Uniform oscillations in spatially extended systems resonate with temporal periodic forcing within the Arnold tongues of single forced oscillators. The Arnold tongues are wedge-like domains in the parameter space spanned by the forcing amplitude and frequency, within which the oscillator’s frequency is locked to a fraction of the forcing frequency. Spatial patterning can modify these domains. We describe here two pattern formation mechanisms affecting frequency locking at half the forcing frequency. The mechanisms are associated with phase-front instabilities and a Turing-like instability of the rest state. Our studies combine experiments on the ruthenium catalyzed light-sensitive Belousov-Zhabotinsky reaction forced by periodic illumination, and numerical and analytical studies of two model systems, the FitzHugh-Nagumo model and the complex Ginzburg-Landau equation, with additional terms describing periodic forcing.
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The chemical concentrations in the two chemical reservoirs separated by the porous membrane were: Reservoir I: bromo-malonic acid, potassium bromate, sulfuric acid; Reservoir II: potassium bromate, Tris(-bipyridyl)dichlororuthenium(II)hexahydrate, sulfuric acid. Each reservoir volume is and the flow rate of chemicals through Reservoir I was , while through Reservoir II it was . Chemicals were premixed before entering each reservoir; a premixer and a premixer fed Reservoirs I and II, respectively. The experiments were conducted at room temperature.
Note that uniform oscillations in the range remain resonant and coexist with nonresonant Bloch waves.
The image data recorded by the camera is processed by frequency filtering with a bandpass filter centered at with a width of . We treat the resulting signal as and plot the phase of the complex amplitude . The front line is defined as and the vortex positions are .
The chemical concentrations in the two chemical reservoirs separated by the porous membrane were: Reservoir I: malonic acid, sodium bromide, potassium bromate, sulfuric acid; Reservoir II: potassium bromate, Tris(-bipyridyl)dichlororuthenium(II)hexahydrate, sulfuric acid. Each reservoir volume is and the flow rate of chemicals through each Reservoir was . Chemicals were premixed before entering each reservoir; a premixer and a premixer fed Reservoirs I and II, respectively. The experiments were conducted at room temperature.
Since we do not know how the parameters of Eq. (1) are related to the oscillatory BZ reaction we cannot determine the sign of . The sign, however, is expected to be positive if for the Ising front is unstable to transverse perturbations.
