分数阶随机系统的非线性动力学行为与控制研究

The fractional calculus is a powerful tool to describle physical systems that have long-term memory and long-range spatial interactions. The fractional-order derivative model can overcome the disadvantages that the experimental findings is not consistent with the result of integer derivative one. The resarch of fractional calculus has been attracted great attention. And the study of nonlinear sciences can be widely applied into many areas,such as atmospheric sicence, physics, transportation and so on. So combined the theory of nonlinear dynamics with ramdom dynamical results, it is very interesting to investigate fractional order dynamics system with stochastic characteristics. In this project, we put forward to some new research idea which can be divided into three parts. The first part is how to find the numerical or approximate solution to some given kinds fractional-order dynamics system with stochastic characteristics. The second one is to analyze the nonlinear dynamical behaviors of the proposed system, such as stochastic stability, bifurcation, breaking bifurcation, period-doubling bifurcation and so forth, and especially to consider whether there exists chaos and find the evolution procedure for some given parameters. The expected investigation will enrich the theory sytem of nonlinear sciences. To improve some characteristics and properties of the given kinds of fractioal order sytem with random parameters. The third one in this project is to inquiry some suitable control method or rule which can realize some requirement in reality. The expected results of this project will provide a theory guarantee that the fractional order model with stochastic characteristics can be widely applied in the fields of atmospheric sciences, secure communication, mechanical vibration, transportation, medical image processing and so on.

分数阶导数具有全局相关性，且能体现系统函数发展的历史依赖过程，分数阶导数模型弥补了经典整数阶模型理论与实验结果吻合不好的严重缺点。因此结合非线性动力学理论和随机动力学理论研究分数阶随机系统的动力学行为具有重要的理论意义和实际应用价值。本项目综合多学科领域知识，引入基于模型求解-动力学行为分析-动力学行为控制的一体化研究思路，利用分数阶微积分理论和随机动力学理论探讨一类分数阶随机系统的精确平稳解的存在性以及系统的随机稳定性；进一步利用非线性分岔理论分析系统依赖于随机参数变化可能产生的随机分岔、破裂分岔、倍周期分岔等行为，特别考虑此类系统中可能产生的混沌动力学行为以及通往混沌的演变过程，这将进一步丰富非线性科学的理论基础。同时，本项目还将结合现代控制理论，提出能够实现改善系统相应品质的控制策略，为分数阶非线性理论在保密通信、机械振动和交通运输等领域的广泛应用提供理论基础支持。

分数阶导数具有全局相关性，且能体现系统函数发展的历史依赖过程，分数阶导数模型弥补了经典整数阶模型理论与实验结果吻合不好的严重缺点。因此结合非线性动力学理论和随机动力学理论研究分数阶随机系统的动力学行为具有重要的理论意义和实际应用价值。本项目综合多学科领域知识，利用分数阶微积分理论和随机动力学理论探讨一类分数阶随机系统的动力学行为；本项目首先从分数阶微积分的基本理论入手，对传统的分数阶Lyapunov方法进行了推广，得到了较一般的分数阶系统的稳定性的判别方法，并以分数阶神经网络系统作为研究对象，对分数阶Hodgkin-Huxley神经元模型相应的动力学性质进行了分析，同时也研究了时滞分数阶Hopfield神经网络模型的稳定性和全局一致稳定性。最后根据取得的结果，给出了分析分数阶神经网络的线性矩阵不等式的条件。并进一步给出分数阶神经网络系统的可控与同步的条件，实现系统不同类型的同步与控制。