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weak damping
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  弱阻尼
    Melnikov method is employed to study the bifurcations and chaos of homoclinic and subharmonic orbits for two pendulum systems. One is with weak damping and parametric excitation and the other is inverted with external driving. The parameter conditions for occurence of bifurcations and chaos are obtained.
    用Melnikov方法研究具有弱阻尼与参数激励的单摆以及倒摆运动的同宿轨道分岔、次谐分岔和混沌现象 ,得到了发生同宿轨道分岔、次谐分岔和混沌的临界参数 ,并将所得理论结果与倒摆运动的实验结果进行了比较与讨论 .
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    This paper discussed the existence of periodic solution of Duffing eguation in weak damping system and hard spring,and second-harmonic wave vibration produced by frequency variance of forced vibration,using Melnikov function,concise estimation of parameter range was also given. These results provided some useful references for analyses of physics experiments and applications of engineering and technology.
    利用Melnikov函数讨论了物理问题中常见的具有弱阻尼与硬弹簧的Duffing方程的周期解的存在性,以及由受迫频率的变化而导致的次谐波振动,给出了参数区域的精确估计,从而给物理实验分析和工程技术应用提供有用参考。
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  weak damping
Moreover the damping is assumed to be nonlinear with no growth assumption at the origin, which allows very weak damping.
      
For nor linear inertial torsion vibrations of the structure of a nematic liquid crystal in a destabilizing magnetic field, there exist low-frequency modes with anomalously weak damping.
      
When the piston velocity is always smaller than the thermal velocity we observe weak damping.
      
We investigate the microscopic mechanisms associated with strong and weak damping in the adiabatic motion of the simple piston with finite mass and finite amount of gas.
      
Strong and Weak Damping in the Adiabatic Motion of the Simple Piston
      
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The damping of channeling radiation and the periodicity of crystal lattice field were considered. The motion equation of particles is simplified to the pendulum equation with weak damping and weak excitation. The chaotic behaviour of the system is discussed, Melnikov function in various cases is calculated, the threshold and possible approach to chaotic condition are derived.

考虑了带电粒子沟道辐射的衰减和晶格场的周期性,把粒子运动方程化为具有弱阻尼和弱激励的摆方程。讨论了带电粒子面沟道辐射的混沌(Chaos)行为,计算了系统的Melnikov函数,导出了系统的阈值和通向混沌的可能途径。

This paper discussed the existence of periodic solution of Duffing eguation in weak damping system and hard spring,and second-harmonic wave vibration produced by frequency variance of forced vibration,using Melnikov function,concise estimation of parameter range was also given. These results provided some useful references for analyses of physics experiments and applications of engineering and technology.

利用Melnikov函数讨论了物理问题中常见的具有弱阻尼与硬弹簧的Duffing方程的周期解的存在性,以及由受迫频率的变化而导致的次谐波振动,给出了参数区域的精确估计,从而给物理实验分析和工程技术应用提供有用参考。

Melnikov method is employed to study the bifurcations and chaos of homoclinic and subharmonic orbits for two pendulum systems. One is with weak damping and parametric excitation and the other is inverted with external driving. The parameter conditions for occurence of bifurcations and chaos are obtained. The analytical results are compared with the experimental ones and some implications are also discussed.

用Melnikov方法研究具有弱阻尼与参数激励的单摆以及倒摆运动的同宿轨道分岔、次谐分岔和混沌现象 ,得到了发生同宿轨道分岔、次谐分岔和混沌的临界参数 ,并将所得理论结果与倒摆运动的实验结果进行了比较与讨论 .

 
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