【作者】 朱刚；

【导师】 魏俊杰；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2013， 博士

【摘要】 时滞微分方程因其对客观现象的描述和刻画比常微分方程更加准确和合理而得到广泛关注和研究，并被应用到众多领域。而分支理论研究的是结构不稳定的系统随参数变化时，当参数经过某些临界值时解的拓扑结构发生变化，分支现象也普遍存在于现实生活中。因此我们在本文中研究时滞微分方程的分支问题，主要是研究了不动点分支和Hopf分支。不动点分支和Hopf分支都是比较常见的分支现象。不动点分支是指当参数经过临界值时系统的平衡点个数或者是稳定性发生变化。Hopf分支是指当参数经过临界值时系统平衡点的稳定性发生反转，并在平衡点附近产生小振幅周期解。通常情况下，不动点分支和Hopf分支的产生总是伴随着系统平衡点稳定性的变化，因此在研究分支问题时我们首先讨论系统平衡点的稳定性，然后研究具体的分支性质，包括不动点分支的类型、Hopf分支的分支方向和从Hopf分支值分支出的周期解的稳定性等。本文研究四类具有实际背景的时滞微分方程，分别是具有一般形式非线性项的单向耦合系统、时滞Rosenzweig-MacArthur型的带有食饵移入项的捕食被捕食模型、带有Mach-Zehnder光电调制器的光电反馈环路系统以及耦合Lang-Kobayashi速率方程。通过分析系统的线性化方程的特征根的分布并结合极限方程的渐近半流的方法，讨论了平衡点的局部稳定性以及产生不动点分支和Hopf分支的条件，并利用Lyapunov泛函和Lassel不变集原理讨论了平衡点的全局稳定性。根据Hopf分支定理和重合度的延展定理证明了周期解的存在性。此外利用Faria和Hassard的规范型方法分别计算了不动点分支和Hopf分支在中心流形上的规范型，进而讨论了分支性质，并根据吴建宏的全局Hopf分支定理证明了分支周期解的全局存在性。更多还原

【Abstract】 Delayed diferential equations are widely used in many fields since it can describethe natural phenomena much better than ordinary diferential equations. Meanwhile, bi-furcation means that for a system with unstable structure, when a certain parameter varies,the topological structure of the solutions changes, which is very common in our dailylives. So here we study the bifurcation of delayed diferential equations.Fixed point bifurcation means the number or the stability of the equilibria change asthe parameter passes through critical value. Hopf bifurcation means that a small ampli-tude periodic solution comes out with the changes of stability for equilibrium point as acertain parameter varies and passes through a critical value. Since the appearance of theHopf bifurcation always accompanies with the change of the stability of the equilibria,so it is inevitable to discuss the stability of the equilibria of the equation when we studybifurcations firstly, and then we study the properties of the bifurcations including the pat-terns of the fixed point bifurcation, the direction of Hopf bifurcation and the stability ofthe bifurcating periodic solutions.We mainly study four practical problems, including a unidirectionally coupled sys-tem with a nonlinear function of general form, delayed Rosenweig-MacArthur modelwith constant rate prey immigration, a delayed-feedback optical loop system with Mach-Zehnder modulator and coupled Lang-Kobayashi rate equation. By analysing the dis-tribution of the characteristic roots of the linearized equation and using the method ofasymptotic semi-flow of limit equation, we study the local stability of the equilibria andget the conditions under which fixed point bifurcation and Hopf bifurcation occurs. Then,using Lyapunov function and Lassel invariant set principle we investigate the global sta-bility of the equilibria. Basing on the Hopf bifurcation theorem and the coincide degreecontinuation theorem, we prove the existence of the periodic solutions. Furthermore, wecalculate the normal form of the fixed point bifurcation and the Hopf bifurcation on thecenter manifold by the method from Faria and Hassard. Finally, we verify the globalexistence of the periodic solutions by global Hopf bifurcation theorem of Wu.更多还原