Transition from spiral waves to defect-mediated turbulence induced by gradient effects in a reaction-diffusion system

Nucleation of spatiotemporal structures from defect turbulence in the two-dimensional complex Ginzburg-Landau equation

We numerically investigate nucleation processes in the transient dynamics of the two-dimensional complex Ginzburg-Landau equation toward its "frozen" state with quasistationary spiral structures. We study the transition kinetics from either the defect turbulence regime or random initial configurations to the frozen state with a well-defined low density of quasistationary topological defects. Nucleation events of spiral structures are monitored using the characteristic length between the emerging shock fronts. We study two distinct situations, namely when the system is quenched either far from the transition limit or near it. In the former deeply quenched case, the average nucleation time for different system sizes is measured over many independent realizations. We employ an extrapolation method as well as a phenomenological formula to account for and eliminate finite-size effects. The nonzero (dimensionless) barrier for the nucleation of single spiral droplets in the extrapolated infinite system size limit suggests that the transition to the frozen state is discontinuous. We also investigate the nucleation of spirals for systems that are quenched close to but beyond the crossover limit and of target waves which emerge if a specific spatial inhomogeneity is introduced. In either of these cases, we observe long, "fat" tails in the distribution of nucleation times, which also supports a discontinuous transition scenario.

Surfactant-induced gradients in the three-dimensional Belousov-Zhabotinsky reaction

Scroll waves are prominent patterns formed in three-dimensional excitable media, and they are frequently considered highly relevant for some types of cardiac arrhythmias. Experimentally, scroll wave dynamics is often studied by optical tomography in the Belousov-Zhabotinsky reaction, which produces CO(2) as an undesired product. Addition of small concentrations of a surfactant to the reaction medium is a popular method to suppress or retard CO(2) bubble formation. We show that in closed reactors even these low concentrations of surfactants are sufficient to generate vertical gradients of excitability which are due to gradients in CO(2) concentration. In reactors open to the atmosphere such gradients can be avoided. The gradients induce a twist on vertically oriented scroll waves, while a twist is absent in scroll waves in a gradient-free medium. The effects of the CO(2) gradients are reproduced by a numerical study, where we extend the Oregonator model to account for the production of CO(2) and for its advection against the direction of gravity. The numerical simulations confirm the role of solubilized CO(2) as the source of the vertical gradient of excitability in reactors closed to the atmosphere.

Control of scroll wave turbulence in a three-dimensional reaction-diffusion system with gradient

In this paper, we summarize our recent experimental and theoretical works on observation and control of scroll wave (SW) turbulence. The experiments were conducted in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system with chemical concentration gradients in one dimension. A spatially homogeneous external forcing was used in the experiments as a control; it was realized by illuminating white light on the light sensitive reaction medium. We observed that, in the oscillatory regime of the system, SW can appear automatically in the gradient system, which will be led to spatiotemporal chaos under certain conditions. A suitable periodic forcing may stabilize inherent turbulence of SW. The mechanism of the transition to SW turbulence is due to the phase twist of SW in the presence of chemical gradients, while modulating the phase twist with a proper periodic forcing can delay this transition. Using the FitzHugh-Nagumo model with an external periodic forcing, we confirmed the control mechanism with numerical simulation. Moreover, we also show in the simulation that adding temporal external noise to the system may have the same control effect. During this process, we observed a new state called "intermittent turbulence," which may undergo a transition into a new type of SW collapse when the noise intensity is further increased. The intermittent state and the collapse could be explained by a random process.

Control of spiral turbulence by periodic forcing in a reaction-diffusion system with gradients

We report an experimental result on successfully controlling spiral turbulence in a reaction-diffusion system. The control is realized by periodic forcing in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system, which has chemical concentration gradients in the third dimension. We observe that, in the oscillatory regime of the system, a suitable periodic forcing may stabilize scroll waves (SWs), which otherwise undergo a transition to spiral turbulence. Relating the spiral phase shift due to gradients and the forcing frequency, the mechanism of the control can be well understood by modulating the phase twist of SWs. We use the FitzHugh-Nagumo model to demonstrate this mechanism.

