Model of the Belousov-Zhabotinsky reaction

The article describes results of the modified model of the Belousov-Zhabotinsky reaction, which resembles rather well the limit set observed upon experimental performance of the reaction in the Petri dish. We discuss the concept of the ignition of circular waves and show that only the asymmetrical ignition leads to the formation of spiral structures. From the qualitative assumptions on the behavior of dynamic systems, we conclude that the Belousov-Zhabotinsky reaction likely forms a regular grid.Comment: 17 pages, 12 figure

Riding a Spiral Wave: Numerical Simulation of Spiral Waves in a Co-Moving Frame of Reference

We describe an approach to numerical simulation of spiral waves dynamics of large spatial extent, using small computational grids.Comment: 15 pages, 14 figures, as accepted by Phys Rev E 2010/03/2

Reaction–diffusion chemistry implementation of associative memory neural network

Unconventional computing paradigms are typically very difficult to program. By implementing efficient parallel control architectures such as artificial neural networks, we show that it is possible to program unconventional paradigms with relative ease. The work presented implements correlation matrix memories (a form of artificial neural network based on associative memory) in reaction–diffusion chemistry, and shows that implementations of such artificial neural networks can be trained and act in a similar way to conventional implementations

Stochastic Turing patterns in the Brusselator model

A stochastic version of the Brusselator model is proposed and studied via the system size expansion. The mean-field equations are derived and shown to yield to organized Turing patterns within a specific parameters region. When determining the Turing condition for instability, we pay particular attention to the role of cross diffusive terms, often neglected in the heuristic derivation of reaction diffusion schemes. Stochastic fluctuations are shown to give rise to spatially ordered solutions, sharing the same quantitative characteristic of the mean-field based Turing scenario, in term of excited wavelengths. Interestingly, the region of parameter yielding to the stochastic self-organization is wider than that determined via the conventional Turing approach, suggesting that the condition for spatial order to appear can be less stringent than customarily believed.Comment: modified version submitted to Phys Rev. E. 5. 3 Figures (5 panels) adde

Thermal switch of oscillation frequency in belousov- zhabotinsky liquid marbles

© 2019 The Authors. External control of oscillation dynamics in the Belousov- Zhabotinsky (BZ) reaction is important for many applications including encoding computing schemes. When considering the BZ reaction, there are limited studies dealing with thermal cycling, particularly cooling, for external control. Recently, liquid marbles (LMs) have been demonstrated as a means of confining the BZ reaction in a system containing a solid-liquid interface. BZ LMs were prepared by rolling 50 ml droplets in polyethylene (PE) powder. Oscillations of electrical potential differences within the marble were recorded by inserting a pair of electrodes through the LM powder coating into the BZ solution core. Electrical potential differences of up to 100mV were observed with an average period of oscillation ca 44 s. BZ LMs were subsequently frozen to 218C to observe changes in the frequency of electrical potential oscillations. The frequency of oscillations reduced upon freezing to 11mHz cf. 23 mHz at ambient temperature. The oscillation frequency of the frozen BZ LM returned to 23 mHz upon warming to ambient temperature. Several cycles of frequency fluctuations were able to be achieved

Quasiperiodic Patterns in Boundary-Modulated Excitable Waves

We investigate the impact of the domain shape on wave propagation in excitable media. Channelled domains with sinusoidal boundaries are considered. Trains of fronts generated periodically at an extreme of the channel are found to adopt a quasiperiodic spatial configuration stroboscopically frozen in time. The phenomenon is studied in a model for the photo-sensitive Belousov-Zabotinsky reaction, but we give a theoretical derivation of the spatial return maps prescribing the height and position of the successive fronts that is valid for arbitrary excitable reaction-diffusion systems.Comment: 4 pages (figures included

Fractional dynamics of coupled oscillators with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range power-wise interaction. The corresponding term in dynamical equations is proportional to $1/|n-m|^{\alpha+1}$. It is shown that the equation of motion in the infrared limit can be transformed into the medium equation with the Riesz fractional derivative of order $\alpha$, when $0<\alpha<2$. We consider few models of coupled oscillators and show how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on $\alpha$. The presence of fractional derivative leads also to the occurrence of localized structures. Particular solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear Schrodinger) equation are derived. These solutions are interpreted as synchronized states and localized structures of the oscillatory medium.Comment: 34 pages, 18 figure

Self-organized stable pacemakers near the onset of birhythmicity

General amplitude equations for reaction-diffusion systems near to the soft onset of birhythmicity described by a supercritical pitchfork-Hopf bifurcation are derived. Using these equations and applying singular perturbation theory, we show that stable autonomous pacemakers represent a generic kind of spatiotemporal patterns in such systems. This is verified by numerical simulations, which also show the existence of breathing and swinging pacemaker solutions. The drift of self-organized pacemakers in media with spatial parameter gradients is analytically and numerically investigated.Comment: 4 pages, 4 figure

Mixed-Mode Oscillations in Three Time-Scale Systems: A Prototypical Example

Mixed-mode dynamics is a complex type of dynamical behavior that is characterized by a combination of small-amplitude oscillations and large-amplitude excursions. Mixed-mode oscillations (MMOs) have been observed both experimentally and numerically in various prototypical systems in the natural sciences. In the present article, we propose a mathematical model problem which, though analytically simple, exhibits a wide variety of MMO patterns upon variation of a control parameter. One characteristic feature of our model is the presence of three distinct time-scales, provided a singular perturbation parameter is sufficiently small. Using geometric singular perturbation theory and geometric desingularization, we show that the emergence of MMOs in this context is caused by an underlying canard phenomenon. We derive asymptotic formulae for the return map induced by the corresponding flow, which allows us to obtain precise results on the bifurcation (Farey) sequences of the resulting MMO periodic orbits. We prove that the structure of these sequences is determined by the presence of secondary canards. Finally, we perform numerical simulations that show good quantitative agreement with the asymptotics in the relevant parameter regime

Buckling of scroll waves

A scroll wave in a sufficiently thin layer of an excitable medium with negative filament tension can be stable nevertheless due to filament rigidity. Above a certain critical thickness of the medium, such scroll wave will have a tendency to deform into a buckled, precessing state. Experimentally this will be seen as meandering of the spiral wave on the surface, the amplitude of which grows with the thickness of the layer, until a break-up to scroll wave turbulence happens. We present a simplified theory for this phenomenon and illustrate it with numerical examples.Comment: 4 pages main text + 5 pages appendix, 4+2 figures and a movie, as accepted by Phys Rev Letters 2012/09/2