一类对称双势阱Duffing系统周期解的衍生与演化

针对自旋单自由度三次强化弹簧—质量振子系统建立的对称双势阱Duffing方程,通过把数值计算与谐波平衡半解析分析相结合,系统分析了该类Duffing系统在谐波强迫激励下周期解随激励频率变化的衍生与演化现象,获得了不同频段谐波强迫激励下系统周期解的类型、周期解的衍生模式与演化规律。分析结果表明,该类Duffing方程存在平衡点临域局部周期解以及鞍结点分岔衍生的对称、反对称与非对称等多种全局周期解;局部或无对称性的全局周期解直接通过倍周期分岔通向混沌运动;全局对称周期一解和反对称周期三次谐波解首先各自发生对称和反对称破缺,再通过倍周期分岔演化为混沌。研究有助于深化对Duffing方程非线性现象及其演化规律的认识。

Periodic Solutions Initiation and Evolving of a Symmetrical Double-Potential Well Duffing System with Forced Harmonic Excitation

For a Duffing equation with symmetrical double potential well,which is derived from a self-rotating mass-spring pendulum with three-order nonlinear stiffness,the periodic solutions type,initiation mode and evolving behavior adapting to the frequency of excitation are investigated meticulously by Harmonic Balance Method and numerical simulation.Some rewarding results are obtained.This Duffing system has local periodic solutions in the region near its equilibrium point and many global periodic solutions with symmetry,antisymmetry and dissymmetry.The global periodic solutions derive from saddle-node bifurcation.The local periodic solutions and the dissymmetrical global periodic solutions evolve directly into chaos passing through double-periodic bifurcation.The symmetrical global 1-periodic solutions evolve into chaos through passing the double-periodic bifurcation only after losting their symmetry by symmetry-breaking,and the antisymmetric global 3-periodic solutions evolve into quasi-periodic solutions through passing the triple-periodic bifurcation also after lost their symmetry by antisymmetry-breaking.The studies help to raise the understanding of the nonlinear phenomenon and the evolution of solutions of the Duffing equation.