Abstract
We study time-periodic forcing of spatially extended patterns near a Turing-Hopf bifurcation point. A symmetry-based normal form analysis yields several predictions, including that (i) weak forcing near the intrinsic Hopf frequency enhances or suppresses the Turing amplitude by an amount that scales quadratically with the forcing strength, and (ii) the strongest effect is seen for forcing that is detuned from the Hopf frequency. To apply our results to specific models, we perform a perturbation analysis on general two-component reaction-diffusion systems, which reveals whether the forcing suppresses or enhances the spatial pattern. For the suppressing case, our results are consistent with features of previous experiments on the chlorine dioxide-iodine-malonic acid chemical reaction. However, we also find examples of the enhancing case, which has not yet been observed in experiment. Numerical simulations verify the predicted dependence on the forcing parameters.
Highlights
Turing patterns, originally conjectured as the basis for biological morphogenesis1͔, arise in such diverse fields as ecology, materials science, and astrophysics2–4͔
A symmetry-based normal form analysis yields several predictions, including thatiweak forcing near the intrinsic Hopf frequency enhances or suppresses the Turing amplitude by an amount that scales quadratically with the forcing strength, andiithe strongest effect is seen for forcing that is detuned from the Hopf frequency
Our results are consistent with features of previous experiments on the chlorine dioxide-iodine-malonic acid chemical reaction
Summary
Originally conjectured as the basis for biological morphogenesis1͔, arise in such diverse fields as ecology, materials science, and astrophysics2–4͔. Many systems that form Turing-type patterns can display a Hopf bifurcation to time-periodic solutions, and the interaction of these instabilities has been implicated in a variety of natural phenomena. Temporal, and spatiotemporal forcing have been shown to induce a transition between patterns29,30͔, to introduce new localized31͔ and complex32͔ patterns, or to suppress patterns33͔ Turing systems that both contain a Hopf instability and are susceptible to external forcing pose an intriguing challenge. In the forced normal form, of special interest is the coefficient of the term that couples the Hopf mode to the Turing mode, whose sign dictates whether the forcing enhances or suppresses Turing patterns. The result of the calculation applies to specific two-component systems such as the Lengyel-Epstein model, the Brusselator, and so forth and determines, as a function of the reaction kinetics, the qualitative effect of the forcing. IV we verify some of our symmetry predictions via numerical simulations
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