数学与统计学院知识库

近年来, 分支问题的研究已成为动力系统中的重要研究课题之一, 并在力学、物理学、化学、生物学、生态学、控制、数值计算、工程技术以及经济学和社会科学中得到广泛的应用. 本文主要考虑由下面偏泛函微分方程描述的二维Lotka-Volterra系统的周期解和分支问题.

在没有扩散影响时, 以时滞τ作为分支参数, 通过分析系统在正平衡点处线性化系统的特征方程, 获得了正平衡点渐近稳定的充分条件以及在它周围分支出周期解的条件. 另外, 通过使用时滞微分方程的规范形理论和中心流形约化, 获得了确定周期解Hopf分支的方向、分支周期解稳定性和周期等性质的显式算法. 通过使用一个拓扑的全局存在性结果, 还给出了分支周期解的全局存在性. 结果表明在时滞的第二个临界值以后, 系统总是存在一个非常数周期解.

当具有扩散影响和Neumann边界条件时, 通过在空间齐次正平衡点处线性化系统和分析相应的特征方程, 研究了正平衡点的渐近稳定性和系统存在Hopf分支的条件. 另外, 通过使用偏泛函微分方程的规范形理论和中心流形约化, 获得了决定空间齐次周期解Hopf分支的方向、分支周期解的稳定性和周期等的显式公式.通过这一显式算法, 给出了分支周期解在中心流形上轨道渐近稳定和不稳定的充分条件. 同时, 通过一些数值模拟验证了文中所得结论的正确性.

在具有扩散影响和Dirichlet边界条件时, 首先利用隐含数定理获得了系统存在正平衡解的条件和正平衡解的渐近表达式. 通过在系统唯一的正平衡解处线性化系统和分析相应的特征方程, 发现在时滞等于零时该正平衡解是渐近稳定的, 当时滞逐渐增大到某个临界值时该正平衡解将失去稳定性而且在该平衡解处分支出空间非齐次的周期解. 随着时滞的进一步增大, 在另一系列临界值处, 虽然正平衡解一直处于不稳定状态, 但系统仍能够在正平衡解处分支出空间非齐次的周期解. 对于Hopf分支的方向和分支周期解的性质, 利用偏泛函微分方程的规范形理论和中心流形约化给出了描述这些性质的显式算法. 根据这一算法, 发现在时滞的第一个临界值处通过Hopf分支产生的周期解在中心流形上是轨道渐近稳定的, 而在其它临界值处通过Hopf分支产生的周期解在中心流形上是不稳定的. 同样对于所得理论结果给予了数值验证.

In recent years, the study of bifurcation problems has been one of important subjects in dynamical systems and has been applied extensively in many fields such as mechanics, physics, chemistry, biology, ecology, control, numerical calculations,engineering technology and economics and social sciences etc. In this thesis we consider mainly periodic solutions and bifurcation problems of the two-dimensional
Lotka-Volterra system described by the following partial functional differential equations.

In the absence of diffusion effects, by regarding the delay τ as the bifurcation parameter and analyzing the characteristic equations of the linearized system of the
original system at the positive equilibrium, the sufficient conditions ensuring that the positive equilibrium is asymptotically stable and the conditions guaranteeing that the system can bifurcate periodic solutions from the positive equilibrium are established. In addition, by applying the normal form theory and the center manifold reduction for delayed differential equations, an explicit algorithm determining the direction of Hopf bifurcation, the stability and period of bifurcated periodic solutions is given. Finally, by using a topological global existence result, we give the global existence of bifurcated periodic solutions and it is found that the system exists always a nonconstant periodic solution after the second critical value of the delay.

In the presence of diffusion effects and Neumann boundary conditions, by linearizing the system at the spatial homogeneous positive equilibrium and analyzing the corresponding characteristic equation, the stability of positive equilibrium is studied and the conditions under which the system undergoes a Hopf bifurcation of periodic solutions are obtained. Furthermore, by using the normal form theory and the center manifold reduction for partial functional differential equations, an explicit algorithm determining the direction of Hopf bifurcation of spatial homogeneous periodic solutions, the stability and period of bifurcated periodic solutions is given. In view of this algorithm, the sufficient conditions ensuring that the bi-
furcated periodic solutions are orbitally asymptotically stable and unstable on the center manifold are obtained. Meanwhile, to verify the theoretical conclusions obtained in this part, some numerical simulations are also included.

When there are diffusion effects and Dirichlet boundary conditions, we firstly obtain the conditions such that the system exists positive equilibrium solutions and give the asymptotic expression of these positive equilibrium solutions. Secondly, by linearizing the system at the unique positive equilibrium solution and analyzing the associated characteristic equation, it is found that the positive equilibrium solution is asymptotically stable when the delay equals to zero and when it is increased to certain critical value, the positive equilibrium solution will loss the stability and a spatially heterohomogeneous periodic solution will bifurcate from this positive equilibrium solution. With the further increase of delay, at another sequence of crit-
ical values of delay, although the positive equilibrium solution is always unstable, a spatially heterohomogeneous periodic solution can also bifurcate from this positive equilibrium solution. For the direction of Hopf bifurcation and the stability and period of bifurcated periodic solutions, we give an explicit algorithm for determining these properties by using the normal form theory and the center manifold reduction for partial functional differential equations. According to this algorithm, it is shown that the bifurcated periodic solutions through Hopf bifurcations at the first critical value of delay are orbitally asymptotically stable on the center manifold and the bifurcated periodic solutions through Hopf bifurcations at other critical values of delay are unstable. Some numerical simulations are also given to verify our theoretical results.