中文摘要：

研究了弱非线性耦合二维各向异性谐振子的奇点稳定性及其在相空间中的轨迹．首先，求得弱非线性耦合二维各向异性谐振子的奇点；其次，分别利用Lyapunov间接法和梯度系统方法讨论该系统的平衡点稳定性；最后，用Matlab方法对系统进行数值模拟，并运用庞加赖截面观察系统在相空间的运动轨迹，发现随着能量的增加系统经历规则运动、规则运动与混沌并存等阶段，最后出现了混沌现象．

英文摘要：

The stability of singular points and their trajectories in phase space of the weak nonlinear coupled two- dimensional anisotropic harmonic oscillator are studied. Firstly, the singular points of the weak nonlinear coupled two-dimensional anisotropic harmonic oscillator are obtained. Based on the Lyapunov indirect method and the gra- dient method, the stability of equilibrium points of this system are then discussed. Finally, numerical simulations are performed by the software Matlab, and Poincare surface of the section are used to study the trajectories of the system in phase space. It is found that, with the increase of energy, the chaos appears finally through two stages of regular motion as well as the coexistence of regular motion and chaos.