Oscillatory Reaction-diffusion Systems Research Articles

Chemical waves and pattern formation in the 1,2-dipalmitoyl-sn-glycero-3-phosphocholine/water lamellar system.

The present work deals with the spatially extended oscillatory Belousov Zhabotinsky reaction-diffusion system carried out in an anisotropic environment of phosphatidylcholines/water binary system, which presents layered aqueous domains separated by lipid bilayers. We report the occurrence of stable Turing patterns, spiral waves, and other exotic structures in phospholipids bilayers that are generally used as a models for cell plasma membranes.

Destruction of stable spiral waves in oscillatory media.

We studied spiral wave dynamics in an oscillatory reaction diffusion system. We find a new phenomenon: without the appearance of any global modulation mode, stable spiral waves break up directly. By investigating the one-dimensional version of the system and the isolated local dynamics, we find that the unstable focus in the local dynamics plays an important role. For different boundary conditions (BCs), we find a transition between spiral waves and traveling waves for periodic BCs and drifting spiral wave for no-flux BC.

Periodic Travelling Wave Selection by Dirichlet Boundary Conditions in Oscillatory Reaction-Diffusion Systems

Periodic travelling waves are a fundamental solution form in oscillatory reaction-diffusion equations. Here I discuss the generation of periodic travelling waves in a reaction-diffusion system of the generic $\lambda$-$\omega$ form. I present numerical results suggesting that when this system is solved on a semi-infinite domain subject to Dirichlet boundary conditions in which the variables are fixed at zero, periodic travelling waves develop in the domain. The amplitude and speed of these waves are independent of the initial conditions, which I generate randomly in numerical simulations. Using a combination of numerical and analytical methods, I investigate the mechanism of periodic travelling wave selection. By looking for an appropriate similarity solution, I reduce the problem to an ODE system. Using this, I derive a formula for the selected speed and amplitude as a function of parameters. Finally, I discuss applications of this work to ecology.

Front explosion in a resonantly forced complex Ginzburg–Landau system

Periodically forced oscillatory reaction–diffusion systems near the Hopf bifurcation can be modeled by the resonantly forced complex Ginzburg–Landau equation. In the 3:1 resonant locking regime this equation has three stable fixed points corresponding to the phase-locked states in the underlying reaction–diffusion system. Phase fronts separate spatial domains containing the phase-locked states. When the Ginzburg–Landau equation parameters lie in the Benjamin–Feir-unstable regime, the phase fronts have a turbulent internal spatio-temporal structure. As the forcing intensity is changed, the intrinsic width of a front grows until, at a critical value, the front “explodes” and the turbulent interfacial zone expands to fill the entire domain. The scaling properties of this transition are explored and it is shown that front width and spatial and temporal correlations diverge as the critical forcing intensity is approached. These results are compared with similar behavior seen in a coupled map model with period-3 local dynamics. The prospects for observation of these phenomena in experiments on periodically forced reaction–diffusion systems are discussed.

Complex Pacemakers and Wave Sinks in Heterogeneous Oscillatory Chemical Systems

We investigate pattern formation in oscillatory reaction-diffusion systems where wave sources and sinks are created by a local shift of the oscillation frequency. General properties of resulting wave patterns in media with positive and negative dispersion are discussed. It is shown that phase slips in the wave patterns develop for strong frequency shifts, indicating the onset of desynchronization in the medium.

Turbulent fronts in resonantly forced oscillatory systems

Phase fronts in the forced complex Ginzburg-Landau equation, a model of a resonantly forced oscillatory reaction-diffusion system, are studied in the 3:1 resonance regime. The focus is on the turbulent (Benjamin-Feir-unstable) regime of the corresponding unforced system; in the forced system, phase fronts between spatially uniform phase-locked states exhibit complex dynamics. In one dimension, for strong forcing, phase fronts move with constant velocity. As the forcing intensity is lowered there is a bifurcation to oscillatory motion, followed by a bifurcation to a regime in which fronts multiply via the nucleation of domains of the third homogeneous phase in the front. In two dimensional systems, rough fronts with turbulent, complex internal structure may arise. For a critical value of the forcing intensity there is a nonequilibrium phase transition in which the turbulent interface grows to occupy the entire system. The phenomena we explore can be probed by experiments on periodically forced light sensitive reaction-diffusion systems.

“Ring-Shaped Model” of the Pacemaker in Oscillatory Reaction-Diffusion System

This paper presents a new model of a pacemaker in the oscillatory reaction-diffusion system called “the ring-shaped model”. The present model has a frequency higher than that of the bulk oscillations in a ring-shaped region. Theoretical and experimental studies have revealed that the “ring-shaped model” explains the experiments with a piece of disk-shaped filter paper as a pacemaker, indicating that the wavelength is a function of the radius of the pacemaker and has a minimum value.

