2,553 research outputs found
Quasi-periodic motions in dynamical systems. Review of a renormalisation group approach
Power series expansions naturally arise whenever solutions of ordinary
differential equations are studied in the regime of perturbation theory. In the
case of quasi-periodic solutions the issue of convergence of the series is
plagued of the so-called small divisor problem. In this paper we review a
method recently introduced to deal with such a problem, based on
renormalisation group ideas and multiscale techniques. Applications to both
quasi-integrable Hamiltonian systems (KAM theory) and non-Hamiltonian
dissipative systems are discussed. The method is also suited to situations in
which the perturbation series diverges and a resummation procedure can be
envisaged, leading to a solution which is not analytic in the perturbation
parameter: we consider explicitly examples of solutions which are only
infinitely differentiable in the perturbation parameter, or even defined on a
Cantor set.Comment: 36 pages, 8 figures, review articl
Resonance tongues and patterns in periodically forced reaction-diffusion systems
Various resonant and near-resonant patterns form in a light-sensitive
Belousov-Zhabotinsky (BZ) reaction in response to a spatially-homogeneous
time-periodic perturbation with light. The regions (tongues) in the forcing
frequency and forcing amplitude parameter plane where resonant patterns form
are identified through analysis of the temporal response of the patterns.
Resonant and near-resonant responses are distinguished. The unforced BZ
reaction shows both spatially-uniform oscillations and rotating spiral waves,
while the forced system shows patterns such as standing-wave labyrinths and
rotating spiral waves. The patterns depend on the amplitude and frequency of
the perturbation, and also on whether the system responds to the forcing near
the uniform oscillation frequency or the spiral wave frequency. Numerical
simulations of a forced FitzHugh-Nagumo reaction-diffusion model show both
resonant and near-resonant patterns similar to the BZ chemical system
Response solutions for beam equations with nonlocal nonlinear damping and Liouvillean frequencies
Response solutions are quasi-periodic ones with the same frequency as the
forcing term. The present work is devoted to the construction of response
solutions for -dimensional beam equations with nonlocal nonlinear damping,
which model frictional mechanisms affecting the bodies based on the average. By
considering in a domain that does not include the origin and
imposing a small quasi-periodic forcing with Liouvillean frequency vector,
which is weaker than the Diophantine or Brjuno one, we can show the existence
of the response solution for such a model. We present an alternative approach
to the contraction mapping principle (cf. [5,33]) through a combination of
reduction and the Nash--Moser iteration technique. The reason behind this
approach lies in the derivative losses caused by the nonlocal nonlinearity.Comment: 21 page
Bifurcations and instabilities in rotating, two-layer fluids: II. ?-plane
International audienceIn this paper, we show that the behavior of weakly nonlinear waves in a 2-layer model of baroclinic instability on a b-plane with varying viscosity is determined by a single, degenerate codimension three bifurcation. In the process, we show how previous studies, using the method of multiple scales to derive evolution equations for the slowly varying amplitude of the growing wave, arise as special limits of the general evolution description
Non-equilibrium steady states of ideal bosonic and fermionic quantum gases
We investigate non-equilibrium steady states of driven-dissipative ideal
quantum gases of both bosons and fermions. We focus on systems of sharp
particle number that are driven out of equilibrium either by the coupling to
several heat baths of different temperature or by time-periodic driving in
combination with the coupling to a heat bath. Within the framework of
(Floquet-)Born-Markov theory, several analytical and numerical methods are
described in detail. This includes a mean-field theory in terms of occupation
numbers, an augmented mean-field theory taking into account also non-trivial
two-particle correlations, and quantum-jump-type Monte-Carlo simulations. For
the case of the ideal Fermi gas, these methods are applied to simple lattice
models and the possibility of achieving exotic states via bath engineering is
pointed out. The largest part of this work is devoted to bosonic quantum gases
and the phenomenon of Bose selection, a non-equilibrium generalization of Bose
condensation, where multiple single-particle states are selected to acquire a
large occupation [Phys. Rev. Lett. 111, 240405 (2013)]. In this context, among
others, we provide a theory for transitions where the set of selected states
changes, describe an efficient algorithm for finding the set of selected
states, investigate beyond-mean-field effects, and identify the dominant
mechanisms for heat transport in the Bose selected state
Chaotic and pseudochaotic attractors of perturbed fractional oscillator
We consider a nonlinear oscillator with fractional derivative of the order
alpha. Perturbed by a periodic force, the system exhibits chaotic motion called
fractional chaotic attractor (FCA). The FCA is compared to the ``regular''
chaotic attractor. The properties of the FCA are discussed and the
``pseudochaotic'' case is demonstrated.Comment: 20 pages, 7 figure
Tunable Modulational Instability Sidebands via Parametric Resonance in Periodically Tapered Optical Fibers
We analyze the modulation instability induced by periodic variations of group
velocity dispersion and nonlinearity in optical fibers, which may be
interpreted as an analogue of the well-known parametric resonance in mechanics.
We derive accurate analytical estimates of resonant detuning, maximum gain and
instability margins, significantly improving on previous literature on the
subject. We also design a periodically tapered photonic crystal fiber, in order
to achieve narrow instability sidebands at a detuning of 35 THz, above the
Raman maximum gain peak of fused silica. The wide tunability of the resonant
peaks by variations of the tapering period and depth will allow to implement
sources of correlated photon pairs which are far-detuned from the input pump
wavelength, with important applications in quantum optics.Comment: 15 pages, 7 figure
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