Forced patterns near a Turing-Hopf bifurcation

TL;DR

This paper investigates how weak periodic forcing near a Turing-Hopf bifurcation affects spatial patterns in reaction-diffusion systems, revealing quadratic scaling effects and potential for pattern enhancement or suppression.

Contribution

It provides a symmetry-based normal form analysis and perturbation approach to predict forcing effects on patterns, including new insights into enhancement phenomena not previously observed.

Findings

Weak forcing near Hopf frequency modulates Turing amplitude quadratically.

Detuned forcing produces the strongest pattern effects.

Numerical simulations confirm the theoretical predictions.

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 explain features of previous experiments on the CDIMA 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.