Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics

Abstract

Spatio-temporal pattern formation in natural systems presents rich nonlinear dynamics which lead to the emergence of periodic nonequilibrium structures. One of the most successful equations currently available for theoretically investigating the behavior of these structures is the Swift-Hohenberg (SH), which contains a bifurcation parameter (forcing) that controls the dynamics by changing the energy landscape of the system. Though a large part of the literature on pattern formation addresses uniformly forced systems, nonuniform forcings are also observed in several natural systems, for instance, in developmental biology and in soft matter applications. In these cases, an orientation effect due to a forcing gradient is a new factor playing a role in the development of patterns, particularly in the class of stripe patterns, which we investigate through the nonuniformly forced SH dynamics. The present work addresses the stability of stripes, and the competition between the orientation effect of the gradient and other bulk, boundary, and geometric effects taking part in the selection of the emerging patterns. A weakly nonlinear analysis shows that stripes tend to align with the gradient, and become unstable when perpendicular to the preferred direction. This analysis is complemented by a numerical work that accounts for other effects, confirming that stripes align, or even reorient from preexisting conditions. However, we observe that the orientation effect does not always prevail in face of competing effects. Other than the cubic SH equation (SH3), the quadratic-cubic (SH23) and cubic-quintic (SH35) equations are also studied. In the SH23 case, not only do forcing gradients lead to stripe orientation, but also interfere in the transition from hexagonal patterns to stripes.