Abstract In this paper, we go beyond what was proposed in theory by Melnikov ( [15] ) to introduce a practical method to calculate the high order splitting distances of stable and unstable manifold in time-periodic equations. Not only we derive integral formula for splitting distances of all orders, but also we develop an analytic theory to evaluate the acquired multiple integrals. We reveal that the dominance of the exponentially small Poincare/Melnikov function for equations of high frequency perturbation is caused by a certain symmetry embedded in the kernel functions of high order Melnikov integrals. This symmetry is beheld by many non-Hamiltonian equations.

中文翻译：

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指数小分裂：直接方法

摘要 本文超越Melnikov ( [15] )在理论上提出的理论，引入了一种实用的方法来计算时间周期方程中稳定和不稳定流形的高阶分裂距离。我们不仅推导出了所有阶次距离分裂的积分公式，而且我们开发了一个解析理论来评估获得的多重积分。我们揭示了高频扰动方程的指数小 Poincare/Melnikov 函数的优势是由嵌入高阶 Melnikov 积分的核函数中的某种对称性引起的。许多非汉密尔顿方程都可以看到这种对称性。