【作者】 胡爱花；

【导师】 徐振源；

【作者基本信息】 江南大学 ， 轻工信息技术与工程， 2010， 博士

【摘要】 本文首先简要介绍了混沌与混沌同步;然后就混沌同步中的一些问题进行了研究,给出了理论分析。包括:噪声与混沌同步之间关系的研究、复杂网络的动力学同步研究以及(离散、非自治、多卷波)混沌系统的同步研究。研究工作概括如下:(1)就噪声与混沌同步的关系,主要研究了两方面的内容。一方面是利用具有一定强度的白噪声作为反馈增益,以此来达到混沌系统广义同步的目的。文中详细推导出了噪声耦合强度函数所需满足的条件;另一方面当混沌系统受到噪声干扰时,基于鲁棒控制方法,设计出反馈项,将混沌同步的误差限制在一个比较小的范围内。同时,进一步考虑了驱动系统和响应系统阶数相同和不相同的两种情况,理论分析给出了同步的充分条件;(2)从三个方面研究了复杂网络的动力学同步问题。首先,研究了复杂网络连接的Chen系统的同步化,所采用的方法不同于Lyapunov函数法和线性近似法。该方法不仅具有相当的普遍性,同时保证了同步化的收敛单调性。以拥有N个Chen系统作为节点的四种复杂网络(环形网络、星形双向耦合网络、小世界网络、全局耦合网络)为研究对象,理论推导出了Chen系统单调同步的参数条件,并对结果进行了比较;其次,利用脉冲牵引控制方法,控制复杂网络中的一部分节点,从而使得复杂网络中的所有节点都同步到了系统的平衡态。设计出的控制方案适用范围广,并且控制所花的代价比较小;最后,研究了复杂网络上动态混沌系统广义同步的存在性问题。以一个具体的复杂网络为研究对象,根据网络中节点的修正系统状态,将广义同步进行分类。文中严格证明了三种类型的广义同步存在和指数吸引的充分条件;(3)对于离散、非自治和多卷波混沌系统,研究内容有:首先,研究了在参数未知情形下离散混沌系统的同步,给出了控制器和参数向量的设计方法,理论证明了该方法的有效性;其次,由于非自治混沌系统的同步流形与时间相关,其广义同步发生的数学物理机制比较复杂。根据响应系统修正方程的状态,将广义同步分成了四类。文中给出了其中三类广义同步存在与指数吸引的条件,并且进一步研究了由于同步流形的不光滑所导致的H?lder连续性问题;最后,以一个具体的多卷波混沌系统为例,讨论了广义同步中多个稳定的混沌吸引子共存的问题。发现在响应系统中,可以存在多个混沌吸引子与驱动系统同步,同时,响应系统最终到达哪一个混沌吸引子,和系统的耦合参数以及初始值的选取有关。本文对上述研究问题都给出了仿真实例,均很好地验证了相应的理论结果。更多还原

【Abstract】 Firstly, a brief description of the chaos and synchronization is given. Then this dissertation mainly studies some issues related to chaos synchronization and proposes theoretical analysis, including the research on relation between noise and chaos synchronization, synchronization in complex dynamical networks, and synchronization of (discrete, non-autonomous, multi-scroll) chaotic systems. The work and contributions are summarized as follows:(1) Two aspects of relation between noise and chaos synchronization are studied. On the one hand, white noise with certain intensity is used as a feedback gain in order to achieve the purpose of generalized synchronization. The conditions required for noise coupling strength function are derived in detail. On the other hand, when chaotic systems are disturbed by noise, based on the robust control method, the feedback terms are designed for limiting the synchronization errors to a relatively small range. In addition, when the orders of driving-response systems are the same or not, two cases are considered and theoretical analysis gives sufficient conditions for synchronization.(2) Three aspects of synchronization in complex dynamical networks are studeid. First of all, synchronization of Chen systems in complex networks is invesrigated. Different with Lyapunov function method and linear approximation method, the approach proposed here can be used universally and ensure the monotonous synchronization. Consider four complex networks, including ring network, star network, small world network and globally coupled network, which all have N Chen systems, the sufficient conditions for synchronization are derived and the results are compared. Secondly, the impulsive pinning controlling method is applied to a small fraction of network nodes, in order that all nodes are synchronized to the equilibrium. The scheme designed here has universal applications and the cost for controlling is low. Finally, the existence of generalized synchronization in complex dynamical networks i s considered. G iven a specific n etwork, t he g eneralized s ynchronization c an be classified a ccording t o the status o f m odified s ystems. T heoretical p roofs a re p roposed to the existence and exponential attractive property of three types of generalized synchronization m anifold.(3) For discrete, non-autonomous and multi-scroll chaotic systems, the follows are studied: Firstly, synchronization of discrete chaotic systems with unknown parameters is researched. The controller and parameters are designed, and theoretical results prove that the method is effective. Secondly, because the synchronization manifold of non-autonomous chaotic systems is related to time, the mathematical mechanism of the synchronization is complicated. The g eneralized s ynchronization i s classified i nto f our t ypes a ccording t o the status o f m odified r esponse s ystems. T he c onditions f or the existence a nd exponential attractive p roperty o f three t ypes o f generalized s ynchronization m anifold a re g iven, and H?lder continuity induced by the non-smooth synchronized manifold is considered. Finally, taking a specific multi-scroll chaotic system for illustration, the phenomenon of multi-stable chaotic attractors in generalized synchronization is discussed. It is found that in response system, there exist multi-stable chaotic attractors, which all synchronize with the driving system. Moreover, which attractor the response system arrives depends on the coupling parameters and initial conditions.For all the above problems, numerical simulations are proposed, which verify the corresponding theoretical results.更多还原