Abstract: The periodic orbits provide a skeleton for the organization of complex chaotic systems, for many important characteristics and dynamic properties of these systems can be determined by solving the periodic orbits, such as the accurate calculation of Lyapunov exponents, estimation of topological entropy and description of a chaotic invariant set. First, the paper reviewed the current commonly used four methods to find periodic orbits, which are NR algorithm, Broyden algorithm, SD algorithm and DL algorithm, and analyzed their characteristics and mutual relations. Second, the paper discussed the advantages, disadvantages and scope of each method with specific examples in detail. Finally, the paper pointed out that DL algorithm is more ideal among the four algorithms, and suggested the future research direction.

Key words: chaos, periodic orbit, iterative algorithm, Poincare section, numerical computation

摘要： 周期轨是混沌系统的基本骨架，系统的很多重要特征和动力学性质都可以通过求解周期轨来确定，例如Lyapunov指数的精确计算、拓扑熵的估计、以及混沌不变集的描述等。综述了目前常用的四种寻找周期轨的方法:NR算法、Broyden算法、SD算法和DL算法，分析其特点及相互关系，然后结合具体计算实例，详细探讨各方法的优缺点和适用范围，得出DL算法是较为理想的搜索算法，并对该领域的研究方向进行了展望。

关键词: 混沌, 周期轨, 迭代算法, 庞加莱截面, 数值计算