分数阶动力系统的二个重要问题

分数阶动力学是经典动力学和分数阶微积分结合的必然产物，它是刻画记忆过程和遗传特性的有效工具。现已发现，分数阶动力学在粘弹性力学、流变学等方面有重要应用。本项目拟研究分数阶动力学的二个重要问题：分数阶微分系统的静态分岔和混沌同步。具体内容如下：(1)建立分数阶微分系统的中心流形定理，将经典的李雅普诺夫-施密特约化方法推广到分数阶微分系统。运用分数阶中心流形定理和推广的李雅普诺夫-施密特约化方法，研究单参数高维分数阶微分系统的跨临界、折叠点、音叉、Hopf等典型静态分岔。建立一套比较完整的静态分岔理论的框架体系。(2)发展分数阶微分系统的稳定性理论，将之应用到分数阶微分系统的混沌同步，重点研究分数阶混沌系统的广义同步和相同步，建立一套比较完整的分数阶微分系统广义同步和相同步的稳定性分析方法。通过本项目的研究，推动分数阶动力学理论的深入发展，为现实应用奠定坚实的理论基础和提供技术支持。

Fractional dynamics is the necessary combination of classical dynamics and fractional calculus.It plays an efficient role in characterizing memory and heredity. It has been found that fractional dynamics can be successfully applied to viscoelastic mechanics and rheology.This program aims to study two important topics, including static bifurcation and chaos synchronization of fractional differential system. For more details, the research contents are scheduled as follows. (1) The center manifold theorem for fractional system is proposed and planed to establish. And the Lyapunov-Schmidt reduction method is generalized to the fractional case. Then by using these two mehtods, the typical static bifurcations of higher-dimensional fractional differential system with sigle parameter are analyzed, such as transcritical bifurcation, fold bifurcation, pitchfork bifurcation, and Hopf bifurcation. Furthermore, a basic theoretical framework for static bifurcations is built. (2) On the other hand, the present project also devotes to developing the stability theory for fractional differential system and to applying the derived stability result to chaos synchronization in coupled fractional differential systems, where the generalized synchronization and phase synchronization are highlighted.The fairly complete analytical method for generalized synchronization and phase synchronization is to provide. Overall, through these studies, the applicants want to develop the theory of fractional dynamics in order to provide solid theoretical fundament and technical support for real applications.