841 research outputs found
Bifurcation structure of cavity soliton dynamics in a VCSEL with saturable absorber and time-delayed feedback
We consider a wide-aperture surface-emitting laser with a saturable absorber
section subjected to time-delayed feedback. We adopt the mean-field approach
assuming a single longitudinal mode operation of the solitary VCSEL. We
investigate cavity soliton dynamics under the effect of time- delayed feedback
in a self-imaging configuration where diffraction in the external cavity is
negligible. Using bifurcation analysis, direct numerical simulations and
numerical path continuation methods, we identify the possible bifurcations and
map them in a plane of feedback parameters. We show that for both the
homogeneous and localized stationary lasing solutions in one spatial dimension
the time-delayed feedback induces complex spatiotemporal dynamics, in
particular a period doubling route to chaos, quasiperiodic oscillations and
multistability of the stationary solutions
Long Tailed Maps as a Representation of Mixed Mode Oscillatory Systems
Mixed mode oscillatory (MMO) systems are known to exhibit some generic
features such as the reversal of period doubling sequences and crossover to
period adding sequences as bifurcation parameters are varied. In addition, they
exhibit a nearly one dimensional unimodal Poincare map with a longtail. We
recover these common features from a general class of two parameter family of
one dimensional maps with a unique critical point that satisfy a few general
constraints that determine the nature of the map. We derive scaling laws that
determine the parameter widths of the dominant windows of periodic orbits
sandwiched between two successive states of RL^k sequence. An example of a two
parameter map with a unique critical point is introduced to verify the
analytical results.Comment: 13 pages and 8 figure
Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation
Complex systems exhibiting critical transitions when one of their governing
parameters varies are ubiquitous in nature and in engineering applications.
Despite a vast literature focusing on this topic, there are few studies dealing
with the effect of the rate of change of the bifurcation parameter on the
tipping points. In this work, we consider a subcritical stochastic Hopf
bifurcation under two scenarios: the bifurcation parameter is first changed in
a quasi-steady manner and then, with a finite ramping rate. In the latter case,
a rate-dependent bifurcation delay is observed and exemplified experimentally
using a thermoacoustic instability in a combustion chamber. This delay
increases with the rate of change. This leads to a state transition of larger
amplitude compared to the one that would be experienced by the system with a
quasi-steady change of the parameter. We also bring experimental evidence of a
dynamic hysteresis caused by the bifurcation delay when the parameter is ramped
back. A surrogate model is derived in order to predict the statistic of these
delays and to scrutinise the underlying stochastic dynamics. Our study
highlights the dramatic influence of a finite rate of change of bifurcation
parameters upon tipping points and it pinpoints the crucial need of considering
this effect when investigating critical transitions
Antispiral waves are sources in oscillatory reaction-diffusion media
Spiral and antispiral waves are studied numerically in two examples of
oscillatory reaction-diffusion media and analytically in the corresponding
complex Ginzburg-Landau equation (CGLE). We argue that both these structures
are sources of waves in oscillatory media, which are distinguished only by the
sign of the phase velocity of the emitted waves. Using known analytical results
in the CGLE, we obtain a criterion for the CGLE coefficients that predicts
whether antispirals or spirals will occur in the corresponding
reaction-diffusion systems. We apply this criterion to the FitzHugh-Nagumo and
Brusselator models by deriving the CGLE near the Hopf bifurcations of the
respective equations. Numerical simulations of the full reaction-diffusion
equations confirm the validity of our simple criterion near the onset of
oscillations. They also reveal that antispirals often occur near the onset and
turn into spirals further away from it. The transition from antispirals to
spirals is characterized by a divergence in the wavelength. A tentative
interpretaion of recent experimental observations of antispiral waves in the
Belousov-Zhabotinsky reaction in a microemulsion is given.Comment: 10 pages, 8 figures, submitted to J. Phys. Chem. B on Feb. 20, 2004.
A short account of the spiral-antispiral criterion has been given in PRL (see
http://link.aps.org/abstract/PRL/v92/e089801
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