哈密尔顿系统作为数学的一个经典研究领域，一直受到广泛关注。一个基本的问题是对其周期解的研究。近几十年来逐渐发展完善的指标及其迭代理论是研究非线性哈密尔顿系统周期解存在性和多重性的一个有力工具。 本文利用指标理论研究了一类具有周期性和偶不变性的特殊哈密尔顿系统的可逆周期解的存在性和多重性。主要结果中利用Maslov-型指标给出了这样系统的可逆周期解个数的一个下界，并通过指标迭代理论得到了由平均指标给出的相应迭代结果。

Hamiltonian dynamic system has received extensive attention as a classical field of mathematics. A basic problem is about its periodic solutions. Gradually developed in recent decades, the index and its iteration theory become a powerful tool to study the existence and multiplicity of periodic solutions for nonlinear Hamiltonian systems.This thesis considers a special class of Hamiltonian systems under certain even and periodic assumptions. Using the index theory, we study the existence and multiplicity of reversible periodic solutions for these systems. The main result gives a low bound of the number of such periodic solutions via Maslov-type index, and obtain several related iteration results by mean index through the index iteration theory.