几类时滞动力系统的分支分析
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摘要
时滞动力系统的动力学性质的研究是一个具有丰富实际背景与广泛应用的领域。在自然科学中，很多过程均可以用时滞系统加以描述与刻画，因此时滞系统动力学性质的研究是一个非常具有实际意义的课题。分支分析是了解动力系统动力学性质的重要途径之一，所谓分支是指当参数变化时流的拓扑结构产生质的改变。其主要包括局部分支，半局部分支和全局分支。Hopf分支是一种常见而重要的分支，它主要研究当参数变化时，系统平衡点的稳定性发生变化，从而在平衡点附近产生小振幅周期解的现象。所以Hopf分支分析也是证明周期解存在性的强有力方法。众所周知，时滞动力系统的相空间一般是无穷维的，其分支的研究既需要经典的动力系统理论，同时也涉及到拓扑、代数、泛函等其它方向的数学知识，也就是说它是一个交叉性很强的领域。
本文主要研究几类具有较强实际背景的时滞动力系统的Hopf分支问题，主要工作如下：
1.研究了具时滞的“食物有限”种群模型的动力学性质。
首先考虑空间齐性的情形。通过分析相应特征方程根的分布情况，得到了关于系统正平衡点的稳定性和Hopf分支存在性的定理。进而运用吴建宏的全局Hopf分支定理，给出了该系统存在全局Hopf分支的条件。其次考虑空间非齐性（带有扩散项）的情形，确切地说，一个具Dirichlet边值的偏泛函微分方程,基于相平面分析思想确定了正稳态解分支的存在性，进而证明了该系统在正稳态解处Hopf分支的存在性，并得到了确定Hopf分支周期解稳定性和分支方向的公式。最后讨论了扩散项对稳定性的影响。
2.研究了一类造血干细胞模型的稳定性和Hopf分支问题。
运用Beretta和Kuang的几何准则，我们首先通过研究一个系数依赖时滞的指数多项式方程根的分布情况，发现系统随时滞变化而出现稳定开关及Hopf分支现象。进而，利用规范型理论和中心流形定理，给出了确定Hopf分支方向和分支周期解稳定性的方法。最后，我们讨论出由Hopf分支产生的非常数周期解在参数于较大范围内变化时仍然存在。
3.研究了具多时滞耦合的Mackey-Glass模型的动力学性质。
由于多时滞的引入，从而特征方程根的分布变得复杂。首先我们通过确定单个时滞的稳定区间，进而讨论其他时滞对特征方程根的分布影响，得到了系统的稳定性随时滞变化而变化情况，并发现了系统在不动点处由于Hopf分支产生而发生“失稳”现象。最后我们给出判断Hopf分支的分支方向和分支周期解稳定性的公式。
4.研究了某个遗传调节系统的动力学性质。
选取τ作为分支参数。首先得到具有现实意义的正平衡点的存在性和唯一性。进而讨论出随着参数变化，该不动点的稳定性情况以及给出于此不动点处发生Hopf分支的充分条件，最后讨论了Hopf分支的属性。
5.研究了位于两个相同区域的单种群年龄结构模型。
由于系统在正齐性不动点处对应的特征方程是一个系数依赖于滞量的特征方程，而具有此类特征方程的系统常常表现出丰富的动力学行为。我们引入Beretta和Kuang的几何准则对其进行讨论，从而得到不动点存在稳定性开关和产生Hopf分支的现象。最后利用Hassard和Kazarinoff等人的理论，导出了关于分支方向、分支周期解稳定性及周期解的振幅和周期变化情况的计算公式。
Delayed dynamical systems have attracted much attention in the past few decadessince many process in natural science can be described by such systems. So it’s of greatsignificance to study the dynamics of delayed dynamical systems. Bifurcation analysis isan important and effective way to obtain dynamics of delayed systems. By bifurcation,we mean that the topological structure of a flow with parameters is changed as parame-ters is cross a critical value. Generally speaking, bifurcations consists of local, semi-localand global bifurcations. Hopf bifurcation is a common and important local bifurcation.It studies the change of stability of equilibria and the consequent occurrence of periodicsolutions with small amplitude near equilibria. Therefore, Hopf bifurcation is an effectiveway to find periodic solutions. As we know, the phase space of delayed systems is usuallyof infinite dimension, and the bifurcation analysis for such systems needs not only classi-cal theory of dynamical systems, but also knowledge of other mathematics branches, suchas topology, algebra and functional analysis. In other words, delayed dynamical systemsis an interdiscipline. In this thesis, we mainly consider several kinds of delayed dynamicalsystems with strongly practical background. The organization is as follows:
