Harald Engel

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.

Traveling waves in the CO oxidation on Pt(110): Theory

A dynamic model designed to describe bistability and kinetic oscillations of the reaction rate during the oxidation of CO on a Pt(110) single crystal surface is extended by incorporating surface diffusion of adsorbed CO in order to analyze the properties of traveling waves propagating on the catalytically active surface. In the range of control parameters (partial pressure of oxygen and carbon monoxide and temperature) which corresponds to excitable dynamics, solitary pulses and periodic wave trains can be triggered. Using both asymptotic and numerical methods, the velocity and shape of the pulses as well as the dispersion relation for periodic wave trains are determined and compared to experimental data where available.

Pattern-formation during the CO oxidation on Pt(110) surfaces under global coupling

A reaction‐diffusion model for CO oxidation on Pt(110) single crystals proposed by Krischer, Eiswirth, and Ertl is supplemented by an equation for the balance of CO partial pressure in the gas phase. This allows us to study the interaction of local and global coupling with the dynamics of the reaction in the oscillatory regime. In absence of global coupling a stability analysis of the homogeneous oscillatory state predicts parameter regions with negative values of the phase diffusion coefficient indicating the possibility of phase turbulence. In the globally coupled system without diffusion we observe the formation of phase‐locked clusters of oscillators and irregular behavior. If both surface diffusion and global coupling through the gas phase are taken into account depending on the range of external parameters we get the following types of structures: phase flips, standing waves, spatially irregular coverage pattern, and the uniformly oscillating surface.

Pulse propagation in a model for the photosensitive Belousov-Zhabotinsky reaction with external noise

We study the dynamics of excitation pulses in a modified Oregonator model for the light-sensitive Belousov-Zhabotinsky (BZ)reaction assuming that the intensity of the applied illumination is a spatiotemporal stochastic field with finite correlation time and correlation length. For a two-component version of the model we discuss the dependence of the pulse speed on the characteristic parameters of the noise in the framework of a small noise approximation up to the first order in the correlation time. In the full three-component model we find enhancement of coherence resonance for suitable chosen correlation time. Based on this observation, we propose a mechanism for noise-enhanced propagation of pulse trains in excitable media subjected to external fluctuations.

On the optimal control of the Schlögl-model

Optimal control problems for a class of 1D semilinear parabolic equations with cubic nonlinearity are considered. This class is also known as the Schlögl model. Main emphasis is laid on the control of traveling wave fronts that appear as typical solutions to the state equation.The well-posedness of the optimal control problem and the regularity of its solution are proved. First-order necessary optimality conditions are established by standard adjoint calculus. The state equation is solved by the implicit Euler method in time and a finite element technique with respect to the spatial variable. Moreover, model reduction by Proper Orthogonal Decomposition is applied and compared with the numerical solution of the full problem. To solve the optimal control problems numerically, the performance of different versions of the nonlinear conjugate gradient method is studied. Various numerical examples demonstrate the capacities and limits of optimal control methods.

An Active Poroelastic Model for Mechanochemical Patterns in Protoplasmic Droplets of Physarum polycephalum

Motivated by recent experimental studies, we derive and analyze a two-dimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a description of an active poroelastic two-phase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The equations for the poroelastic medium are obtained from continuum force balance and include the relevant mechanical fields and an incompressibility condition for the two-phase medium. The reaction-diffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stress. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a mechanochemical coupling. A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets.

Modeling of the Modulation by Buffers of Ca2+ Release through Clusters of IP3 Receptors

Intracellular Ca(2+) release is a versatile second messenger system. It is modeled here by reaction-diffusion equations for the free Ca(2+) and Ca(2+) buffers, with spatially discrete clusters of stochastic IP(3) receptor channels (IP(3)Rs) controlling the release of Ca(2+) from the endoplasmic reticulum. IP(3)Rs are activated by a small rise of the cytosolic Ca(2+) concentration and inhibited by large concentrations. Buffering of cytosolic Ca(2+) shapes global Ca(2+) transients. Here we use a model to investigate the effect of buffers with slow and fast reaction rates on single release spikes. We find that, depending on their diffusion coefficient, fast buffers can either decouple clusters or delay inhibition. Slow buffers have little effect on Ca(2+) release, but affect the time course of the signals from the fluorescent Ca(2+) indicator mainly by competing for Ca(2+). At low [IP(3)], fast buffers suppress fluorescence signals, slow buffers increase the contrast between bulk signals and signals at open clusters, and large concentrations of buffers, either fast or slow, decouple clusters.

Controlling the position of traveling waves in reaction-diffusion systems.

We present a method to control the position as a function of time of one-dimensional traveling wave solutions to reaction-diffusion systems according to a prespecified protocol of motion. Given this protocol, the control function is found as the solution of a perturbatively derived integral equation. Two cases are considered. First, we derive an analytical expression for the space (x) and time (t) dependent control function f(x,t) that is valid for arbitrary protocols and many reaction-diffusion systems. These results are close to numerically computed optimal controls. Second, for stationary control of traveling waves in one-component systems, the integral equation reduces to a Fredholm integral equation of the first kind. In both cases, the control can be expressed in terms of the uncontrolled wave profile and its propagation velocity, rendering detailed knowledge of the reaction kinetics unnecessary.

Coherence resonance in a chemical excitable system driven by coloured noise

We investigate how the temporal correlation in excitable systems driven by external noise affects the coherence of the systems response. The coupling to the fluctuating environment is introduced via fluctuations of a bifurcation parameter that controls the local dynamics of the light-sensitive Belousov–Zhabotinsky reaction and of its numerical description, the Oregonator model. Both systems are brought from a highly incoherent regime to a coherent one by an appropriate choice of the correlation time and keeping noise variance constant. This effect has been found both for an Ornstein–Uhlenbeck process and for a dichotomous telegraph signal. In the latter case, we are able to connect the optimal correlation time, for which the system behaviour is most coherent, with a characteristic time scale of the system.

Reaction fronts and pulses in the CO oxidation on Pt : theoretical analysis

Abstract A set of coupled nonlinear reaction-diffusion equations was formulated for the oxidation of CO on low-index plane Pt surfaces and solved for their spatiotemporal behaviour using continuation techniques and the method of global connections. A 2-variable model for Pt(111), consisting of the CO and O coverages as a function of time and space, gave rise to reaction fronts. In order to model the behaviour of Pt(111) or Pt(100) a third equation describing the respective adsorbate-driven phase transition had to be added. The resulting 3-variable system predicts extended regions of excitability, where pulses can be triggered. The front — respectively pulse — velocities as well as the critical radii for the nucleation were computed and compared to experimental data where available.