自然界中各种斑图结构和各种各样的集体行为十分普遍，研究、掌握时空斑图和系统的集体行为背后的机制具有重要的理论和应用价值。对动力学系统的集体行为与斑图现象的研究很早就成为非线性科学研究的重要课题。自然界大量的过程都是周期过程，极限环运动是最典型的非线性周期过程。通过耦合极限环系统来研究时空斑图和系统的集体行为是本文的主题。第一章简单介绍了非线性科学的基本知识，对非线性动力系统解的形式和稳定性分析以及分岔做了介绍。然后对非线性动力系统的两个振荡解，即极限环振荡和混沌现象做了初步介绍，第四节又介绍了耦合非线性系统的同步现象。第二章对耦合极限环系统即Kuramoto模型进行了详细介绍，为了使Kuramoto模型更接近于真实的自然和社会现象，许多研究者又对Kuramoto模型进行了推广，包括耦合拓扑结构的变化，模型中的无序，时间延迟行为，存在外场的情况，有噪声的情况，更普遍的周期耦合方程和惯性效应，本章主要介绍耦合强度变化的情况和考虑振幅的Kuramoto模型。Kuramoto模型可以用来解释许多物理、化学以及生物上的问题，本章又对Kuramoto模型的应用进行了详细介绍，包括在神经系统、约瑟夫森结、激光序列、电荷密度波和化学振荡中的应用。第三章详细地研究了考虑振幅的耦合极限环系统的振幅死亡现象。非线性频率吸引和反应耦合是Cross等人在研究耦合的纳米电子器件系统时引入的两个参量，本章是在考虑非线性频率吸引和反应耦合的考虑振幅的耦合极限环系统的基础上，对振幅死亡现象进行了详细地研究。本章通过两个量：归一化非相干能和活振子比例，来描述系统的振幅死亡现象。然后为了了解振幅死亡产生的具体过程又对系统的相位动力学、振子在复平面上的分布、振幅动力学和振幅分布概率进行了考察。第四章讨论了二维Kuramoto模型的斑图动力学。Kuramoto模型假设所有的极限环振子之间是相互吸引的，而事实上个体之间的耦合是十分复杂的，因此人们在Kura-moto模型中引入相移，使振子之间的相互作用从相互吸引到相互排斥，或者介于吸引和排斥之间。本章主要研究了相移不同时系统斑图的变化，以及在二维相移耦合极限环系统中加入缺陷时系统的斑图结构。

All kinds of spatiotemporal patterns and collective behaviors are ubiquitous in nature and society. It is both practically and theoretically significant to research and understand the mechanism of spatiotemporal patterns and collectivebehaviors. Collective behaviors and spatiotemporal patterns of coupled dynamic systems have recently received a great deal ofattention in nonlinear science. Many natural processes are periodic processes, limit-cycle oscillation is the most typicalnonlinear periodic process. The thesisaims to investigate the spatiotemporal patterns and the collective behaviors through coupled limit-cycle oscillators systems.In Chapter 1 we simply review the basic knowledge of nonlinear science, the solutions of nonlinear dynamic systems and the stability and bifurcation of these solutions. Then we simply recall the two oscillatory solutions of nonlinear dynamicalsystems, namely limit-cycle oscillation and chaos. In section four, we review the synchronization phenomenon of coupled nonlineardynamical systems.In chapter 2, we introduce the coupled limit- cycle oscillators system namely Kuramoto model in detail. In order to make the Kuramoto model similar to the realistic phenomena in nature and society, scientists have given some variations of Kuramoto model, such as the variations of coupling mode, models with disorder, time-delaycoupling, external fields, multiplicative noise, more general periodic coupling functions and Kuramoto model with inertia. Inthis chapter we mainly introduce the variation of coupling strength model and Kuramoto model with amplitude. Kuramoto modelcan be used to explain some physical, chemical and biological phenomena. Then we introduce some applications of Kuramoto modelat length, for example the applications in Neuralnetworks, Josephson-junction arrays, laser arrays, charge density waves and chemical oscillators.In chapter 3, the amplitude death of coupled limit-cycle oscillatorssystems are studied in detail. Reactive coupling and nonlinear frequency pulling are considered by Cross et al. when they research the coupled nanoelectromechanical systems. In this Chapter we investigate the amplitude death phenomenon of coupled limit-cycle oscillators systems with amplitude effects by considering dissipative coupling and nonlinear frequency pulling. We use two parameters to study the amplitude death phenomenon, which are normalized mean incoherent energy and percentage of active oscillators. Then in order to investigate the process of amplitude death, we study the snapshots of all oscillators on the complex plane of Z for different couplingstrengths, amplitude dynamics, phase dynamics and the distribution probability of amplitude.In chapter 4,we investigate the spatiotemporal patterns of two dimensional Kuramoto model. In Kuramoto model, the interactions between any two limit-cycle oscillators are pulling. In fact the interactions between individuals are complex. So it is important to introduce the phase shift which induces the interaction between oscillators from phase pulling to phase repulsive coupling. We mainly investigate spatiotemporal patterns with different phase shift and spatiotemporal patterns in two dimensional phase-shifted coupled systems by considering impure coupling.