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Bifurcation analysis for periodic patterns appearing in nonlinear dynamical systems

非线性动力系统中出现的周期模式的分岔分析

基本信息

批准号：
12440026
负责人：
OGAWA Toshiyuki
金额：
$ 3.01万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440026/
关键词：
periodic traveling wave mode interaction modulated wave Eckhaus instability gradient skew gradient system global bifurcation numerical verification

项目摘要

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.

研究周期性模式的结构（周期性平稳解或波廓为周期性的行波解）。更准确地说，我们研究这些解的分叉、分叉解二次分叉的稳定性以及它们周围的动力学。首先，我们研究扰动可积系统的周期解的行为，这最初是一个描述倾斜平面上液体层上的波动的物理问题。事实证明，这些方法适用于更一般的非线性波现象，例如 Swift-Hohenberg 方程，它是一个简单的热对流模型。通过数学严格的范式分析（有时称为“弱非线性分析”），我们可以证明这些方程存在非平凡的稳定混合模式解，这对应于调制波解。对于 Swift-Hohenberg 方程，我们通过数值模拟找到了脉冲的局域模式以及混合模式解。随后我们通过数值验证技术证明了这些解的存在性。另一方面，我们研究了梯度/斜梯度系统中周期性模式的稳定性。因此，我们可以证明激活剂-抑制剂系统和 Swift-Hohenberg 方程是斜梯度，并且这些系统中周期性滚动模式的动力学受 Eckhaus 不稳定性和 Zigzag 不稳定性控制。