Bifurcation analysis for periodic patterns appearing in nonlinear dynamical systems

研究了周期波型(周期驻定解或波型为周期的行波解)的结构。更准确地说，我们研究了这些解的分叉，二次分叉解的稳定性以及它们周围的动力学。首先，我们研究了一类摄动可积系统的周期解的性态，它最初是一个描述倾斜平面上液体层上波动的物理问题。这些方法适用于更一般的非线性波动现象，如Swift-Hohenberg方程，它是热对流的一个简单模型。通过严格的数学规范形分析，有时被称为“弱非线性分析”，我们可以证明这些方程的非平凡稳定混合模解的存在性，这与调制波解相对应。在Swift-Hohenberg方程的情况下，通过数值模拟得到了脉冲的局域模式和混合模解。另一方面，我们研究了梯度/斜梯度系统中周期模式的稳定性。结果表明，激活剂-抑制剂系统和Swift-Hohenberg方程是斜梯度的，并且这些系统中的周期滚动图的动力学受埃克豪斯不稳定性和之字形不稳定性控制。

Structures of periodic patterns (periodic stationary solutions or traveling wave solution whose wave profile is periodic) are studied. More precisely, we study bifurcations of these solutions, stability of bifiucated solutions secondary bifurcation and the dynamics around them. First we study the behavior of periodic solutions to a perturbed integrable systems which is originally a physical problem that describe wave motion on a liquid layer over an inclined plane. Nest, these method turns out to be applicable to more general nonlinear wave phenomena, such as the Swift-Hohenberg equation which is a simple model of thermal convection. By the mathematical rigorous normal form analysis, sometimes referred to as "weak nonlinear analisys", we could show the existence of non-trivial stable mixed mode solutions to these equations and this corresponds to the modulated wave solutions. In the case of Swift-Hohenberg equation we found a localized patterns of pulses as well as the mixed mode solution by using the numerical simulation. And later we prove the existence of these solutions by numerical verification technique.On the other hand we study the stability of periodic patterns in the context of gradient/skew gradient systems. As a result, we could show that activator-inhibitor system and the Swift-Hohenberg equations are skew gradient and also the dynamics of periodic roll patterns in these system are controlled by the Eckhaus instability and Zigzag instability.