Traveling pulse on a periodic background in parametrically driven systems

Parametric instability-induced synchronization in chemical oscillations and spatiotemporal patterns

We consider a model reaction-diffusion system with two coupled layers in which one of the components in a layer is parametrically driven by a periodic force. On perturbation of a homogeneous stable steady state, the system exhibits parametric instability inducing synchronization in temporal oscillation at half the forcing frequency in absence of diffusion and spatiotemporal patterns in presence of diffusion, when strength of parametric forcing and the strength of coupling are kept above their critical thresholds. We have formulated a general scheme to derive analytically the critical thresholds and dispersion relation to locate the unstable spatial modes lying between the tilted Arnold tongue in the amplitude-frequency plot. Full numerical simulations on Gierer-Meinhardt activator-inhibitor model corroborate our theoretical analysis on parametric instability-induced antiphase synchronization in chemical oscillation and spatiotemporal pattern formation, between the two layers.

Transition from nonradiative to radiative oscillons in parametrically driven systems

Nonequilibrium systems exhibit particle-type solutions. Oscillons are one of the best-known localized states of systems with time-dependent forcing or parametrically driven systems. We investigate the transition from nonradiative to radiative oscillons in the parametrically driven sine-Gordon model in two spatial dimensions. The bifurcation takes place when the strength of the forcing (frequency) increases (decreases) above a certain threshold. As a result of this transition, the oscillon emits radially symmetric evanescent waves. Numerically, we provide the phase diagram and show the supercritical nature of this transition. For small oscillations, based on the amplitude equation approach, the sine-Gordon equation with time-dependent forcing is transformed into the parametrically driven damped nonlinear Schrödinger model in two spatial dimensions. This amplitude equation exhibits a transition between nonradiative to radiative localized structures, consistently. Both models show quite good agreement.

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.