Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model.
Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model.
复制标题
空间非均匀 Swift-Hohenberg 模型中延迟引起的局部结构脱钉。
作者：
F. Tabbert;C. Schelte;M. Tlidi;S. Gurevich
我们报告的动力学的本地化结构的非均匀Swift-Hohenberg模型描述图案的形成在横向平面的光学腔。这个真实的序参量方程在接近与双稳性相关的二阶临界点处是有效的。光腔由非均匀空间高斯泵浦光照明，并受到延时反馈。高斯注入光束通过对局域结构施加吸引力打破了系统的平移对称性。我们发现，本地化的结构可以被钉扎到中心的不均匀性，抑制延迟诱导的漂移分叉，已报告在特定的情况下，注入是均匀的，假设一个连续波操作。在非均匀空间泵浦光的作用下，我们对局域解进行了稳定性分析，以确定由时滞反馈引起的不同的不稳定区域。特别是，我们预测的双臂螺旋的形成，以及振荡和脱钉动力学引起的相互作用的吸引的不均匀性和不稳定的时间延迟反馈。通过数值延拓技术研究了从振荡解到脱钉解的过渡。在分析上，我们使用序参数的方法来推导出一个正常形式的延迟诱导的Hopf分岔导致振荡的解决方案。此外，我们模型的相互作用的吸引的不均匀性和不稳定的时间延迟描述本地化的解决方案作为一个过阻尼粒子在一个潜在的不均匀性产生的。在这种情况下，时间延迟反馈充当驱动力。比较结果，从后一种方法与完整的Swift-Hohenberg模型，我们表明，该方法不仅提供了一个有益的描述的脱钉动力学，但也是数值准确的整个参数制度。
We report on the dynamics of localized structures in an inhomogeneous Swift-Hohenberg model describing pattern formation in the transverse plane of an optical cavity. This real order parameter equation is valid close to the second-order critical point associated with bistability. The optical cavity is illuminated by an inhomogeneous spatial Gaussian pumping beam and subjected to time-delayed feedback. The Gaussian injection beam breaks the translational symmetry of the system by exerting an attracting force on the localized structure. We show that the localized structure can be pinned to the center of the inhomogeneity, suppressing the delay-induced drift bifurcation that has been reported in the particular case where the injection is homogeneous, assuming a continuous wave operation. Under an inhomogeneous spatial pumping beam, we perform the stability analysis of localized solutions to identify different instability regimes induced by time-delayed feedback. In particular, we predict the formation of two-arm spirals, as well as oscillating and depinning dynamics caused by the interplay of an attracting inhomogeneity and destabilizing time-delayed feedback. The transition from oscillating to depinning solutions is investigated by means of numerical continuation techniques. Analytically, we use an order parameter approach to derive a normal form of the delay-induced Hopf bifurcation leading to an oscillating solution. Additionally we model the interplay of an attracting inhomogeneity and destabilizing time delay by describing the localized solution as an overdamped particle in a potential well generated by the inhomogeneity. In this case, the time-delayed feedback acts as a driving force. Comparing results from the later approach with the full Swift-Hohenberg model, we show that the approach not only provides an instructive description of the depinning dynamics, but also is numerically accurate throughout most of the parameter regime.