This paper investigates periodic bifurcation solutions of a mechanical system which involves avan der Pol type damping and a hysteretic damper representing restoring force. This systemhas recently been studied based on the singularity theory for bifurcations of smooth functions.However, the results do not actually take into account the property of nonsmoothness involvedin the system. In particular, the transition varieties due to constraint boundaries were ignored,resulting in failure in _nding some important bifurcation solutions. To reveal all possible bifurcationpatterns for such systems, a new method is developed in this paper. With this method,a continuous, piecewise smooth bifurcation problem can be transformed into several subbifurcationproblems with either single-sided or double-sided constraints. Further, the constrainedbifurcation problems are converted to unconstrained problems and then singularity theory isemployed to _nd transition varieties. Explicit formulas are applied to reconsider the mechanicalsystem. Numerical simulations are carried out to verify analytical predictions. Moreover, symbolicnotation for a sequence of bifurcations is introduced to easily show the characteristics ofbifurcations, and also the comparison of di_erent bifurcations. The method developed in thispaper can be easily extended to study bifurcation problems with other types of nonsmoothness.

讨论双频内共振系统的Normal Fbrm及其降维问题．利用发展的Normal Fbrm直接方 法，导出了任意双频内共振系统Normaj Fbrm的一般形式．指出Poincaré共振项分为内共振 项和非内共振性两类，并定义了内共振项的阶．提出了一种普遍适用的降维变换，并证明了该 变换可将任意双频内共振系统的Normal F0rm方程降到3维．应用举例表明，该变换不仅适 用于半单问题，也适于非半单问题(即强1：l内共振系统)．

在文[1]基础上，提出一种仅知道派生线性系统零实部特征值时求解非线性系统非半单分叉Normal Form的方法。通过适当的分类，将要求解的线性代数方程组分为若干相互独立的方程组。将所求系数向量按字典序列排列后，各独立方程组的系数矩阵是上三角矩阵。在非共振情形，各系数向量可按反字典序列递推求出，在共振情形，根据文中的二个定理，巧妙地由一简单的常数矩阵的最大秩子矩阵，定位其系数矩阵的满秩子矩阵，解决了这类方程组的降维简化。通过消元法，把简化后的方程化成类似于半单分叉Normal Form求解过程中方程的形式，其解法也类似。该方法非常易于在计算机代数软件平台上程序化。

对非线性模态的定义、构造方法、非线性模态的特点及其研究进展进行了全面综述，指出了该领域研究中存在的一些问题和今后的主要研究方向．

引入不可分偶数维不变流形的概念来定义非线性模态。在此基础上，揭示出了一种新的模态——耦合非线性模态，并对实际系统中各种可能的模态进行了分类。这种分类可能是新的构筑非线性模态理论的框架。用此方法构造非线性模态，得到的模态振子具有范式的形式，形式最简、却能反映原系统在平衡点附近的主要动力学行为，且易于得到非线性频率及非线性稳定性等方面的信息。不仅适用于分析一般的多自由度系统，还可用于分析奇数维系统;不仅可构造内共振系统的非耦合模态，还可用于构造内共振耦合模态。从掌握的资料看，以前的方法还不能解决上述所有问题.