双频内共振系统的Normal Form及其简化

讨论双频内共振系统的 Normal Form及其降维问题.利用发展的 Normal Form直接方法,导出了任意双频内共振系统 Normal Form的一般形式.指出 Poincare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 Form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:1:1内共振系统）.

NORMAL FORMS AND THEIR SIMPLIFICATION OF NONLINEAR SYSTEMS WITH INTERNAL RESONANCE BETWEEN TWO FREQUENCIES

The Normal Forms and their simplification of the nonlinear systems with internal resonance between two frequencies are discussed. By using the method for solving the Normal Form of a nonlinear systems developed by us, the general Normal Form is derived out for the nonlinear systems with an arbitrary kind of internal resonance between two frequencies. To distinguish their effects, the nonlinear terms in Normal Forms are divided into two classes, the non-internal-resonant ones and the internal-resonant ones. It is proved that, under the universal transformation proposed in the paper, the Normal Forms of all the two-frequency-resonant nonlinear system become new dynamical systems of dimension 3. Three application examples, i.e. nonlinear systems of 1 degree of freedom excited by multiple harmonics, 2:3 internally resonant systems and strong 1:1 internally resonant systems, are used to show that the transformation is useful in simplifying the Normal Forms either for semi-simple systems or nonsemi-simple systems.