92 research outputs found
Turing Instability and Pattern Formation in an Activator-Inhibitor System with Nonlinear Diffusion
In this work we study the effect of density dependent nonlinear diffusion on
pattern formation in the Lengyel--Epstein system. Via the linear stability
analysis we determine both the Turing and the Hopf instability boundaries and
we show how nonlinear diffusion intensifies the tendency to pattern formation;
%favors the mechanism of pattern formation with respect to the classical linear
diffusion case; in particular, unlike the case of classical linear diffusion,
the Turing instability can occur even when diffusion of the inhibitor is
significantly slower than activator's one. In the Turing pattern region we
perform the WNL multiple scales analysis to derive the equations for the
amplitude of the stationary pattern, both in the supercritical and in the
subcritical case. Moreover, we compute the complex Ginzburg-Landau equation in
the vicinity of the Hopf bifurcation point as it gives a slow spatio-temporal
modulation of the phase and amplitude of the homogeneous oscillatory solution.Comment: Accepted for publication in Acta Applicandae Mathematica
Bifurcation in the Lengyel–Epstein system for the coupled reactors with diffusion
AbstractThe main goal of this paper is to continue the investigations of the important system of Fengqi et al. (2008). The occurrence of Turing and Hopf bifurcations in small homogeneous arrays of two coupled reactors via diffusion-linked mass transfer which described by a system of ordinary differential equations is considered. I study the conditions of the existence as well as stability properties of the equilibrium solutions and derive the precise conditions on the parameters to show that the Hopf bifurcation occurs. Analytically I show that a diffusion driven instability occurs at a certain critical value, when the system undergoes a Turing bifurcation, patterns emerge. The spatially homogeneous equilibrium loses its stability and two new spatially non-constant stable equilibria emerge which are asymptotically stable. Numerically, at a certain critical value of diffusion the periodic solution gets destabilized and two new spatially nonconstant periodic solutions arise by Turing bifurcation
Competition of spatial and temporal instabilities under time delay near codimension-two Turing-Hopf bifurcations
Competition of spatial and temporal instabilities under time delay near the
codimension-two Turing-Hopf bifurcations is studied in a reaction-diffusion
equation. The time delay changes remarkably the oscillation frequency, the
intrinsic wave vector, and the intensities of both Turing and Hopf modes. The
application of appropriate time delay can control the competition between the
Turing and Hopf modes. Analysis shows that individual or both feedbacks can
realize the control of the transformation between the Turing and Hopf patterns.
Two dimensional numerical simulations validate the analytical results.Comment: 13 pages, 6 figure
Harmonic vs. subharmonic patterns in a spatially forced oscillating chemical reaction
The effects of a spatially periodic forcing on an oscillating chemical
reaction as described by the Lengyel-Epstein model are investigated. We find a
surprising competition between two oscillating patterns, where one is harmonic
and the other subharmonic with respect to the spatially periodic forcing. The
occurrence of a subharmonic pattern is remarkable as well as its preference up
to rather large values of the modulation amplitude. For small modulation
amplitudes we derive from the model system a generic equation for the envelope
of the oscillating reaction that includes an additional forcing contribution,
compared to the amplitude equations known from previous studies in other
systems. The analysis of this amplitude equation allows the derivation of
analytical expressions even for the forcing corrections to the threshold and to
the oscillation frequency, which are in a wide range of parameters in good
agreement with the numerical analysis of the complete reaction equations. In
the nonlinear regime beyond threshold, the subharmonic solutions exist in a
finite range of the control parameter that has been determined by solving the
reaction equations numerically for various sets of parameters.Comment: 14 pages, 11 figure
Turing Instability in a Boundary-fed System
The formation of localized structures in the chlorine dioxide-idodine-malonic
acid (CDIMA) reaction-diffusion system is investigated numerically using a
realistic model of this system. We analyze the one-dimensional patterns formed
along the gradients imposed by boundary feeds, and study their linear stability
to symmetry-breaking perturbations (Turing instability) in the plane transverse
to these gradients. We establish that an often-invoked simple local linear
analysis which neglects longitudinal diffusion is inappropriate for predicting
the linear stability of these patterns. Using a fully nonuniform analysis, we
investigate the structure of the patterns formed along the gradients and their
stability to transverse Turing pattern formation as a function of the values of
two control parameters: the malonic acid feed concentration and the size of the
reactor in the dimension along the gradients. The results from this
investigation are compared with existing experiments.Comment: 41 pages, 18 figures, to be published in Physical Review
Instabilities and Patterns in Coupled Reaction-Diffusion Layers
We study instabilities and pattern formation in reaction-diffusion layers
that are diffusively coupled. For two-layer systems of identical two-component
reactions, we analyze the stability of homogeneous steady states by exploiting
the block symmetric structure of the linear problem. There are eight possible
primary bifurcation scenarios, including a Turing-Turing bifurcation that
involves two disparate length scales whose ratio may be tuned via the
inter-layer coupling. For systems of -component layers and non-identical
layers, the linear problem's block form allows approximate decomposition into
lower-dimensional linear problems if the coupling is sufficiently weak. As an
example, we apply these results to a two-layer Brusselator system. The
competing length scales engineered within the linear problem are readily
apparent in numerical simulations of the full system. Selecting a :1
length scale ratio produces an unusual steady square pattern.Comment: 13 pages, 5 figures, accepted for publication in Phys. Rev.
Experimental evidence of localized oscillations in the photosensitive chlorine dioxide-iodine-malonic acid reaction
The interaction between Hopf and Turing modes has been the subject of active research in recent years. We present here experimental evidence of the existence of mixed Turing-Hopf modes in a two-dimensional system. Using the photosensitive chlorine dioxide-iodine-malonic acid reaction (CDIMA) and external constant background illumination as a control parameter, standing spots oscillating in amplitude and with hexagonal ordering were observed. Numerical simulations in the Lengyel-Epstein model for the CDIMA reaction confirmed the results
Dynamics of Turing patterns under spatio-temporal forcing
We study, both theoretically and experimentally, the dynamical response of
Turing patterns to a spatio-temporal forcing in the form of a travelling wave
modulation of a control parameter. We show that from strictly spatial
resonance, it is possible to induce new, generic dynamical behaviors, including
temporally-modulated travelling waves and localized travelling soliton-like
solutions. The latter make contact with the soliton solutions of P. Coullet
Phys. Rev. Lett. {\bf 56}, 724 (1986) and provide a general framework which
includes them. The stability diagram for the different propagating modes in the
Lengyel-Epstein model is determined numerically. Direct observations of the
predicted solutions in experiments carried out with light modulations in the
photosensitive CDIMA reaction are also reported.Comment: 6 pages, 5 figure
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