一类非线性相对转动系统的组合共振分岔与混沌

研究一类具有异宿轨道的非线性相对转动系统的分岔与混沌运动. 应用耗散系统的拉格朗日方程建立一类组合谐波激励作用下非线性相对转动系统的动力学方程. 利用多尺度法求解相对转动系统发生组合共振时满足的分岔响应方程并进行奇异性分析, 得到了系统稳态响应的转迁集. 根据相对转动系统异宿轨道参数方程, 求解了异宿轨道的Melnikov函数, 并给出了系统发生Smale马蹄变换意义下混沌的临界条件. 最后采用数值方法, 通过分岔图, 最大Lyapunov指数图, 相轨迹图和庞加莱截面图研究系统参数对混沌运动的影响.

Combination resonance bifurcations and chaos of some nonlinear relative rotation system

The bifurcation and chaotic motion of some nonlinear relative rotation system with heteroclinic orbit is studied. By using dissipative system Lagrange equation, the dynamics equation of nonlinear relative rotation system under combination harmonic excitations is established. Firstly, the bifurcation response equation of relative rotation system under combination resonance is deduced with the method of multiple scales. Singularity analysis is employed to obtain the transition set of steady motion. Secondly, the Melnikov function of heteroclinic orbit is solved according to heteroclinic orbit parameter equation of relative rotation system. The critical condition of chaos about Smale commutation is given. Finally, numerical method is employed to analyze the influences of different system parameters on chaotic motion by bifurcation diagram, the maximum Lyapunov, phase trajectory and Poincare map.