168 research outputs found
Turing pattern formation in the Brusselator system with nonlinear diffusion
In this work we investigate the effect of density dependent nonlinear
diffusion on pattern formation in the Brusselator system. Through linear
stability analysis of the basic solution we determine the Turing and the
oscillatory instability boundaries. A comparison with the classical linear
diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern
formation. We study the process of pattern formation both in 1D and 2D spatial
domains. Through a weakly nonlinear multiple scales analysis we derive the
equations for the amplitude of the stationary patterns. The analysis of the
amplitude equations shows the occurrence of a number of different phenomena,
including stable supercritical and subcritical Turing patterns with multiple
branches of stable solutions leading to hysteresis. Moreover we consider
traveling patterning waves: when the domain size is large, the pattern forms
sequentially and traveling wavefronts are the precursors to patterning. We
derive the Ginzburg-Landau equation and describe the traveling front enveloping
a pattern which invades the domain. We show the emergence of radially symmetric
target patterns, and through a matching procedure we construct the outer
amplitude equation and the inner core solution.Comment: Physical Review E, 201
Entropy production in a mesoscopic chemical reaction system with oscillatory and excitable dynamics
Stochastic thermodynamics of chemical reaction systems has recently gained
much attention. In the present paper, we consider such an issue for a system
with both oscillatory and excitable dynamics, using catalytic oxidation of
carbon monoxide on the surface of platinum crystal as an example. Starting from
the chemical Langevin equations, we are able to calculate the stochastic
entropy production P along a random trajectory in the concentration state
space. Particular attention is paid to the dependence of the time averaged
entropy productionP on the system sizeN in a parameter region close to the
deterministic Hopf bifurcation.In the large system size (weak noise) limit, we
find that P N^{\beta} with {\beta}=0 or 1 when the system is below or abovethe
Hopf bifurcation, respectively. In the small system size (strong noise) limit,
P always increases linearly with N regardless of the bifurcation parameter.
More interestingly,P could even reach a maximum for some intermediate system
size in a parameter region where the corresponding deterministic system shows
steady state or small amplitude oscillation. The maximum value of P decreases
as the system parameter approaches the so-called CANARD point where the maximum
disappears.This phenomenon could be qualitativelyunderstood by partitioning the
total entropy production into the contributions of spikes and of small
amplitude oscillations.Comment: 13 pages, 6 figure
Low-lying bifurcations in cavity quantum electrodynamics
The interplay of quantum fluctuations with nonlinear dynamics is a central
topic in the study of open quantum systems, connected to fundamental issues
(such as decoherence and the quantum-classical transition) and practical
applications (such as coherent information processing and the development of
mesoscopic sensors/amplifiers). With this context in mind, we here present a
computational study of some elementary bifurcations that occur in a driven and
damped cavity quantum electrodynamics (cavity QED) model at low intracavity
photon number. In particular, we utilize the single-atom cavity QED Master
Equation and associated Stochastic Schrodinger Equations to characterize the
equilibrium distribution and dynamical behavior of the quantized intracavity
optical field in parameter regimes near points in the semiclassical
(mean-field, Maxwell-Bloch) bifurcation set. Our numerical results show that
the semiclassical limit sets are qualitatively preserved in the quantum
stationary states, although quantum fluctuations apparently induce phase
diffusion within periodic orbits and stochastic transitions between attractors.
We restrict our attention to an experimentally realistic parameter regime.Comment: 13 pages, 10 figures, submitted to PR
Finite-size and correlation-induced effects in Mean-field Dynamics
The brain's activity is characterized by the interaction of a very large
number of neurons that are strongly affected by noise. However, signals often
arise at macroscopic scales integrating the effect of many neurons into a
reliable pattern of activity. In order to study such large neuronal assemblies,
one is often led to derive mean-field limits summarizing the effect of the
interaction of a large number of neurons into an effective signal. Classical
mean-field approaches consider the evolution of a deterministic variable, the
mean activity, thus neglecting the stochastic nature of neural behavior. In
this article, we build upon two recent approaches that include correlations and
higher order moments in mean-field equations, and study how these stochastic
effects influence the solutions of the mean-field equations, both in the limit
of an infinite number of neurons and for large yet finite networks. We
introduce a new model, the infinite model, which arises from both equations by
a rescaling of the variables and, which is invertible for finite-size networks,
and hence, provides equivalent equations to those previously derived models.
The study of this model allows us to understand qualitative behavior of such
large-scale networks. We show that, though the solutions of the deterministic
mean-field equation constitute uncorrelated solutions of the new mean-field
equations, the stability properties of limit cycles are modified by the
presence of correlations, and additional non-trivial behaviors including
periodic orbits appear when there were none in the mean field. The origin of
all these behaviors is then explored in finite-size networks where interesting
mesoscopic scale effects appear. This study leads us to show that the
infinite-size system appears as a singular limit of the network equations, and
for any finite network, the system will differ from the infinite system
Entropy production of cyclic population dynamics
Entropy serves as a central observable in equilibrium thermodynamics.
However, many biological and ecological systems operate far from thermal
equilibrium. Here we show that entropy production can characterize the behavior
of such nonequilibrium systems. To this end we calculate the entropy production
for a population model that displays nonequilibrium behavior resulting from
cyclic competition. At a critical point the dynamics exhibits a transition from
large, limit-cycle like oscillations to small, erratic oscillations. We show
that the entropy production peaks very close to the critical point and tends to
zero upon deviating from it. We further provide analytical methods for
computing the entropy production which agree excellently with numerical
simulations.Comment: 4 pages, 3 figures and Supplementary Material. To appear in Phys.
Rev. Lett.
Amplitude equations near pattern forming instabilities for strongly driven ferromagnets
A transversally driven isotropic ferromagnet being under the influence of a
static external and an uniaxial internal anisotropy field is studied. We
consider the dissipative Landau-Lifshitz equation as the fundamental equation
of motion and treat it in ~dimensions. The stability of the spatially
homogeneous magnetizations against inhomogeneous perturbations is analyzed.
Subsequently the dynamics above threshold is described via amplitude equations
and the dependence of their coefficients on the physical parameters of the
system is determined explicitly. We find soft- and hard-mode instabilities,
transitions between sub- and supercritical behaviour, various bifurcations of
higher codimension, and present a series of explicit bifurcation diagrams. The
analysis of the codimension-2 point where the soft- and hard-mode instabilities
coincide leads to a system of two coupled Ginzburg-Landau equations.Comment: LATeX, 25 pages, submitted to Z.Phys.B figures available via
[email protected] in /pub/publications/frank/zpb_95
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