Competing resonances in spatially forced pattern-forming systems

Abstract

Spatial periodic forcing can entrain a pattern-forming system in the same way as temporal periodic forcing can entrain an oscillator. The forcing can lock the pattern's wave number to a fraction of the forcing wave number within tonguelike domains in the forcing parameter plane, it can increase the pattern's amplitude, and it can also create patterns below their onset. We derive these results using a multiple-scale analysis of a spatially forced Swift-Hohenberg equation in one spatial dimension. In two spatial dimensions the one-dimensional forcing can induce a symmetry-breaking instability that leads to two-dimensional (2D) patterns, rectangular or oblique. These patterns resonate with the forcing by locking their wave-vector component in the forcing direction to half the forcing wave number. The range of this type of 2:1 resonance overlaps with the 1:1 resonance tongue of stripe patterns. Using a multiple-scale analysis in the overlap region we show that the 2D patterns can destabilize the 1:1 resonant stripes even at exact resonance. This result sheds new light on the use of spatial periodic forcing for controlling patterns.

Resonance patterns in spatially forced Rayleigh–Bénard convection

“…For slightly larger ε, theory predicts that the amplitude of one of these modes dominates the other and an oblique roll pattern emerges. The same behaviour was also found in simulations of the parametrically forced Swift-Hohenberg equation by Manor, Hagberg & Meron (2008) and Mau et al (2013). Possible reasons for the discrepancy will be discussed in § 4.…”

Spatial forcing of pattern-forming systems that lack inversion symmetry

“…(14). Whether the 3 : 1 resonance is a good representative of higher resonances in its basic properties-a subcritical pattern forming instability and multiplicity of n translationally symmetric, stable even-phase solutions-remains an open question.…”

“…We note that Eq. (14) reduces to known equations when the inversion symmetry is reintroduced (λ = 0) [13,14].…”

“…(1) can be obtained by studying constant solutions of the amplitude Eq. (14) or its equivalent form Eq. (17).…”

“…The SH equation can be viewed as the simplest model that captures a nonuniform stationary instability, i.e., an instability that renders a uniform state unstable and gives rise to a stationary periodic pattern. In a series of recent works we used the SH equation to study spatial parametric forcing as a means of stabilizing patterns, extending their existence range, controlling their wavenumbers and amplitudes, and creating new patterns [11][12][13][14].…”