【作者】 过榴晓；

【导师】 徐振源；

【作者基本信息】 江南大学 ， 轻工信息与技术工程， 2009， 博士

【摘要】 近年来,混沌及混沌同步吸引了越来越多的科学家和工程技术人员的广泛重视和研究。本文首先简单介绍了混沌、混沌同步的控制和复杂网络动力学同步的发展背景。然后就两个混沌系统间的广义同步和动态网络模型的同步方面进行了深入的研究与探讨。具体的研究工作概括如下:(1)研究了单向耦合的2个不同的混沌系统两类光滑广义同步流形的存在性。即在响应系统的修正方程在具有渐近稳定平衡点或轨道渐近稳定的周期解的情况下,一定的条件下,可将广义同步化流形存在性问题转化为Lipschitz函数族的压缩不动点问题,理论上严格证明了该广义同步化流形的不变性和指数吸引性。进一步的,研究了对于单向耦合的2个不同的非线性混沌系统两类H?lder连续广义同步流形的存在性。特定条件下,将广义同步化流形存在性问题转化为H?lder连续函数空间上的Schauder不动点问题,证明广义同步流形的不光滑性和多值性。(2)研究了更复杂的相互线性耦合的非线性混沌动力系统X和Y广义同步流形的存在性。根据相互耦合的修正系统的性质,将可能的七种情况转化为其中两类的特殊情况分析,即对于Y系统修正方程具有混沌态而X系统修正方程具有渐近稳定平衡点或轨道渐近稳定周期解情况下,理论证明两类双向耦合的光滑广义同步化流形的不变性和指数吸引性。类似的,利用H?lder连续函数空间上的Schauder不动点问题,数学上严格证明了广义H?lder连续同步流形的不光滑性和多值性。(3)建立了网络间脉冲耦合形式控制实现驱动-响应动态网络的投影同步和投影聚类同步模型,应用脉冲微分方程理论的比较系统方法,给出了驱动-响应网络系统可以实现投影同步和聚类投影同步的数学的充分条件,证明了脉冲间隔可以取任意有限正值。并且由牵制控制可以调节响应网络相对于驱动节点的比例因子数值和各聚类的比例因子数值。(4)提出并研究了线性控制驱动-响应网络的多因子投影同步模型,基于Lyapunov函数理论方法和矩阵左特征值理论,得到响应网络在对称或不对称耦合下的实现多因子投影同步的充分条件。(5)建立了投影同步的一个新的一般的网络模型(耦合矩阵非对称且无驱动阵子),利用左特征值理论,设计了耦合强度的一个自适应规则(adaptive scheme)来实现网络多因子投影同步。本文对所有研究的问题给出了数值仿真例子,仿真结果均很好地验证了相应的理论分析结果。更多还原

【Abstract】 In the rencent years, the study on chaos and chaos synchronization has attracted increased attention from scientists and engineers. Firstly in this dissertation, a brief introduction to the historical backgrounds of chaos and complex networks, research progress and typical methods for chaotic synchronization are given. Then the dissertation mainly focuses on the generalized synchronization of two chaotic different systems and projective synchronization of complex dynamical networks. The author’s main research work and contributions are as follows:(1) The existence of two types generalized synchronization(GS) of chaotic unidirectional coupled systems is studied. Based on the modified system approach, GS is classified into three types: equilibrium GS, periodic GS, and C-GS, when the modified system has an asymptotically stable equilibrium, asymptotically stable limit cycles, and chaotic attractors, respectively. The existence of the first two type GS can be converted to the problem of compression fixed point under certain conditions. Strict theoretical proofs are given to exponential attractive property of generalized synchronization manifold. By using the Schauder Fixed Point Theorem in H?lder continuous function space, the existence of two types H?lder continuous inertial manifolds, equilibrium GS and periodic GS, with certain conditions will be theoretically proved.(2) Furthermore, we studies the existence of two types generalized synchronization of bi-directionally coupled chaotic oscillators. More complex situations appear. When the modified system of Y oscillator is chaotic, while the modified system of X oscillator collapses to a stable equilibrium point or periodic orbit, the existence of generalized synchronization can be converted to the problem of Lipschitz function family’s compression fixed point under certain conditions. Proofs are also given to the existence of bi-directionally coupled smooth and H?lder continuous generalized synchronization manifold.(3) Impulsive projective (cluster) synchronization in 1 + N coupled chaotic systems are investigated with the drive-response dynamical network (DRDN) model. Based on impulsive stability theory, some simple but less conservative criteria are achieved for projective synchronization in DRDNs. It is worth noting that any width of impulsive intervalΔ≥0 can be arbitrarily selected to implement projective synchronization(PS) with desired scaling factor under certain conditions. Furthermore, impulsive pinning scheme is also adopted to direct the (cluster) scaling factor onto the desired value.(4) By using simple linear control, some sufficient criteria for projective synchronization with different scaling factors of the response networks of symmetrical and asymmetrical coupling are given based on Lyapunov function method.(5) A new model of N coupled chaotic systems networks without the drive oscillator is established. By designing an adaptive scheme, sufficient criteria for the projective synchronization with different scaling factors PSDF in symmetrical and asymmetrical coupled networks are obtained based on the Lyapunov function method and the left eigenvalue theory.For all the above theoretical results, we proposed numerical simulations which verity the corresponding theoretical results.更多还原