标题： 横截性同宿轨道与双曲平衡点在Hamilton系统中的研究，通过Hamilton–Jacobi方程 显示英文标题

标题： Transversality of homoclinic orbits to hyberbolic equilibria in a Hamiltonian system, via the Hamilton--Jacobi equation

摘要： 我们研究了一个具有两个自由度的哈密顿系统，其中存在一个具有环路或同宿轨道的双曲平衡点（或者，作为替代方案，由异宿轨道连接的两个双曲平衡点），这是为了理解近可积哈密顿系统在二重共振附近的性态所迈出的一步。 我们提供了一种构造性的方法来研究双曲点的不稳定和稳定不变流形是否沿着环路在其共同能量水平上横截相交。 对于所考虑的系统，我们以一个黎卡提方程的解作为不变流形在垂直于环路方向上的斜率，给出了横截性的充要条件。 我们方法的关键点在于用生成函数表示不变流形，这些生成函数是哈密顿-雅可比方程的解。 在一些例子中，我们表明只需分析黎卡提方程的相图即可，而无需显式求解它。 最后，我们在微扰情况下考虑了类似的问题。 如果未扰动环路的不变流形重合，则我们面临的是分离曲线分裂的问题。 在这种情况下，黎卡提方程被一个梅尔尼科夫势取代，该势定义为一个积分，提供了存在扰动环路及其横截性的条件。 这同样通过一个具体的例子进行了说明。 显示英文摘要

摘要： We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop, inside their common energy level. For the system considered, we establish a necessary and sufficient condition for the transversality, in terms of a Riccati equation whose solutions give the slope of the invariant manifolds in a direction transverse to the loop. The key point of our approach is to write the invariant manifolds in terms of generating functions, which are solutions of the Hamilton--Jacobi equation. In some examples, we show that it is enough to analyse the phase portrait of the Riccati equation without solving it explicitly. Finally, we consider an analogous problem in a perturbative situation. If the invariant manifolds of the unperturbed loop coincide, we have a problem of splitting of separatrices. In this case, the Riccati equation is replaced by a Mel'nikov potential defined as an integral, providing a condition for the existence of a perturbed loop and its transversality. This is also illustrated with a concrete example.