含三次耦合项的两自由度Duffing系统的共振及混沌行为

研究了一类含三次耦合项的两自由度Duffing系统的动力学行为。首先应用多尺度方法近似求解系统的一阶稳态响应。通过讨论系统的主共振和1∶1内共振,分析了三次耦合项对系统响应的影响。随后研究系统随外加周期力强度变化的分岔过程,发现除了常见的倍周期分岔通向混沌外,还存在一种直接由周期运动进入混沌的突发路径。结合对系统的最大Lyapunov指数,相轨图及Poincar啨映射的分析验证了上述结论。

Resonance and Chaos Behavior in a Two-Degree-of-Freedom Duffing System with Cubic Coupled Terms

The dynamical behaviors for a class of two-degree-of-freedom Duffing system with cubic coupled terms are evaluated, where multiple scale method is adopted to find the first-order steady-state response of the system. The effect of cubic terms on system response is investigated by analyzing the principal resonance and internal resonance, then the bifurcation process is analyzed in terms of the intensity of external force. The numerical results show that besides the route to chaos through period doubling, the system exists a kind of sudden transition route between periodic motion and chaotic vibrations, which are further verified by the top Lyapunov exponent, phase portrait and Poincar mapping analysis on dynamical behavior of the system.