Spiral waves in linearly coupled reaction-diffusion systems

The dynamics of spiral waves in a pair of linearly coupled reaction-diffusion systems is investigated. We find that the spiral dynamics depends on the coupling strength between the two subsystems. When the coupling strength is weak, the frequency and wavelength of the spiral wave in each subsystem remain unchanged. The interaction between the two subsystems induces the drift of spiral waves. When the coupling strength is strong, synchronization between the two subsystems is established. The two subsystems play different roles in the collective dynamics: one subsystem is always dominant and enslaves the other.

Negative-Tension Instability of Scroll Waves and Winfree Turbulence in the Oregonator Model

Excitable media support self-organized scroll waves which can be unstable and give rise to three-dimensional wave chaos. Winfree turbulence of scroll waves results from the negative-tension instability of scroll waves; it plays an important role in the cardiac tissue where it may lead to ventricular fibrillation. By numerical simulations of the Oregonator model, we show that this instability and, thus, the Winfree turbulence may also be observed in the Belousov-Zhabotinsky reaction. The region in the parameter space, where the instability takes place, is determined, and a relationship between the negative-tension instability and the meandering behavior of spiral waves is found. The application of global periodic forcing to control such turbulence in the Oregonator model is discussed.

Chemical Turbulence and Line Defects Induced by Gradient Effects in a Three-Dimensional Reaction−Diffusion System

We report experimental observations of chemical turbulence and line defects in a three-dimensional (3-D) Belousov-Zhabotinsky (BZ) reaction-diffusion system. Transitions from spiral waves to 3-D chemical turbulence to line defects are observed. These transitions are caused by concentration gradients across the third dimension in the 3-D reaction medium, indicating the observed line defects have a 3-D structure. The line defects come out of the 3-D turbulent state, and become smooth with the increase of the control parameter. Simulation with the two-variable Oregonator model in the 3-D system reproduces similar line defects.

Spontaneous scroll ring creation and scroll instability in oscillatory medium with gradients

Scroll waves in a quasi-three-dimensional reaction-diffusion medium with a gradient in the third dimension are studied by numerical simulations using the Fitzhugh-Nagumo model. Under a simple initial condition with only one straight filament, we show a spontaneous creation of twisted scroll rings surrounding the initial filament when the control parameter is increased across a threshold. We find that due to the presence of the gradient, the difference of oscillation frequencies in the third dimension is the underlying cause of the phenomenon. Further increase of the control parameter will lead to the spiral breakups, as a result of the interaction of filaments. Observations in the experiments conducted in the same type of medium with the Belousov-Zhabotinsiky reaction qualitatively agree with our simulation results.

Spiral breakup due to mechanical deformation in excitable media

To address the problem of how cardiac muscle contraction affects the dynamics of rotating spiral waves, spiral breakup induced by mechanical deformation in excitable media is studied in two partial-differential-equation models. It is shown that spirals begin to break up at omega=0.5 omega(0) when we increase the amplitude of the mechanical deformation gradually. Our numerical results point to a new mechanism of transition from spirals to spatiotemporal chaos, in which the anisotropic time-dependent diffusion coefficient is essential.

Chemical waves with line defects in the Belousov-Zhabotinsky reaction

We report our experimental observations of line-defected chemical waves in a quasi-two-dimensional reaction-diffusion system of Belousov-Zhabotinsky reaction. The observed line defects are explicit, which can be directly monitored in real time. In the parameter space, the state of the chemical waves with line defects is located between two regimes of the defect-mediated turbulence. The line defects appear in target waves as well as in spiral waves. We demonstrate that the line defects come out in traveling waves as the later reorganize their spatial topologies to adapt to the change in the local dynamics from simple to complex oscillations or vice versa.