309 research outputs found
Transitions between nonsymmetric and symmetric steady states near a triple eigenvalue
We examine the existence of nonuniform steady-state solutions of a certain class of reaction-diffusion
equations. Our analysis concentrates on the case where the first bifurcation is near a triple eigenvalue. We derive the conditions for a continuous transition between nonsymmetric and symmetric solutions when the bifurcation parameter progressively increases from zero. Finally, we give an example of a four variables model which presents the possibility of a triple eigenvalue
Imperfect Bifurcation Near a Double Eigenvalue: Transitions Between Nonsymmetric and Symmetric Patterns
We examine the existence of nonsymmetric and symmetric steady state solutions of a general class of reaction-diffusion equations.
Our study consists of two parts: (i) By analyzing the bifurcation from a uniform reference state to nonuniform regimes, we demonstrate the existence of a unique symmetric solution (basic wave number two) which becomes linearly stable when it surpasses a critical amplitude. (We assume that the first bifurcation point corresponds to the emergence of the simplest nonsymmetric steady state solutions.) (ii) This result is not affected when a parameter is nonuniformly distributed in the system. However, one of the two possible branches of nonsymmetric solutions may disappear from the bifurcation diagram.
Our analysis is motivated by the fact that experimental observations of pattern transitions during morphogenesis are interpreted in terms of the dynamics of stable concentration gradients. We have shown that in addition to the values of the physico-chemical parameters, these structures can be selected by two different mechanisms: (i) the linear stability of the nonuniform patterns, (ii) the effects of a small and nonuniform variation of a parameter in the spatial domain
Synchronization of tunable asymmetric square-wave pulses in delay-coupled optoelectronic oscillators
Identification of delays and discontinuity points of unknown systems by using synchronization of chaos
In this paper we present an approach in which synchronization of chaos is
used to address identification problems. In particular, we are able to
identify: (i) the discontinuity points of systems described by piecewise
dynamical equations and (ii) the delays of systems described by delay
differential equations. Delays and discontinuities are widespread features of
the dynamics of both natural and manmade systems. The foremost goal of the
paper is to present a general and flexible methodology that can be used in a
broad variety of identification problems.Comment: 11 pages, 3 figure
Mesa-type patterns in the one-dimensional Brusselator and their stability
The Brusselator is a generic reaction-diffusion model for a tri-molecular
chemical reaction. We consider the case when the input and output reactions are
slow. In this limit, we show the existence of -periodic, spatially bi-stable
structures, \emph{mesas}, and study their stability. Using singular
perturbation techniques, we find a threshold for the stability of mesas.
This threshold occurs in the regime where the exponentially small tails of the
localized structures start to interact. By comparing our results with Turing
analysis, we show that in the generic case, a Turing instability is followed by
a slow coarsening process whereby logarithmically many mesas are annihilated
before the system reaches a steady equilibrium state. We also study a
``breather''-type instability of a mesa, which occurs due to a Hopf
bifurcation. Full numerical simulations are shown to confirm the analytical
results.Comment: to appear, Physica
Coexisting periodic attractors in injection locked diode lasers
We present experimental evidence for coexisting periodic attractors in a
semiconductor laser subject to external optical injection. The coexisting
attractors appear after the semiconductor laser has undergone a Hopf
bifurcation from the locked steady state. We consider the single mode rate
equations and derive a third order differential equation for the phase of the
laser field. We then analyze the bifurcation diagram of the time periodic
states in terms of the frequency detuning and the injection rate and show the
existence of multiple periodic attractors.Comment: LaTex, 14 pages, 6 postscript figures include
Simulation of the switching of an all-optical flip-flop based on a SOA/DFB-laser diode optical feedback scheme
All-Optical flip-flop operation using a SOA and DFB laser diode in an optical feedback scheme is presented. The device is based on the gain difference in the SOA and DFB laser diode for the different stable states. Flip-flop operation is shown for set and reset pulse widths of 10 and 150ps of 5W and 3.5mW respectively
Variable-delay feedback control of unstable steady states in retarded time-delayed systems
We study the stability of unstable steady states in scalar retarded
time-delayed systems subjected to a variable-delay feedback control. The
important aspect of such a control problem is that time-delayed systems are
already infinite-dimensional before the delayed feedback control is turned on.
When the frequency of the modulation is large compared to the system's
dynamics, the analytic approach consists of relating the stability properties
of the resulting variable-delay system with those of an analogous distributed
delay system. Otherwise, the stability domains are obtained by a numerical
integration of the linearized variable-delay system. The analysis shows that
the control domains are significantly larger than those in the usual
time-delayed feedback control, and that the complexity of the domain structure
depends on the form and the frequency of the delay modulation.Comment: 13 pages, 8 figures, RevTeX, accepted for publication in Physical
Review
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