325 research outputs found
Chaotic dynamics of three-dimensional H\'enon maps that originate from a homoclinic bifurcation
We study bifurcations of a three-dimensional diffeomorphism, , that has
a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers
(\lambda e^{i\vphi}, \lambda e^{-i\vphi}, \gamma), where
and . We show that in a
three-parameter family, g_{\eps}, of diffeomorphisms close to , there
exist infinitely many open regions near \eps =0 where the corresponding
normal form of the first return map to a neighborhood of a homoclinic point is
a three-dimensional H\'enon-like map. This map possesses, in some parameter
regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that
this homoclinic bifurcation leads to a strange attractor. We also discuss the
place that these three-dimensional H\'enon maps occupy in the class of
quadratic volume-preserving diffeomorphisms.Comment: laTeX, 25 pages, 6 eps figure
Quantum Breaking Time Scaling in the Superdiffusive Dynamics
We show that the breaking time of quantum-classical correspondence depends on
the type of kinetics and the dominant origin of stickiness. For sticky dynamics
of quantum kicked rotor, when the hierarchical set of islands corresponds to
the accelerator mode, we demonstrate by simulation that the breaking time
scales as with the transport exponent
that corresponds to superdiffusive dynamics. We discuss also other
possibilities for the breaking time scaling and transition to the logarithmic
one with respect to
Heteroclinic intersections between Invariant Circles of Volume-Preserving Maps
We develop a Melnikov method for volume-preserving maps with codimension one
invariant manifolds. The Melnikov function is shown to be related to the flux
of the perturbation through the unperturbed invariant surface. As an example,
we compute the Melnikov function for a perturbation of a three-dimensional map
that has a heteroclinic connection between a pair of invariant circles. The
intersection curves of the manifolds are shown to undergo bifurcations in
homologyComment: LaTex with 10 eps figure
Chaotic Advection and the Emergence of Tori in the K\"uppers-Lortz State
Motivated by the roll-switching behavior observed in rotating
Rayleigh-B\'enard convection, we define a K\"uppers-Lortz (K-L) state as a
volume-preserving flow with periodic roll switching. For an individual roll
state, the Lagrangian particle trajectories are periodic. In a system with
roll-switching, the particles can exhibit three-dimensional, chaotic motion. We
study a simple phenomenological map that models the Lagrangian dynamics in a
K-L state. When the roll axes differ by in the plane of rotation,
we show that the phase space is dominated by invariant tori if the ratio of
switching time to roll turnover time is small. When this parameter approaches
zero these tori limit onto the classical hexagonal convection patterns, and, as
it gets large, the dynamics becomes fully chaotic and well-mixed. For
intermediate values, there are interlinked toroidal and poloidal structures
separated by chaotic regions. We also compute the exit time distributions and
show that the unbounded chaotic orbits are normally diffusive. Although the map
presumes instantaneous switching between roll states, we show that the
qualitative features of the flow persist when the model has smooth, overlapping
time-dependence for the roll amplitudes (the Busse-Heikes model).Comment: laTeX, 23 pages, 7 figure
Quantum Poincar\'e Recurrences
We show that quantum effects modify the decay rate of Poincar\'e recurrences
P(t) in classical chaotic systems with hierarchical structure of phase space.
The exponent p of the algebraic decay P(t) ~ 1/t^p is shown to have the
universal value p=1 due to tunneling and localization effects. Experimental
evidence of such decay should be observable in mesoscopic systems and cold
atoms.Comment: revtex, 4 pages, 4 figure
Asymptotic Statistics of Poincar\'e Recurrences in Hamiltonian Systems with Divided Phase Space
By different methods we show that for dynamical chaos in the standard map
with critical golden curve the Poincar\'e recurrences P(\tau) and correlations
C(\tau) asymptotically decay in time as P ~ C/\tau ~ 1/\tau^3. It is also
explained why this asymptotic behavior starts only at very large times. We
argue that the same exponent p=3 should be also valid for a general chaos
border.Comment: revtex, 4 pages, 3 ps-figure
Resonance Zones and Lobe Volumes for Volume-Preserving Maps
We study exact, volume-preserving diffeomorphisms that have heteroclinic
connections between a pair of normally hyperbolic invariant manifolds. We
develop a general theory of lobes, showing that the lobe volume is given by an
integral of a generating form over the primary intersection, a subset of the
heteroclinic orbits. Our definition reproduces the classical action formula in
the planar, twist map case. For perturbations from a heteroclinic connection,
the lobe volume is shown to reduce, to lowest order, to a suitable integral of
a Melnikov function.Comment: ams laTeX, 8 figure
Ulam method for the Chirikov standard map
We introduce a generalized Ulam method and apply it to symplectic dynamical
maps with a divided phase space. Our extensive numerical studies based on the
Arnoldi method show that the Ulam approximant of the Perron-Frobenius operator
on a chaotic component converges to a continuous limit. Typically, in this
regime the spectrum of relaxation modes is characterized by a power law decay
for small relaxation rates. Our numerical data show that the exponent of this
decay is approximately equal to the exponent of Poincar\'e recurrences in such
systems. The eigenmodes show links with trajectories sticking around stability
islands.Comment: 13 pages, 13 figures, high resolution figures available at:
http://www.quantware.ups-tlse.fr/QWLIB/ulammethod/ minor corrections in text
and fig. 12 and revised discussio
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