral arms in the
case of a barred spiral galaxy with two pattern speeds is the manifold theory,
where the orbits that support the spiral density wave are chaotic, and are
related to the manifolds emanating from the Lagrangian points L_1 and L_2 at
the end of the bar. In the present study, we consider an alternative scenario
in the case where the bar rotates fast enough in comparison with the spiral
arms and the bar potential can be considered as a perturbation of the spiral
potential. In this case, the stable elliptical orbits that support the spiral
density wave (in the case of grand design galaxies) are transformed into
quasiperiodic orbits (or 2D tori) with a certain thickness. The superposition
of these perturbed preccesing ellipses for all the energy levels of the
Hamiltonian creates a slightly perturbed symmetrical spiral density wave.Comment: 9 pages, 7 figure
Stickiness in Chaos
We distinguish two types of stickiness in systems of two degrees of freedom
(a) stickiness around an island of stability and (b) stickiness in chaos, along
the unstable asymptotic curves of unstable periodic orbits. We studied these
effects in the standard map with a rather large nonlinearity K=5, and we
emphasized the role of the asymptotic curves U, S from the central orbit O and
the asymptotic curves U+U-S+S- from the simplest unstable orbit around the
island O1. We calculated the escape times (initial stickiness times) for many
initial points outside but close to the island O1. The lines that separate the
regions of the fast from the slow escape time follow the shape of the
asymptotic curves S+,S-. We explained this phenomenon by noting that lines
close to S+ on its inner side (closer to O1) approach a point of the orbit 4/9,
say P1, and then follow the oscillations of the asymptotic curve U+, and escape
after a rather long time, while the curves outside S+ after their approach to
P1 follow the shape of the asymptotic curves U- and escape fast into the
chaotic sea. All these curves return near the original arcs of U+,U- and
contribute to the overall stickiness close to U+,U-. The isodensity curves
follow the shape of the curves U+,U- and the maxima of density are along U+,U-.
For a rather long time the stickiness effects along U+,U- are very pronounced.
However after much longer times (about 1000 iterations) the overall stickiness
effects are reduced and the distribution of points in the chaotic sea outside
the islands tends to be uniform.Comment: 28 pages, 12 figure
Harassment Origin for Kinematic Substructures in Dwarf Elliptical Galaxies?
We have run high resolution N-body models simulating the encounter of a dwarf
galaxy with a bright elliptical galaxy. The dwarf absorbs orbital angular
momentum and shows counter-rotating features in the external regions of the
galaxy. To explain the core-envelope kinematic decoupling observed in some
dwarf galaxies in high-density environments requires nearly head-on collisions
and very little dark matter bound to the dwarf. These kinematic structures
appear under rather restrictive conditions. As a consequence, in a cluster like
Virgo ~1% of dwarf galaxies may present counter-rotation formed by harassment.Comment: 10 pages, 7 figures; Accepted for publication in Astronomy and
Astrophysic
Periodic Orbits and Escapes in Dynamical Systems
We study the periodic orbits and the escapes in two different dynamical
systems, namely (1) a classical system of two coupled oscillators, and (2) the
Manko-Novikov metric (1992) which is a perturbation of the Kerr metric (a
general relativistic system). We find their simple periodic orbits, their
characteristics and their stability. Then we find their ordered and chaotic
domains. As the energy goes beyond the escape energy, most chaotic orbits
escape. In the first case we consider escapes to infinity, while in the second
case we emphasize escapes to the central "bumpy" black hole. When the energy
reaches its escape value a particular family of periodic orbits reaches an
infinite period and then the family disappears (the orbit escapes). As this
family approaches termination it undergoes an infinity of equal period and
double period bifurcations at transitions from stability to instability and
vice versa. The bifurcating families continue to exist beyond the escape
energy. We study the forms of the phase space for various energies, and the
statistics of the chaotic and escaping orbits. The proportion of these orbits
increases abruptly as the energy goes beyond the escape energy.Comment: 28 pages, 23 figures, accepted in "Celestial Mechanics and Dynamical
Astronomy
Boxy/peanut/X bulges, barlenses and the thick part of galactic bars: What are they and how did they form?
Bars have a complex three-dimensional shape. In particular their inner part
is vertically much thicker than the parts further out. Viewed edge-on, the
thick part of the bar is what is commonly known as a boxy-, peanut- or X- bulge
and viewed face-on it is referred to as a barlens. These components are due to
disc and bar instabilities and are composed of disc material. I review here
their formation, evolution and dynamics, using simulations, orbital structure
theory and comparisons to observations.Comment: 21 pages, 7 figures, invited review to appear in "Galactic Bulges",
E. Laurikainen, R. Peletier, D. Gadotti, (eds.), Springe
Manifold spirals in barred galaxies with multiple pattern speeds
In the manifold theory of spiral structure in barred galaxies, the usual assumption is that the spirals rotate with the same pattern speed as the bar. Here, we generalize the manifold theory under the assumption that the spirals rotate with a different pattern speed than the bar. More generally, we consider the case in which one or more modes, represented by the potentials V2, V3, etc., coexist in the galactic disk in addition to the barâs mode Vbar, but the modes rotate with pattern speeds, Ω2, Ω3, etc., which are incommensurable between themselves and with Ωbar. Through a perturbative treatment (assuming that V2, V3, etc. are small with respect to Vbar), we then show that the unstable Lagrangian points L1 and L2 of the pure bar model (Vbar,âΩbar) are continued in the full model as periodic orbits, in the case of one extra pattern speed, or as epicyclic âLissajous-likeâ unstable orbits, in the case of more than one extra pattern speeds. We use GL1 and GL2 to denote the continued orbits around the points L1 and L2. Furthermore, we show that the orbits GL1 and GL2 are simply unstable. As a result, these orbits admit invariant manifolds, which can be regarded as the generalization of the manifolds of the L1 and L2 points in the single pattern speed case. As an example, we computed the generalized orbits GL1, GL2, and their manifolds in a Milky-Way-like model in which bar and spiral pattern speeds were assumed to be different. We find that the manifolds produce a time-varying morphology consisting of segments of spirals or âpseudoringsâ. These structures are repeated after a period equal to half the relative period of the imposed spirals with respect to the bar. Along one period, the manifold-induced time-varying structures are found to continuously support at least some part of the imposed spirals, except at short intervals around specific times at which the relative phase of the imposed spirals with respect to the bar is equal to ±Ï/2. The connection of these effects to the phenomenon of recurrent spirals is discussed