几类分数阶微分动力系统的稳定性问题研究

The fractional differential dynamical systems are being extensively applied to many fields of science and engineering since fractional calculus provides a powerful tool for the description of memory and hereditary effects in various substances. So, the qualitative theorem analysis of fractional differential dynamical systems is a meaningful research topic. Based on the applicant's research work on stability of linear fractional differential systems, this project mainly aims at nonlinear fractional differential dynamical systems with Caputo and Riemann-Liouville derivative. By employing the equivalent system, the fixed point theorem, the comparison principle and fractional Lyapunov direct method, we will study the following research contents: 1) The stability of Caputo type nonlinear fractional differential dynamical system with the same order lying in (1,2); 2) The stability of Riemann-Liouville type nonlinear fractional differential dynamical system with the multiple-rational orders lying in (0,2); 3) The stability of Riemann-Liouville type nonlinear fractional differential dynamical system with the same order lying in (0,2). Results of this project can not only enrich the qualitative theory of fractional differential dynamical systems, but also provide the theoretical foundation for many actual problems (such as the mechanical property of complex viscoelastic material).

分数阶微积分可以很好地描述物质的记忆和遗传性质，使得分数阶微分动力系统越来越广泛地应用于科学和工程的众多领域。因此，分数阶微分动力系统的定性理论分析是一个非常有意义的研究课题。基于申请人已完成的分数阶线性微分系统的稳定性研究工作，本课题主要针对Caputo型和Riemann-Liouville型非线性分数阶微分动力系统，利用等价系统、不动点定理、比较原理、分数阶Lyapunov函数法等研究以下内容：1）阶位于1到2之间任意实数的Caputo型非线性同分数阶微分动力系统的稳定性；2）阶位于0到2之间有理数的Riemann-Liouville型多分数阶微分动力系统的稳定性；3）阶位于0到2之间任意实数的Riemann-Liouville型同分数阶微分动力系统的稳定性。本项目的研究不仅可以丰富分数阶微分动力系统的定性理论，还可以为大量实际问题（如复杂黏弹性材料的力学特性）提供有力的理论支撑。

分数阶微积分可以很好地描述物质的记忆和遗传性质，使得分数阶微分系统越来越广泛地应用于科学和工程的众多领域。因此，分数阶微分系统的定性理论分析是一个非常有意义的研究课题。基于申请人已完成的分数阶线性微分系统的稳定性研究工作，本项目主要研究了几类分数阶微分系统的动力学行为。具体内容为：1）推导了Caputo型变系数时滞分数阶微分系统的有限时间稳定性的充分条件；2）分析了一个分数阶随机禽流感传染病模型的持久性与灭绝性，并证明遍历的平稳分布的存在性，同时在数值上进行了验证；3）利用特征根分布给出了几类分数阶（时滞）反应扩散捕食模型的Hopf分支的存在性，通过规范型理论和中心流形定理给出决定分支周期解的稳定性和Hopf分支方向的算法，且借助于数值模拟来例证所得的理论结果。