中文摘要：

在这份报纸，周期的解决方案光滑、不连续(SD ) 振荡器，它是一个强烈荒谬的非线性的系统为有粘滞的抑制和外部泛音刺激的系统被讨论。四个维的平均方法被使用完全的 Jacobian 采用椭圆形的积分直接获得从夸张僵绳和不受搅乱的系统的非夸张的中心区分的使不安的主要回答。这些周期的答案的稳定性被用 Lyapunov 方法检验四个维的平均方程分析。此处介绍的结果这份报纸是有效的因为两个变光滑(> 0 ) 并且不连续(= 0 ) 提供答案给平均定理惊人地为在未击中的系统的外部强迫的中等力量的盒子失败的问题的阶段由霍姆斯分析了。数字计算为这个特别系统与理论预言和分析的优秀效率显示出一个好协议，它也建议分析对强烈非线性的系统适用。

英文摘要：

In this paper, the periodic solutions of the smooth and discontinuous （SD） oscillator, which is a strongly irra- tional nonlinear system are discussed for the system having a viscous damping and an external harmonic excitation. A four dimensional averaging method is employed by using the complete Jacobian elliptic integrals directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle and the non-hyperbolic centres of the un- perturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equa- tion using Lyapunov method. The results presented herein this paper are valid for both smooth （e 〉 0） and discontin- uous （ce = 0） stages providing the answer to the question why the averaging theorem spectacularly fails for the case of medium strength of external forcing in the Duffing system analysed by Holmes. Numerical calculations show a good agreement with the theoretical predictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems.