Chirikov’s celebrated criterion of resonance overlap has been widely used in celestial mechanics and Hamiltonian dynamics to detect global instability, but is rarely rigorous. We introduce two simple Hamiltonian systems, each depending on two parameters measuring, respectively, the distance to resonance overlap and nonintegrability. Within some thin region of the parameter plane, classical perturbation theory shows the existence of global instability and symbolic dynamics, thus illustrating Chirikov’s criterion.

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关于共振重叠开始的评论

奇里科夫著名的共振重叠准则已广泛应用于天体力学和哈密顿动力学中以检测全局不稳定性，但很少是严格的。我们引入两个简单的哈密顿系统，每个系统取决于两个参数，分别测量共振重叠的距离和不可积性。在参数平面的某些薄区域内，经典微扰理论表明存在全局不稳定性和符号动力学，从而说明了奇里科夫准则。