含有非线性不可公度分数阶节点的复杂网络的全局同步控制研究

Synchronization of complex networks with fractional-order nodes is on the cutting edge of the research fields. However, this research mainly focuses on the situation where the complex networks are composed of the commensurate nodes. There are still no any existing results about the global synchronization of complex networks with nonlinear incommensurate fractional-order nodes. This project concentrates on the global synchronization of complex networks with nonlinear incommensurate fractional-order nodes, and is aimed to complete the following work: (1) Propose a global synchronization control strategy for complex networks consisting of nonlinear incommensurate fractional-order nodes with the same structures, when all the nodes are controllable; (2) Propose a global pinning synchronization control strategy for complex networks consisting of nonlinear incommensurate fractional-order nodes with the same structures, when part of the nodes are controllable; (3) Propose a global synchronization control strategy for complex networks consisting of nonlinear incommensurate fractional-order nodes with the different structures, when all the nodes are controllable; (4) Propose a global pinning synchronization control strategy for complex networks consisting of nonlinear incommensurate fractional-order nodes with the different structures, when part of the nodes are controllable. Meanwhile, the project is also aimed to develop new Lyapunov stability theory for nonlinear incommensurate fractional-order systems. All the theoretical results will be validated by a simulation platform, which is the byproduct of this project.

含有分数阶节点的复杂网络同步问题是前沿研究课题，目前的研究主要集中在网络中含有可公度分数阶节点的情况，而对含有非线性不可公度分数阶节点的复杂网络的全局同步问题至今尚未见有相关研究。本项目致力于研究由非线性不可公度分数阶节点组成的复杂网络的全局同步问题，拟完成如下内容：（1）对于有相同节点的非线性不可公度分数阶复杂网络，提出节点全部可控时的全局同步控制策略；（2）对于有相同节点的非线性不可公度分数阶复杂网络，提出节点部分可控时的全局牵制同步控制策略；（3）对于有不同节点结构的非线性不可公度分数阶复杂网络，提出节点全部可控时的全局同步控制策略；（4）对于有不同节点结构的非线性不可公度分数阶复杂网络，提出节点部分可控时的全局牵制同步控制策略。此外，在解决上述问题的同时，本项目还拟建立关于非线性不可公度分数阶系统的Lyapunov稳定性理论。本项目也将构建仿真实验平台对理论成果加以验证。

本项目研究了不可公度分数阶非线性系统的稳定性判别准则，提出了针对不可公度非线性分数阶系统的Lyapunov函数的一种适用形式。对于分数阶系统中的隐藏吸引子和共存吸引子的存在情况进行了深入研究，得到了一些非常有意义的结论，特别是，我们发现了一个新的无平衡点的分数阶超混沌系统，该系统具有非常丰富的动力学特性，不仅具有多个吸引子，还有隐藏吸引子存在。利用MultiSim软件实现了这个新型分数阶系统的电路设计，为以后的工程应用提供了准备。