1. Dynamics of a food-limited population model.
For the spatially homogeneous case, we first establish a theorem on the stabilityof the positive equilibrium and the existence of Hopf bifurcation via analyzing the dis-tribution of roots to the corresponding characteristic equation. Then we get a sufficientcondition for global Hopf bifurcation to exist by Wu’s global Hopf bifurcation theorem.For the spatially inhomogeneous case (more precisely, a partial functional differentialequation with Dirichlet boundary condition), we obtain the existence for bifurcation ofpositive steady states employed the idea of phase plane analysis, and further prove theexistence of Hopf bifurcation at positive steady states as well as the formula for deter-mining the stability of bifurcated periodic solutions and the direction of Hopf bifurcation.Finally, we present a discussion on the effect of diffusion on stability.
2. Hopf bifurcation of a haematopoietic stem cell model with a single delay.
We first study the distribution of roots to the characteristic equation, which is a expo- nential polynomial equation with delay-dependent coefficients. Consequently, we obtainthe stability threshold and the existence of Hopf bifurcation. Then we derive an explicitformula for determining the stability and direction of periodic solutions bifurcated fromHopf bifurcations using the normal form theory combined with center manifold argu-ments. The persistence of periodic solution induced by Hopf bifurcation as parametersvary in a large range is confirmed.
3. Dynamics of a coupled Mackey-Glass electronic circuits model with multipledelays.
The distribution of roots to the characteristic equation is complicated because ofmultiple delays. To analyze it, we first figure out the stability interval of one delay, andthen discuss the effects of other delays. Consequently, we find a sequence of Hopf bifur-cation points and derive an explicit algorithm for determining the direction and stability ofbifurcated periodic solutions by appealing to the normal form theory and center manifoldargument.
4. Hopf bifurcation in a genetic regulatory model with delay.
The existence and uniqueness of positive equilibrium are established. Choosing timedelay τ as the parameter, we obtain a sufficient condition for Hopf bifurcation to happen.
5. Dynamics of an age-structured population model of a single species living in twoidentical patches.
This is a delayed system arose in population biology. It has rich dynamical behavior,which is still largely opened. We here perform some bifurcation to this model. Someproperties of Hopf bifurcation are obtained by the normal form theory coupled with thecenter manifold theorem.
本文主要研究几类具有较强实际背景的时滞动力系统的Hopf分支问题，主要工作如下：
1.研究了具时滞的“食物有限”种群模型的动力学性质。
首先考虑空间齐性的情形。通过分析相应特征方程根的分布情况，得到了关于系统正平衡点的稳定性和Hopf分支存在性的定理。进而运用吴建宏的全局Hopf分支定理，给出了该系统存在全局Hopf分支的条件。其次考虑空间非齐性（带有扩散项）的情形，确切地说，一个具Dirichlet边值的偏泛函微分方程,基于相平面分析思想确定了正稳态解分支的存在性，进而证明了该系统在正稳态解处Hopf分支的存在性，并得到了确定Hopf分支周期解稳定性和分支方向的公式。最后讨论了扩散项对稳定性的影响。
2.研究了一类造血干细胞模型的稳定性和Hopf分支问题。
运用Beretta和Kuang的几何准则，我们首先通过研究一个系数依赖时滞的指数多项式方程根的分布情况，发现系统随时滞变化而出现稳定开关及Hopf分支现象。进而，利用规范型理论和中心流形定理，给出了确定Hopf分支方向和分支周期解稳定性的方法。最后，我们讨论出由Hopf分支产生的非常数周期解在参数于较大范围内变化时仍然存在。
3.研究了具多时滞耦合的Mackey-Glass模型的动力学性质。
由于多时滞的引入，从而特征方程根的分布变得复杂。首先我们通过确定单个时滞的稳定区间，进而讨论其他时滞对特征方程根的分布影响，得到了系统的稳定性随时滞变化而变化情况，并发现了系统在不动点处由于Hopf分支产生而发生“失稳”现象。最后我们给出判断Hopf分支的分支方向和分支周期解稳定性的公式。
4.研究了某个遗传调节系统的动力学性质。
选取τ作为分支参数。首先得到具有现实意义的正平衡点的存在性和唯一性。进而讨论出随着参数变化，该不动点的稳定性情况以及给出于此不动点处发生Hopf分支的充分条件，最后讨论了Hopf分支的属性。
5.研究了位于两个相同区域的单种群年龄结构模型。
由于系统在正齐性不动点处对应的特征方程是一个系数依赖于滞量的特征方程，而具有此类特征方程的系统常常表现出丰富的动力学行为。我们引入Beretta和Kuang的几何准则对其进行讨论，从而得到不动点存在稳定性开关和产生Hopf分支的现象。最后利用Hassard和Kazarinoff等人的理论，导出了关于分支方向、分支周期解稳定性及周期解的振幅和周期变化情况的计算公式。
