非线性振动链中的共振能量交换和极限相轨迹：从小系统到大系统
Resonant energy exchange in nonlinear oscillatory chains and Limiting Phase Trajectories: from small to large systems
摘要：
我们提出了一种充分的分析方法来描述弱耦合振子和振动链之间具有强能量交换的非线性振动。介绍了与完全能量交换相对应的极限相轨迹的基本概念。首先，我们提出一个简单的非线性振子振动的解析描述。我们发现，两个动力学的转变发生在系统中。第一种对应于振子反相振动的分岔。第二种是由于LPT与划分两个稳定定态的分界线重合，导致LPT的相位和时间行为发生质的变化。下一个考虑的问题涉及到密集的周期性非线性系统中的模态间交换有限（n>2）自由度。我们考虑两种极限情况。当粒子数不足够大时，考虑二维积分流形中非线性简正模之间的能量交换。当粒子数增加时，相邻积分流形之间的能量交换变得重要，导致类似于一维连续介质中呼吸子的局域激发的形成。
英文摘要：
We present an adequate analytical approach to the description of nonlinear vibration with strong energy exchange between weakly coupled oscillators and oscillatory chains. The fundamental notion of the limiting phase trajectory (LPT) corresponding to complete energy exchange is introduced. At first we propose a simple analytical description of vibrations of nonlinear oscillators. We show that two dynamical transitions occur in the system. First of them corresponds to the bifurcation of anti-phase vibrations of oscillators. And the second one is caused by coincidence of LPT with separatrix dividing two stable stationary states and leads to qualitative change in both phase and temporal behavior of the LPT. Next problem under consideration relates to intensive intermodal exchange in the periodic nonlinear systems with finite (n>2) number of degrees of freedom. We consider two limiting cases. If the number of particles is not large enough, the energy exchange between nonlinear normal modes in two-dimensional integral manifolds is considered. When the number of the particles increases the energy exchange between neighbor integral manifolds becomes important that leads to formation of the localized excitations resembling the breathers in the one-dimensional continuum media.
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