Werner Horsthemke

Sustained spiral waves in a continuously fed unstirred chemical reactor

Previous experiments on the self‐organization of spatial patterns in chemical reactors have generally been restricted to closed (batch) reactors. In such reactors the system relaxes irreversibly and uncontrollably towards thermodynamic equilibrium. It is difficult to make comparisons with existing theories which address asymptotic (long time) behavior because of the transient nature of the spatial patterns and the lack of well‐defined control parameters in such experiments. We report a novel disk‐shaped reactor that can be maintained far from thermodynamic equilibrium indefinitely by a continuous feed of reagents. Chemical patterns are formed inside a thin layer of inert gel that suppresses any convective motion. The feed to the gel is uniform and normal to the plane in which pattern can form. The reactor is used to conduct the first quantitative study of transitions between well‐defined states with different patterns.

CHEMICAL PATTERN FORMATION WITH EQUAL DIFFUSION COEFFICIENTS

Abstract We show that time-independent spatial patterns can form in a reaction-diffusion system even when the different chemical species have equal diffusion coefficients. The patterns organize in response to finite amplitude perturbations and are connected to the branch of uniform steady states by a branch of unstable steady states.

Phase diagrams of Noise Induced Transitions: Exact Results for a Class Of External Coloured Noise

By using the formula for the steady state probability distribution of fluctuations in a non-linear system under the influence of two-level Markovian noise, the existence of phase transitions of such a system with the variation of intensity and correlation time of the noise is shown. Explicit results for the Verhulst model and a model for population genetics are given and compared with the previous results for the white noise case.

White and coloured external noise and transition phenomena in nonlinear systems

It is shown that in a system whose phenomenological description does not present any instability a transition can be induced by external noise. The class of systems in which such a phenomenon can occur is determined.

Phase transition induced by external noise

Abstract We demonstrate on a chemical model that, even above the critical point, phase transitions can be induced solely by the effect of external noise.

Turing instabilities with nearly equal diffusion coefficients

We show that if a Turing instability occurs in a reaction–diffusion system with a nearly scalar diffusion matrix, then the parameters of the corresponding well‐mixed system are necessarily such that the well‐mixed system has at least two eigenvalues near zero. Conversely, if the corresponding well‐mixed system is sufficiently close to a coalescence point of Hopf and saddle‐node bifurcations (two eigenvalues are zero at such a point), and if the spatial domain is sufficiently large, then there exists a nearly scalar diffusion matrix such that a Turing instability occurs. These results imply that information on bifurcation loci from experiments in continuous‐flow stirred tank reactor suffices to locate regions of parameter space where Turing instabilities are likely to occur; no knowledge of the reaction mechanism and rate constants is needed. In order to illustrate these results, we have analyzed a six‐step model of the Belousov–Zhabotinskii reaction due to Showalter, Noyes, and Bar‐Eli. In this model, the...

Pseudo-regular oscillations induced by external noise

An examination of the effect of noise on a general system at a saddle-node bifurcation has revealed that, in the limit of weak noise, the probability density of the time to pass through the saddle-node has a universal shape, the specific kinetics of the particular system serving only to set the time scale. This probability density is displayed and its salient features are explicated. In the case of a saddle-node bifurcation leading to relaxation oscillations, this analysis leads to the prediction of the existence of noise-induced oscillations which appear much less random than might at first be expected. The period of these oscillations has a well-defined, nonzero most probable value, the inverse of which is a noise-induced frequency. This frequency can be detected as a peak in power spectra from numerical simulations of such a system. This is the first case of the prediction and detection of a noise-induced frequency of which the authors are aware.

Bistability in fluctuating environments. Implications in tumor immunology

We analyze under different environmental conditions the occurrence of bistability, i.e. of two simultaneously stable steady states, in a biological model system which describes the immunological interactions of neoplastic-target cells and cytotoxic-effector cells. As a result of environmental fluctuations such complex biological systems may undergo drastic modifications of their steady state properties. In particular, when the variance of fluctuations increases around a well defined mean value, transition phenomena appear which are absent in the usual bifurcation diagrams. The properties of the non-fluctuating systems can no longer be considered as a first approximation of the properties of the real system. Interestingly, in the case of the model, these transitions correspond to a rejection of tumor cells.

Turing patterns in an open reactor

Steady spatial chemical patterns have been found in model reaction–diffusion systems but have not yet been observed in any laboratory experiments. The reasons for this are discussed and the need for open reactors is stressed. A model open reactor is investigated in order to guide the experimental search for steady patterns. Specifically, Turing bifurcations in this reactor are studied for a simple autocatalytic chemistry (the Gray–Scott model) in order to determine the effects of varying diffusion coefficients, chemical time scales, and residence time. A description of all the steady‐state bifurcations from an initially homogeneous state is obtained. The Liapunov–Schmidt reduction is used to determine the stability of the bifurcating solutions and a steady‐state continuation technique is used to follow stable and unstable branches of bifurcating solutions.

Coloured noise induced transitions: exact results for external dichotomous markovian noise

A general method to calculate explicitly the stationary probability of nonlinear systems subjected to a special case of coloured noise is presented. For a simple model system the “phase diagram” for the various noise-induced transitions is determined.