二维非线性临界解析动态系统的局部渐近稳定性

Locally asymptotic stability of 2-dimension nonlinear analytic dynamic systems in critical cases

考察具有一对共轭纯虚数特征值的二维非线性临界解析动态系统的局部渐近稳定性.首先在非奇异线性坐标变换和时间尺度变换下,将其化成标准形式.之后,运用形式级数法的思想,通过构造多组线性方程组,给出了确定该系统的李雅普诺夫函数的方法,并得到了判别系统局部渐近稳定和不稳定的充分条件.最后通过示例说明该判别条件的有效性.

The locally asymptotic stability of a 2-dimension nonlinear analytic dynamic system with a pair of conjugated imaginary eigenvalues is studied. The system is firstly simplified to a standard form by using the non-singular linear coordinate transformation and the time scale transformation. Next, based on the idea of formal progression, a method is developed to determine the Lyapunov function for this standard form by constructing several sets of linear equations. Finally, a sufficient condition of locally asymptotic stability for the system is obtained. The validity is shown by two examples at the end of this paper.