Amplitude equations for breathing spiral waves in a forced reaction-diffusion system

Radially evolving spiral wave patterns in the Gierer-Meinhardt reaction-diffusion system

Spiral wave formation in spatially extended systems is a fascinating phenomenon that has garnered significant attention in reaction-diffusion systems. In this study, we explore the emergence of spiral wave-like patterns in the Gierer-Meinhardt reaction-diffusion model. By employing a multiple-time scale perturbation technique, we derive amplitude equations that reveal the conditions for spiral wave formation. Notably, our analysis shows that the amplitude of these spiral waves varies with the radial distance, introducing a distinctive feature to this pattern. Our theoretical predictions are further substantiated by numerical simulations, which confirm the emergence of spiral wave structures and validate the distinct radial dependence of their amplitude.

Unravelling diverse spatiotemporal orders in chlorine dioxide-iodine-malonic acid reaction-diffusion system through circularly polarized electric field and photo-illumination

Designing and predicting self-organized pattern formation in out-of-equilibrium chemical and biochemical reactions holds fundamental significance. External perturbations like light and electric fields exert a crucial influence on reaction-diffusion systems involving ionic species. While the separate impacts of light and electric fields have been extensively studied, comprehending their combined effects on spatiotemporal dynamics is paramount for designing versatile spatial orders. Here, we theoretically investigate the spatiotemporal dynamics of chlorine dioxide-iodine-malonic acid reaction-diffusion system under photo-illumination and circularly polarized electric field (CPEF). By applying CPEF at varying intensities and frequencies, we observe the predominant emergence of oscillating hexagonal spot-like patterns from homogeneous stable steady states. Furthermore, our study unveils a spectrum of intriguing spatiotemporal instabilities, encompassing stripe-like patterns, oscillating dumbbell-shaped patterns, spot-like instabilities with square-based symmetry, and irregular chaotic patterns. However, when we introduce periodic photo-illumination to the hexagonal spot-like instabilities induced by CPEF in homogeneous steady states, we observe periodic size fluctuations. Additionally, the stripe-like instabilities undergo alternating transitions between hexagonal spots and stripes. Notably, within the Turing region, the interplay between these two external influences leads to the emergence of distinct superlattice patterns characterized by hexagonal-and square-based symmetry. These patterns include parallel lines of spots, target-like formations, black-eye patterns, and other captivating structures. Remarkably, the simple perturbation of the system through the application of these two external fields offers a versatile tool for generating a wide range of pattern-forming instabilities, thereby opening up exciting possibilities for future experimental validation.

Deciphering electric field induced spatial pattern formation in the photosensitive chlorine-dioxide iodine malonic acid reaction and the Brusselator reaction-diffusion systems

Reaction-diffusion systems involving ionic species are susceptible to an externally applied electric field. Depending on the charges on the ionic species and the intensity of the applied electric field, diverse spatiotemporal patterns can emerge. We here considered two prototypical reaction-diffusion systems that follow activator-inhibitor kinetics: the photosensitive chlorine dioxide-iodine-malonic acid (CDIMA) reaction and the Brusselator model. By theoretical investigation and numerical simulations, we unravel how and to what extent an externally applied electric field can induce and modify the dynamics of these two systems. Our results show that both the uni- and bi-directional electric fields may induce Turing-like stationary patterns from a homogeneous uniform state resulting in horizontal, vertical, or bent stripe-like inhomogeneity in the photosensitive CDIMA system. In contrast, in the Brusselator model, for the activator and the inhibitor species having the same positive or negative charges, the externally applied electric field cannot develop any spatiotemporal instability when the diffusion coefficients are identical. However, various spatiotemporal patterns emerge for the same opposite charges of the interacting species, including moving spots and stripe-like structures, and a phenomenon of wave-splitting is observed. Moreover, the same sign and different magnitudes of the ionic charges can give rise to Turing-like stationary patterns from a homogeneous, stable, steady state depending upon the intensity of the applied electric field in the case of the Brusselator model. Our findings open the possibilities for future experiments to verify the predictions of electric field-induced various spatiotemporal instabilities in experimental reaction-diffusion systems.

Spatiotemporal antiresonance in coupled reaction-diffusion systems

We present a theoretical study of the spatiotemporal antiresonance in a system of two diffusively coupled chemical reactions, one of which is driven by an external periodic forcing. Although antiresonance is well known in various physical systems, the phenomenon in coupled chemical reactions has largely been overlooked. Based on the linearized dynamics around the steady state of the two-component coupled reaction-diffusion systems we have derived the general analytical expressions for the amplitude-frequency response functions of the driven and undriven components of the system. Our theoretical analysis is well corroborated by detailed numerical simulations on coupled Gray-Scott reaction-diffusion systems exhibiting antiresonance dip in the amplitude-frequency response curve as a result of destructive interference between the coupling and the periodic external forcing imparting differential stability of the two subsystems. This leads to the emergence of spatiotemporal patterns in an undriven subsystem, while the driven one settles down to a homogeneously stable steady state.