Delayed dynamical systems have attracted much attention in the past few decadessince many process in natural science can be described by such systems. So it’s of greatsignificance to study the dynamics of delayed dynamical systems. Bifurcation analysis isan important and effective way to obtain dynamics of delayed systems. By bifurcation,we mean that the topological structure of a flow with parameters is changed as parame-ters is cross a critical value. Generally speaking, bifurcations consists of local, semi-localand global bifurcations. Hopf bifurcation is a common and important local bifurcation.It studies the change of stability of equilibria and the consequent occurrence of periodicsolutions with small amplitude near equilibria. Therefore, Hopf bifurcation is an effectiveway to find periodic solutions. As we know, the phase space of delayed systems is usuallyof infinite dimension, and the bifurcation analysis for such systems needs not only classi-cal theory of dynamical systems, but also knowledge of other mathematics branches, suchas topology, algebra and functional analysis. In other words, delayed dynamical systemsis an interdiscipline. In this thesis, we mainly consider several kinds of delayed dynamicalsystems with strongly practical background. The organization is as follows:
1. Dynamics of a food-limited population model.
For the spatially homogeneous case, we first establish a theorem on the stabilityof the positive equilibrium and the existence of Hopf bifurcation via analyzing the dis-tribution of roots to the corresponding characteristic equation. Then we get a sufficientcondition for global Hopf bifurcation to exist by Wu’s global Hopf bifurcation theorem.For the spatially inhomogeneous case (more precisely, a partial functional differentialequation with Dirichlet boundary condition), we obtain the existence for bifurcation ofpositive steady states employed the idea of phase plane analysis, and further prove theexistence of Hopf bifurcation at positive steady states as well as the formula for deter-mining the stability of bifurcated periodic solutions and the direction of Hopf bifurcation.Finally, we present a discussion on the effect of diffusion on stability.
2. Hopf bifurcation of a haematopoietic stem cell model with a single delay.
We first study the distribution of roots to the characteristic equation, which is a expo- nential polynomial equation with delay-dependent coefficients. Consequently, we obtainthe stability threshold and the existence of Hopf bifurcation. Then we derive an explicitformula for determining the stability and direction of periodic solutions bifurcated fromHopf bifurcations using the normal form theory combined with center manifold argu-ments. The persistence of periodic solution induced by Hopf bifurcation as parametersvary in a large range is confirmed.
3. Dynamics of a coupled Mackey-Glass electronic circuits model with multipledelays.
The distribution of roots to the characteristic equation is complicated because ofmultiple delays. To analyze it, we first figure out the stability interval of one delay, andthen discuss the effects of other delays. Consequently, we find a sequence of Hopf bifur-cation points and derive an explicit algorithm for determining the direction and stability ofbifurcated periodic solutions by appealing to the normal form theory and center manifoldargument.
4. Hopf bifurcation in a genetic regulatory model with delay.
The existence and uniqueness of positive equilibrium are established. Choosing timedelay τ as the parameter, we obtain a sufficient condition for Hopf bifurcation to happen.
5. Dynamics of an age-structured population model of a single species living in twoidentical patches.
This is a delayed system arose in population biology. It has rich dynamical behavior,which is still largely opened. We here perform some bifurcation to this model. Someproperties of Hopf bifurcation are obtained by the normal form theory coupled with thecenter manifold theorem.
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