Master equation approach to synchronization in diffusion-coupled nonlinear oscillators

We study the influence of internal fluctuations on phase synchronization in oscillatory reaction-diffusion systems through a master equation approach. In the limit of large system size, the probability density is analyzed by means of the eikonal approximation. This approximation yields a Hamilton-Jacobi equation for the stochastic potential, which may be reduced to coupled nonlinear diffusion equations for the phase of oscillation and (conjugate) "momentum." We give explicit expressions for the coefficients of these equations in terms of averages over the deterministic periodic orbit. For one-dimensional systems, we obtain an explicit solution for the stationary stochastic potential: the width in phase, which is defined as the root mean square fluctuation in phase, characterizes the roughness of phase locking, and diverges with the system size L according to a power law w approximately Lalpha, with alpha=1/2. To study higher-dimensional systems, we show that the eikonal approximations of the diffusion-coupled oscillator problem and the Kardar-Parisi-Zhang (KPZ) equation (in the limit of small noise intensity) are equivalent. The KPZ equation governs some forms of surface growth, and the height of a growing front corresponds to the phase (the 2pi periodicity in phase is ignored) in the diffusion-coupled oscillator problem. From the equivalence, we obtain the result that spatially synchronized states may exist only in systems with a spatial dimension greater than or equal to 3; for dimensions 1 and 2, a "rough" state exists in which the width (in phase) diverges algebraically with the system size, alpha>0.

Resonantly forced inhomogeneous reaction-diffusion systems.

The dynamics of spatiotemporal patterns in oscillatory reaction-diffusion systems subject to periodic forcing with a spatially random forcing amplitude field are investigated. Quenched disorder is studied using the resonantly forced complex Ginzburg-Landau equation in the 3:1 resonance regime. Front roughening and spontaneous nucleation of target patterns are observed and characterized. Time dependent spatially varying forcing fields are studied in the 3:1 forced FitzHugh-Nagumo system. The periodic variation of the spatially random forcing amplitude breaks the symmetry among the three quasi-homogeneous states of the system, making the three types of fronts separating phases inequivalent. The resulting inequality in the front velocities leads to the formation of "compound fronts" with velocities lying between those of the individual component fronts, and "pulses" which are analogous structures arising from the combination of three fronts. Spiral wave dynamics is studied in systems with compound fronts. (c) 2000 American Institute of Physics.

Spiral waves in oscillatory media with an applied electric field

Spiral waves in oscillatory reaction-diffusion systems under the influence of a uniform, time-independent electric field are modeled by the complex Ginzburg-Landau equation extended to include a convective term with complex coefficient. Results for the spiral drift, deformation, and frequency shift due to the electric field are obtained. The coefficient of the additional convective term is derived from the original reaction-diffusion system. The equation provides a good qualitative model of experimentally seen distortion of spiral waves in the presence of an applied electric field.

Perturbation theory for the breakdown of mean-field kinetics in oscillatory reaction-diffusion systems

Spatially distributed, nonequilibrium chemical systems described by a Markov chain model are considered. The evolution of such systems arises from a combination of local birth-death reactive events and random walks executed by the particles on a lattice. The parameter γ, the ratio of characteristic time scales of reaction and diffusion, is used to gauge the relative contributions of these two processes to the overall dynamics. For the case of relatively fast diffusion, i.e., γ≪1, an approximate solution to the Markov chain in the form of a perturbation expansion in powers of γ is derived. Kinetic equations for the average concentrations that follow from the solution differ from the mass-action law and contain memory terms. For a reaction-diffusion system with Willamowski-Rössler reaction mechanism, we further derive the following two results: (a) in the limit of γ→0, these memory terms vanish and the mass-action law is recovered; (b) the memory kernel is found to assume a simple exponential form. A comparison with numerical results from lattice gas automaton simulations is also carried out.

A comparison of two numerical methods for oscillatory reaction-diffusion systems

Reaction-diffusion systems whose kinetics contain a stable limit cycle are an established class of models for a range of oscillatory biological and chemical phenomena. In this paper, the author compares two numerical methods for calculating the oscillatory wake solutions generated by spatially localized perturbations for one particular reaction-diffusion system, of λ-ω type. The two methods are a semi-implicit, or implicit-explicit, finite difference scheme based on the Crank-Nicolson algorithm, and the method of lines with Gear's method. Though both solutions ultimately converge to a common solution, the approach to this final solution is very different in the two cases. The results provide a clear illustration of the care required in numerical solution of oscillatory reaction-diffusion equations.

Phase waves in oscillatory media

Abstract Oscillation phase spatial distributions influence the behavior of an oscillatory reaction-diffusion system. The dependence of the frequency of oscillations on the constant phase gradient is obtained. It is shown that for small enough values of the phase gradient the frequency is constant and is equal to that of homogeneous bulk oscillations. This can explain target patterns observed in BZ systems. The case of spatially non-uniform phase gradient is also considered. It is shown that there exist solutions corresponding to a moving stationary phase pattern which may be regarded as a secondary solitary wave superimposed on the primary phase waves. This approach helps to explain the details of such transitional processes as collision and displacement of waves with differing frequencies.

