Volterra级数与摄动法的内在关系研究

摄动法是求解弱非线性方程的经典定量方法之一，而Volterra级数是一种可表征弱非线性系统的经典数学模型．研究表明，对于受迫多项式非线性系统，只有当其线性派生系统是阻尼耗散系统，由对系统输入进行小参数摄动求得的稳态输出响应才和由Volterra级数给出的解完全一致．结论意味着：Volterra级数并不能表示任意的多项式非线性系统，而仅适合表征线性派生系统是阻尼耗散系统的多项式非线性系统．

A study about the relationship between the Volterra series and the perturbation method

The perturbation method is one of the classical methods for solving weak nonlinear equations, and the Volterra series is a classical mathematical tool for modeling the systems with weak nonlinearity. It is revealed in this study that, for a forced polynomial nonlinear system, only when its derived linear system is a damped dissipative system, the steady output responses calculated by using the perturbation method and by using the Volterra series would coincide with each other. This means that the Volterra series is valid for modeling only the polynomial nonlinear systems whose derived linear system is a damped dissipative system.