441 research outputs found
Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations
We study the dynamics of the five-parameter quadratic family of
volume-preserving diffeomorphisms of R^3. This family is the unfolded normal
form for a bifurcation of a fixed point with a triple-one multiplier and also
is the general form of a quadratic three-dimensional map with a quadratic
inverse. Much of the nontrivial dynamics of this map occurs when its two fixed
points are saddle-foci with intersecting two-dimensional stable and unstable
manifolds that bound a spherical ``vortex-bubble''. We show that this occurs
near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at
least in its normal form, an elliptic invariant circle. We develop a simple
algorithm to accurately compute these elliptic invariant circles and their
longitudinal and transverse rotation numbers and use it to study their
bifurcations, classifying them by the resonances between the rotation numbers.
In particular, rational values of the longitudinal rotation number are shown to
give rise to a string of pearls that creates multiple copies of the original
spherical structure for an iterate of the map.Comment: 53 pages, 29 figure
Entropy and Correlations in Lattice Gas Automata without Detailed Balance
We consider lattice gas automata where the lack of semi-detailed balance
results from node occupation redistribution ruled by distant configurations;
such models with nonlocal interactions are interesting because they exhibit
non-ideal gas properties and can undergo phase transitions. For this class of
automata, mean-field theory provides a correct evaluation of properties such as
compressibility and viscosity (away from the phase transition), despite the
fact that no H-theorem strictly holds. We introduce the notion of locality -
necessary to define quantities accessible to measurements - by treating the
coupling between nonlocal bits as a perturbation. Then if we define
operationally ``local'' states of the automaton - whether the system is in a
homogeneous or in an inhomogeneous state - we can compute an estimator of the
entropy and measure the local channel occupation correlations. These
considerations are applied to a simple model with nonlocal interactions.Comment: 13 pages, LaTeX, 5 PostScript figures, uses psfig. Submitted to Int.
J. Mod. Phys.
Simulating Three-Dimensional Hydrodynamics on a Cellular-Automata Machine
We demonstrate how three-dimensional fluid flow simulations can be carried
out on the Cellular Automata Machine 8 (CAM-8), a special-purpose computer for
cellular-automata computations. The principal algorithmic innovation is the use
of a lattice-gas model with a 16-bit collision operator that is specially
adapted to the machine architecture. It is shown how the collision rules can be
optimized to obtain a low viscosity of the fluid. Predictions of the viscosity
based on a Boltzmann approximation agree well with measurements of the
viscosity made on CAM-8. Several test simulations of flows in simple geometries
-- channels, pipes, and a cubic array of spheres -- are carried out.
Measurements of average flux in these geometries compare well with theoretical
predictions.Comment: 19 pages, REVTeX and epsf macros require
Probing the local dynamics of periodic orbits by the generalized alignment index (GALI) method
As originally formulated, the Generalized Alignment Index (GALI) method of
chaos detection has so far been applied to distinguish quasiperiodic from
chaotic motion in conservative nonlinear dynamical systems. In this paper we
extend its realm of applicability by using it to investigate the local dynamics
of periodic orbits. We show theoretically and verify numerically that for
stable periodic orbits the GALIs tend to zero following particular power laws
for Hamiltonian flows, while they fluctuate around non-zero values for
symplectic maps. By comparison, the GALIs of unstable periodic orbits tend
exponentially to zero, both for flows and maps. We also apply the GALIs for
investigating the dynamics in the neighborhood of periodic orbits, and show
that for chaotic solutions influenced by the homoclinic tangle of unstable
periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during
which their amplitudes change by many orders of magnitude. Finally, we use the
GALI method to elucidate further the connection between the dynamics of
Hamiltonian flows and symplectic maps. In particular, we show that, using for
the computation of GALIs the components of deviation vectors orthogonal to the
direction of motion, the indices of stable periodic orbits behave for flows as
they do for maps.Comment: 17 pages, 9 figures (accepted for publication in Int. J. of
Bifurcation and Chaos
Escaping from nonhyperbolic chaotic attractors
We study the noise-induced escape process from chaotic attractors in
nonhyperbolic systems. We provide a general mechanism of escape in the low
noise limit, employing the theory of large fluctuations. Specifically, this is
achieved by solving the variational equations of the auxiliary Hamiltonian
system and by incorporating the initial conditions on the chaotic attractor
unambiguously. Our results are exemplified with the H{\'e}non and the Ikeda map
and can be implemented straightforwardly to experimental data.Comment: replaced with published versio
Core Collapse via Coarse Dynamic Renormalization
In the context of the recently developed "equation-free" approach to
computer-assisted analysis of complex systems, we extract the self-similar
solution describing core collapse of a stellar system from numerical
experiments. The technique allows us to side-step the core "bounce" that occurs
in direct N-body simulations due to the small-N correlations that develop in
the late stages of collapse, and hence to follow the evolution well into the
self-similar regime.Comment: 5 pages, 3 figure
Quadratic Volume Preserving Maps
We study quadratic, volume preserving diffeomorphisms whose inverse is also
quadratic. Such maps generalize the Henon area preserving map and the family of
symplectic quadratic maps studied by Moser. In particular, we investigate a
family of quadratic volume preserving maps in three space for which we find a
normal form and study invariant sets. We also give an alternative proof of a
theorem by Moser classifying quadratic symplectic maps.Comment: Ams LaTeX file with 4 figures (figure 2 is gif, the others are ps
Nambu-Hamiltonian flows associated with discrete maps
For a differentiable map that has
an inverse, we show that there exists a Nambu-Hamiltonian flow in which one of
the initial value, say , of the map plays the role of time variable while
the others remain fixed. We present various examples which exhibit the map-flow
correspondence.Comment: 19 page
Statistical Mechanics of the Fluctuating Lattice Boltzmann Equation
We propose a new formulation of the fluctuating lattice Boltzmann equation
that is consistent with both equilibrium statististical mechanics and
fluctuating hydrodynamics. The formalism is based on a generalized lattice-gas
model, with each velocity direction occupied by many particles. We show that
the most probable state of this model corresponds to the usual equilibrium
distribution of the lattice Boltzmann equation. Thermal fluctuations about this
equilibrium are controlled by the mean number of particles at a lattice site.
Stochastic collision rules are described by a Monte Carlo process satisfying
detailed balance. This allows for a straightforward derivation of discrete
Langevin equations for the fluctuating modes. It is shown that all
non-conserved modes should be thermalized, as first pointed out by Adhikari et
al.; any other choice violates the condition of detailed balance. A
Chapman-Enskog analysis is used to derive the equations of fluctuating
hydrodynamics on large length and time scales; the level of fluctuations is
shown to be thermodynamically consistent with the equation of state of an
isothermal, ideal gas. We believe this formalism will be useful in developing
new algorithms for thermal and multiphase flows.Comment: Submitted to Physical Review E-11 pages Corrected Author(s) field on
submittal for
Interference phenomena in scalar transport induced by a noise finite correlation time
The role played on the scalar transport by a finite, not small, correlation
time, , for the noise velocity is investigated, both analytically and
numerically. For small 's a mechanism leading to enhancement of
transport has recently been identified and shown to be dominating for any type
of flow. For finite non-vanishing 's we recognize the existence of a
further mechanism associated with regions of anticorrelation of the Lagrangian
advecting velocity. Depending on the extension of the anticorrelated regions,
either an enhancement (corresponding to constructive interference) or a
depletion (corresponding to destructive interference) in the turbulent
transport now takes place.Comment: 8 pages, 3 figure
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