Turbulence and standing waves in oscillatory chemical reactions with global coupling

Using the model of the complex Ginzburg–Landau equation with global coupling, the influence of long-range interactions on the turbulent state of oscillatory reaction–diffusion systems is investigated. Experimental realizations of such a system are, e.g., oscillatory reactions on single crystal surfaces where some of the phenomena we simulate have been observed experimentally. We find that strong global coupling suppresses turbulence by transforming it into a pattern of standing waves or into uniform oscillations. Weaker global coupling gives rise to an intermittent turbulent state which retains partial synchrony.

Chemical turbulence and standing waves in a surface reaction model: The influence of global coupling and wave instabilities.

Among heterogeneously catalyzed chemical reactions, the CO oxidation on the Pt(110) surface under vacuum conditions offers probably the greatest wealth of spontaneous formation of spatial patterns. Spirals, fronts, and solitary pulses were detected at low surface temperatures (T<500 K), in line with the standard phenomenology of bistable, excitable, and oscillatory reaction-diffusion systems. At high temperatures (T greater, similar 540 K), more surprising features like chemical turbulence and standing waves appeared in the experiments. Herein, we study a realistic reaction-diffusion model of this system, with respect to the latter phenomena. In particular, we deal both with the influence of global coupling through the gas phase on the oscillatory reaction and the possibility of wave instabilities under excitable conditions. Gas-phase coupling is shown to either synchronize the oscillations or to yield turbulence and standing structures. The latter findings are closely related to clustering in networks of coupled oscillators and indicate a dominance of the global gas-phase coupling over local coupling via surface diffusion. In the excitable regime wave instabilities in one and two dimensions have been discovered. In one dimension, pulses become unstable due to a vanishing of the refractory zone. In two dimensions, turbulence can also emerge due to spiral breakup, which results from a violation of the dispersion relation.

Unidirectional waves on rings: Models for chiral preference of circumnutating plants

Twining plants exhibit a striking oscillation of their stems in their quest for a support. The oscillations, called circumnutation, have periods generally of 1–5 hr, and virtually all species have a preferred direction of twining. I seek to explain these chiral asymmetries in plant behavior by hypothesizing a chiral asymmetry in plant anatomy. Such asymmetries already exist, for example, in phyllotaxis. I explore wave phenomena on asymmetric but isotropic rings, and seek systems which will only support (stable) waves in one direction around the ring, and not in the other. Simulations indicate that (1) oscillatory reaction-diffusion systems do not support unidirectional waves on rings; (2) excitable reaction-diffusion systems do support unidirectional waves on rings; and (3) unidirectional phase-locking (discrete unidirectional waves) occurs in rings of coupled oscillators. Thus, chiral asymmetries of circumnutating plants cannot be explained by continuum oscillator phenomena, but can be explained by general discrete oscillators, or excitable phenomena on the continuum.

Conditions for the occurence of degeneracy between Turing and phase instabilities

The relation between the conditions for the onset of Turing and phase instabilities is studied in reaction-diffusion systems. Whether these conditions are mutually exclusive crucially depends on the amplitude-frequency relation of the oscillator. Degeneracy between the two diffusion-induced instabilities can occur when the A-ω relation is positive. One way of classifying oscillatory reaction-diffusion systems arises from the present investigation.

Inhomogeneous perturbations and phase resetting in an oscillatory reaction–diffusion system

The response of the Brusselator reaction–diffusion system to inhomogeneous perturbations is studied. The main focus of this work is on a spatial generalization of the phase resetting problem. A randomly chosen fraction p of an initially homogeneous oscillatory system is locally perturbed and driven off the limit cycle. The asymptotic local phase is monitored and averaged over local regions and realizations of the perturbation process. From this information a phase response curve can be constructed which depends both on the local stimulus amplitude and on p. The system exhibits two qualitatively different kinds of response depending on the stimulus amplitude and the phase at which the perturbation is applied. It either relaxes to a spatially homogeneous oscillatory state or develops persistent spatial patterns. The origin of this behavior is discussed.

Phase resetting dynamics for a discrete reaction–diffusion model

The results of a study of spatial pattern formation in a two-dimensional oscillatory reaction–diffusion system are presented. The calculations are carried out on a discrete model of the Brusselator reaction. The system responds to inhomogeneous perturbations in two different ways. For most perturbations it relaxes back to a spatially homogeneous state with a phase shift. However, special perturbations produce persistent structures which consist of spiral waves and target patterns. The nature of these spatio-temporal states is discussed.

The Uniqueness and Stability of the Rest State for Strongly Coupled Oscillators

In this paper it is shown that an unstable steady state can be stabilized in the presence of sufficient inhomogeneity and strong diffusion for a continuum and discrete model. These results are applied to a pair of coupled oscillators and to an oscillatory reaction-diffusion system. It is also shown that for the reaction-diffusion system, the trivial rest state is the unique phase-locked solution. Some additional numerical results are presented that illustrate the nature of this bifurcation for a realistic model oscillator.