1,823 research outputs found
The ratchet effect and the transporting islands in the chaotic sea
We study directed transport in a classical deterministic dissipative system.
We consider the generic case of mixed phase space and show that large ratchet
currents can be generated thanks to the presence, in the Hamiltonian limit, of
transporting stability islands embedded in the chaotic sea. Due to the
simultaneous presence of chaos and dissipation the stationary value of the
current is independent of initial conditions, except for initial states with
very small measure.Comment: 5 pages, 6 figure
Unsteady undular bores in fully nonlinear shallow-water theory
We consider unsteady undular bores for a pair of coupled equations of
Boussinesq-type which contain the familiar fully nonlinear dissipationless
shallow-water dynamics and the leading-order fully nonlinear dispersive terms.
This system contains one horizontal space dimension and time and can be
systematically derived from the full Euler equations for irrotational flows
with a free surface using a standard long-wave asymptotic expansion.
In this context the system was first derived by Su and Gardner. It coincides
with the one-dimensional flat-bottom reduction of the Green-Naghdi system and,
additionally, has recently found a number of fluid dynamics applications other
than the present context of shallow-water gravity waves. We then use the
Whitham modulation theory for a one-phase periodic travelling wave to obtain an
asymptotic analytical description of an undular bore in the Su-Gardner system
for a full range of "depth" ratios across the bore. The positions of the
leading and trailing edges of the undular bore and the amplitude of the leading
solitary wave of the bore are found as functions of this "depth ratio". The
formation of a partial undular bore with a rapidly-varying finite-amplitude
trailing wave front is predicted for ``depth ratios'' across the bore exceeding
1.43. The analytical results from the modulation theory are shown to be in
excellent agreement with full numerical solutions for the development of an
undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9
figure
Transient times, resonances and drifts of attractors in dissipative rotational dynamics
In a dissipative system the time to reach an attractor is often influenced by
the peculiarities of the model and in particular by the strength of the
dissipation. In particular, as a dissipative model we consider the spin-orbit
problem providing the dynamics of a triaxial satellite orbiting around a
central planet and affected by tidal torques. The model is ruled by the
oblateness parameter of the satellite, the orbital eccentricity, the
dissipative parameter and the drift term. We devise a method which provides a
reliable indication on the transient time which is needed to reach an attractor
in the spin-orbit model; the method is based on an analytical result, precisely
a suitable normal form construction. This method provides also information
about the frequency of motion. A variant of such normal form used to
parametrize invariant attractors provides a specific formula for the drift
parameter, which in turn yields a constraint - which might be of interest in
astronomical problems - between the oblateness of the satellite and its orbital
eccentricity.Comment: 21 pages, 7 figures, colo
Pulses and Snakes in Ginzburg--Landau Equation
Using a variational formulation for partial differential equations (PDEs)
combined with numerical simulations on ordinary differential equations (ODEs),
we find two categories (pulses and snakes) of dissipative solitons, and analyze
the dependence of both their shape and stability on the physical parameters of
the cubic-quintic Ginzburg-Landau equation (CGLE). In contrast to the regular
solitary waves investigated in numerous integrable and non-integrable systems
over the last three decades, these dissipative solitons are not stationary in
time. Rather, they are spatially confined pulse-type structures whose envelopes
exhibit complicated temporal dynamics. Numerical simulations reveal very
interesting bifurcations sequences as the parameters of the CGLE are varied.
Our predictions on the variation of the soliton amplitude, width, position,
speed and phase of the solutions using the variational formulation agree with
simulation results.Comment: 30 pages, 14 figure
- …