含平方项非线性动力系统的分岔研究

Bifurcation Analysis of Non-linear Dynamic System with a Quadratic Term

用规范形理论共轭算子法研究了含平方项非线性系统,获得了方程的稳态渐近解.然后用普适开折理论分析了含平方项非线性Duffing系统的分岔响应方程,分析了余维2分岔,得到了转迁集和分岔图.得到的结果与数值模拟结果相一致.

The normal form theory was extended to study the non-linear dynamical system with a quadratic term. The unknown variable frequency ω_1 was defined as the basic frequency of the weakly nonlinear system with a quadratic term, and the nonlinear transformation was appropriately supposed. With the present method, the non-linear one-degree-of-freedom oscillators′ steady periodic solutions can be obtained easily, and the transition set and bifurcation diagram can be achieved by the universal unfolding theory. The bifurcation of codimension two for the nonlinear system was analysed. The results obtained by the method coincided very well with those obtained by numerical integration for the system. It is obvious that the proposed normal form method is very easy, simple and valid.