Harmonic versus subharmonic patterns in a spatially forced oscillating chemical reaction

Effects of square spatial periodic forcing on oscillatory hexagon patterns in coupled reaction-diffusion systems

One of the central issues in pattern formation is understanding the response of pattern-forming systems to an external stimulus. While significant progress has been made in systems with only one instability, much less is known about the response of complex patterns arising from the interaction of two or more instabilities. In this paper, we consider the effects of square spatial periodic forcing on oscillatory hexagon patterns in a two-layer coupled reaction diffusion system which undergoes both Turing and Hopf instabilities. Two different types of additive forcings, namely direct and indirect forcing, have been applied. It is shown that the coupled system exhibits different responses towards the spatial forcing under different forcing types. In the indirect case, the oscillatory hexagon pattern transitions into other oscillatory Turing patterns or resonant Turing patterns, depending on the forcing wavenumber and strength. In the direct forcing case, only non-resonant Turing patterns can be obtained. Our results may provide new insight into the modification and control of spatio-temporal patterns in multilayered systems, especially in biological and ecological systems.

On the bandwidth of stable nonlinear stripe patterns in finite size systems

Nonlinear stripe patterns occur in many different systems, from the small scales of biological cells to geological scales as cloud patterns. They all share the universal property of being stable at different wavenumbers q, i.e., they are multistable. The stable wavenumber range of the stripe patterns, which is limited by the Eckhaus- and zigzag instabilities even in finite systems for several boundary conditions, increases with decreasing system size. This enlargement comes about because suppressing degrees of freedom from the two instabilities goes along with the system reduction, and the enlargement depends on the boundary conditions, as we show analytically and numerically with the generic Swift-Hohenberg (SH) model and the universal Newell-Whitehead-Segel equation. We also describe how, in very small system sizes, any periodic pattern that emerges from the basic state is simultaneously stable in certain parameter ranges, which is especially important for the Turing pattern in cells. In addition, we explain why below a certain system width, stripe patterns behave quasi-one-dimensional in two-dimensional systems. Furthermore, we show with numerical simulations of the SH model in medium-sized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stable patterns for different combinations of boundary conditions.

Monochromatic waves induced by large-scale parametric forcing

We study the formation and stability of monochromatic waves induced by large-scale modulations in the framework of the complex Ginzburg-Landau equation with parametric nonresonant forcing dependent on the spatial coordinate. In the limiting case of forcing with very large characteristic length scale, analytical solutions for the equation are found and conditions of their existence are outlined. Stability analysis indicates that the interval of existence of a monochromatic wave can contain a subinterval where the wave is stable. We discuss potential applications of the model in rheology, fluid dynamics, and optics.

Design and control of patterns in reaction-diffusion systems

We discuss the design of reaction-diffusion systems that display a variety of spatiotemporal patterns. We also consider how these patterns may be controlled by external perturbation, typically using photochemistry or temperature. Systems treated include the Belousov-Zhabotinsky (BZ) reaction, the chlorite-iodide-malonic acid and chlorine dioxide-malonic acid-iodine reactions, and the BZ-AOT system, i.e., the BZ reaction in a water-in-oil reverse microemulsion stabilized by the surfactant sodium bis(2-ethylhexyl) sulfosuccinate (AOT).

Influence of parametric forcing on the nonequilibrium dynamics of wave patterns

We investigate analytically and numerically the effect of inhomogeneities on the nonequilibrium dynamics of wave patterns in the framework of a complex Ginzburg-Landau equation (CGLE) with parametric, nonresonant forcing periodic in space and time. It is found that the forcing results in occurrence of traveling waves with different dispersion properties. In the limiting case of forcing with very large wavelength, the waves have essentially anharmonic spatial structure. We consider the influence of modulations on the development of an intermittent chaos and show that the parametric forcing may completely suppress the appearance of chaotic patterns. The relations between this and other pattern-forming systems are discussed. The results obtained are applied to describe the dynamics of thermal Rossby waves influenced by surface topography.