4,020 research outputs found
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
Effective transport barriers in nontwist systems
In fluids and plasmas with zonal flow reversed shear, a peculiar kind of transport barrier appears in the shearless region, one that is associated with a proper route of transition to chaos. These barriers have been identified in symplectic nontwist maps that model such zonal flows. We use the so-called standard nontwist map, a paradigmatic example of nontwist systems, to analyze the parameter dependence of the transport through a broken shearless barrier. On varying a proper control parameter, we identify the onset of structures with high stickiness that give rise to an effective barrier near the broken shearless curve. Moreover, we show how these stickiness structures, and the concomitant transport reduction in the shearless region, are determined by a homoclinic tangle of the remaining dominant twin island chains. We use the finite-time rotation number, a recently proposed diagnostic, to identify transport barriers that separate different regions of stickiness. The identified barriers are comparable to those obtained by using finite-time Lyapunov exponents.FAPESPCNPqCAPESMCT/CNEN (Rede Nacional de Fusao)Fundacao AraucariaUS Department of Energy DE-FG05-80ET-53088Physic
Characterizing Weak Chaos using Time Series of Lyapunov Exponents
We investigate chaos in mixed-phase-space Hamiltonian systems using time
series of the finite- time Lyapunov exponents. The methodology we propose uses
the number of Lyapunov exponents close to zero to define regimes of ordered
(stickiness), semi-ordered (or semi-chaotic), and strongly chaotic motion. The
dynamics is then investigated looking at the consecutive time spent in each
regime, the transition between different regimes, and the regions in the
phase-space associated to them. Applying our methodology to a chain of coupled
standard maps we obtain: (i) that it allows for an improved numerical
characterization of stickiness in high-dimensional Hamiltonian systems, when
compared to the previous analyses based on the distribution of recurrence
times; (ii) that the transition probabilities between different regimes are
determined by the phase-space volume associated to the corresponding regions;
(iii) the dependence of the Lyapunov exponents with the coupling strength.Comment: 8 pages, 6 figure
Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space
We investigate the dynamics of chaotic trajectories in simple yet physically
important Hamiltonian systems with non-hierarchical borders between regular and
chaotic regions with positive measures. We show that the stickiness to the
border of the regular regions in systems with such a sharply divided phase
space occurs through one-parameter families of marginally unstable periodic
orbits and is characterized by an exponent \gamma= 2 for the asymptotic
power-law decay of the distribution of recurrence times. Generic perturbations
lead to systems with hierarchical phase space, where the stickiness is
apparently enhanced due to the presence of infinitely many regular islands and
Cantori. In this case, we show that the distribution of recurrence times can be
composed of a sum of exponentials or a sum of power-laws, depending on the
relative contribution of the primary and secondary structures of the hierarchy.
Numerical verification of our main results are provided for area-preserving
maps, mushroom billiards, and the newly defined magnetic mushroom billiards.Comment: To appear in Phys. Rev. E. A PDF version with higher resolution
figures is available at http://www.pks.mpg.de/~edugal
Separation of particles leading to decay and unlimited growth of energy in a driven stadium-like billiard
A competition between decay and growth of energy in a time-dependent stadium
billiard is discussed giving emphasis in the decay of energy mechanism. A
critical resonance velocity is identified for causing of separation between
ensembles of high and low energy and a statistical investigation is made using
ensembles of initial conditions both above and below the resonance velocity.
For high initial velocity, Fermi acceleration is inherent in the system.
However for low initial velocity, the resonance allies with stickiness hold the
particles in a regular or quasi-regular regime near the fixed points,
preventing them from exhibiting Fermi acceleration. Also, a transport analysis
along the velocity axis is discussed to quantify the competition of growth and
decay of energy and making use distributions of histograms of frequency, and we
set that the causes of the decay of energy are due to the capture of the orbits
by the resonant fixed points
Characterizing weak chaos in nonintegrable Hamiltonian systems: the fundamental role of stickiness and initial conditions
Weak chaos in high-dimensional conservative systems can be characterized
through sticky effect induced by invariant structures on chaotic trajectories.
Suitable quantities for this characterization are the higher cummulants of the
finite time Lyapunov exponents (FTLEs) distribution. They gather the {\it
whole} phase space relevant dynamics in {\it one} quantity and give
informations about ordered and random states. This is analyzed here for
discrete Hamiltonian systems with local and global couplings. It is also shown
that FTLEs plotted {\it versus} initial condition (IC) and the nonlinear
parameter is essential to understand the fundamental role of ICs in the
dynamics of weakly chaotic Hamiltonian systems.Comment: 7 pages, 6 figures, submitted for publicatio
Instabilities and stickiness in a 3D rotating galactic potential
We study the dynamics in the neighborhood of simple and double unstable
periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic
type. In order to visualize the four dimensional spaces of section we use the
method of color and rotation. We investigate the structure of the invariant
manifolds that we found in the neighborhood of simple and double unstable
periodic orbits in the 4D spaces of section. We consider orbits in the
neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis
(the rotational axis of our system). Close to the transition points from
stability to simple instability, in the neighborhood of the bifurcated simple
unstable x1v2 periodic orbits we encounter the phenomenon of stickiness as the
asymptotic curves of the unstable manifold surround regions of the phase space
occupied by rotational tori existing in the region. For larger energies, away
from the bifurcating point, the consequents of the chaotic orbits form clouds
of points with mixing of color in their 4D representations. In the case of
double instability, close to x1v2 orbits, we find clouds of points in the four
dimensional spaces of section. However, in some cases of double unstable
periodic orbits belonging to the z-axis family we can visualize the associated
unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky
to this surface for long times (of the order of a Hubble time or more). Among
the orbits we studied we found those close to the double unstable orbits of the
x1v2 family having the largest diffusion speed.Comment: 29pages, 25 figures, accepted for publication in the International
Journal of Bifurcation and Chao
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