Spatio-temporal pattern formation in complex systems presents rich nonlinear dynamics which leads to the emergence of periodic nonequilibrium structures. One of the most prominent equations for the theoretical and numerical study of the evolution of these textures is the Swift–Hohenberg (SH) equation, which presents a bifurcation parameter (forcing) that controls the dynamics by changing the energy landscape of the system, and has been largely employed in phase-field models. 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 forcing gradients 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 amplitude instability of stripe textures induced by forcing gradients, 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 suggests that stripes are stable with respect to small amplitude perturbations when aligned with the gradient, and become unstable to such perturbations when when aligned perpendicularly to the gradient. This analysis is vastly complemented by a numerical work that accounts for other effects, confirming that forcing gradients drive stripe alignment, or even reorient them from preexisting conditions. However, we observe that the orientation effect does not always prevail in the face of competing effects, whose hierarchy is suggested to depend on the magnitude of the forcing gradient. 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.

复杂系统中的时空模式形成呈现出丰富的非线性动力学，导致周期性非平衡结构的出现。对这些纹理的演化进行理论和数值研究的最突出的方程之一是 Swift-Hohenberg (SH) 方程，它提出了一个分岔参数（强迫），它通过改变系统的能量景观来控制动力学，并且已大量用于相场模型。虽然大部分关于模式形成的文献都针对均匀受迫系统，但在几个自然系统中也观察到了非均匀强迫，例如，在发育生物学和软物质应用中。在这些情况下，由强制梯度引起的方向效应是在模式发展中发挥作用的新因素，特别是在条纹图案类中，我们通过非均匀强迫的 SH 动力学进行研究。目前的工作解决了由强制梯度引起的条纹纹理的振幅不稳定性，以及梯度的方向效应与参与新兴图案选择的其他体积、边界和几何效应之间的竞争。弱非线性分析表明，条纹在与梯度对齐时相对于小幅度扰动是稳定的，而当与梯度垂直对齐时，对此类扰动变得不稳定。这种分析得到了解释其他影响的数值工作的极大补充，确认强制梯度驱动条纹对齐，甚至从预先存在的条件重新定向它们。然而，我们观察到定向效应在面对竞争效应时并不总是占优势，建议其等级取决于强迫梯度的大小。除了三次 SH 方程 (SH3)，还研究了二次-三次 (SH23) 和三次-五次 (SH35) 方程。在 SH23 情况下，强制梯度不仅会导致条纹取向，还会干扰从六边形图案到条纹的过渡。