Emergent spatiotemporal instabilities in reactive spatially extended systems by thermodiffusion

Thermodiffusion or thermophoresis or Soret effect, i.e., mass-transport induced by thermal gradient, has immense application in segregation of species in two or multicomponent gaseous, liquid, or colloidal mixtures. Here, we show that an external thermal gradient can be effectively utilized in creation and modification of patterns in spatially extended systems. We consider Brusselator and chlorine-dioxide iodine malonic acid (CDIMA) reaction-diffusion systems, which follow activator-inhibitor kinetics subjected to an external thermal gradient. We find that the conspicuous interaction of emergent thermodiffusive flux with reaction kinetics and diffusion can lead to various spatiotemporal instabilities in these two models. Specifically, our result reveals formation of Turing-like spatial patterns even for equal diffusivities of the activator and inhibitor components in the Brusselator model under the influence of differential thermodiffusion, whereas formation of such stationary patterns in the CDIMA system from a homogenous stable steady state, which is also stable under differential diffusion, requires the same sign and magnitude of Soret coefficients. However, with equal diffusivities of the components of the CDIMA system and without starch in the medium, our result identifies formation of drifting spiral waves which finally disappears at longer times under the influence of thermodiffusion. We also show formation of propagating patterns of spotlike or stripelike heterogeneity in both the model systems under appropriate conditions. Our study provides a route to pattern formation beyond Turing space and reveals remarkable influence of thermodiffusion to modify the pattern types just by employing an external thermal gradient which also opens up the possibility to set up new related experiments.

Nonequilibrium transition and pattern formation in a linear reaction-diffusion system with self-regulated kinetics

We consider a reaction-diffusion system with linear, stochastic activator-inhibitor kinetics where the time evolution of concentration of a species at any spatial location depends on the relative average concentration of its neighbors. This self-regulating nature of kinetics brings in spatial correlation between the activator and the inhibitor. An interplay of this correlation in kinetics and disparity of diffusivities of the two species leads to symmetry breaking non-equilibrium transition resulting in stationary pattern formation. The role of initial noise strength and the linear reaction terms has been analyzed for pattern selection.

Differential-flow-induced transition of traveling wave patterns and wave splitting

We have analyzed the differential flow-induced instability in the presence of diffusive transport in a reaction-diffusion system following activator-inhibitor kinetics. The conspicuous interaction of differential flow and differential diffusivity that leads to pattern selection during transition of the traveling waves from stripes to rotating spots propagating in hexagonal arrays subsequent to wave splitting has been explored on the basis of a few-mode Galerkin scheme.

Parametric spatiotemporal oscillation in reaction-diffusion systems

We consider a reaction-diffusion system in a homogeneous stable steady state. On perturbation by a time-dependent sinusoidal forcing of a suitable scaling parameter the system exhibits parametric spatiotemporal instability beyond a critical threshold frequency. We have formulated a general scheme to calculate the threshold condition for oscillation and the range of unstable spatial modes lying within a V-shaped region reminiscent of Arnold's tongue. Full numerical simulations show that depending on the specificity of nonlinearity of the models, the instability may result in time-periodic stationary patterns in the form of standing clusters or spatially localized breathing patterns with characteristic wavelengths. Our theoretical analysis of the parametric oscillation in reaction-diffusion system is corroborated by full numerical simulation of two well-known chemical dynamical models: chlorite-iodine-malonic acid and Briggs-Rauscher reactions.

Dichotomous-noise-induced pattern formation in a reaction-diffusion system

We consider a generic reaction-diffusion system in which one of the parameters is subjected to dichotomous noise by controlling the flow of one of the reacting species in a continuous-flow-stirred-tank reactor (CSTR) -membrane reactor. The linear stability analysis in an extended phase space is carried out by invoking Furutzu-Novikov procedure for exponentially correlated multiplicative noise to derive the instability condition in the plane of the noise parameters (correlation time and strength of the noise). We demonstrate that depending on the correlation time an optimal strength of noise governs the self-organization. Our theoretical analysis is corroborated by numerical simulations on pattern formation in a chlorine-dioxide-iodine-malonic acid reaction-diffusion system.