不可积Hamilton系统的跨临界分岔
Transcritical bifurcations in non-integrable Hamiltonian systems
摘要：
本文研究了不可积二维哈密顿系统中周期轨道的跨临界分岔。我们讨论了它们的存在性准则和它们的一些性质，使用最近的数学描述的跨临界分支的家庭辛映射。然后，我们提出了跨临界分岔的一类广义Hénon-Heiles哈密顿的数值例子，并说明其稳定性和各种扰动下的哈密顿展开。我们证明了对于含有直线振动轨道的哈密顿量，这些轨道的跨临界分岔是典型的情况下，也发生在没有任何离散对称性，而其isoerythropy-pitchfork分岔是一个例外。我们确定这两种类型的分岔的规范形式，并推导出所需的统一近似，包括跨临界分岔轨道的量子哈密顿量的状态密度的半经典跟踪公式。我们计算了一个具体的例子中的粗粒态密度半经典和量子力学，并找到很好的协议的结果。
英文摘要：
We report on transcritical bifurcations of periodic orbits in non-integrable two-dimensional Hamiltonian systems. We discuss their existence criteria and some of their properties using a recent mathematical description of transcritical bifurcations in families of symplectic maps. We then present numerical examples of transcritical bifurcations in a class of generalized Hénon-Heiles Hamiltonians and illustrate their stabilities and unfoldings under various perturbations of the Hamiltonians. We demonstrate that for Hamiltonians containing straight-line librating orbits, the transcritical bifurcation of these orbits is the typical case which occurs also in the absence of any discrete symmetries, while their isochronous pitchfork bifurcation is an exception. We determine the normal forms of both types of bifurcations and derive the uniform approximation required to include transcritically bifurcating orbits in the semiclassical trace formula for the density of states of the quantum Hamiltonian. We compute the coarse-grained density of states in a specific example both semiclassically and quantum mechanically and find excellent agreement of the results.
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