Abstract

In slow–fast systems, fast variables change at a rate of the order of one, and slow variables, at a rate of the order of \(\varepsilon\ll 1\). The system obtained for \(\varepsilon=0\) is said to be frozen. If the frozen (fast) system has one degree of freedom, then in the region where the level curves of the frozen Hamiltonian are closed there exists an adiabatic invariant. A. Neishtadt showed that near a separatrix of the frozen system the adiabatic invariant exhibits quasirandom jumps of order \(\varepsilon\). In this paper we partially extend Neishtadt’s result to the multidimensional case. We show that if the frozen system has a hyperbolic critical point possessing several transverse homoclinics, then for small \(\varepsilon\) there exist trajectories shadowing homoclinic chains. The slow variables evolve in a quasirandom way, shadowing trajectories of systems with Hamiltonians similar to adiabatic invariants. This paper extends the work of V. Gelfreich and D. Turaev, who considered similar phenomena away from critical points of the frozen Hamiltonian.

中文翻译：

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慢速哈密顿系统的同宿集附近的局部绝热不变量

摘要

在慢速系统中，快速变量的变化速率为1，而慢速变量的变化速率为\（\ varepsilon \ ll 1 \）。为\（\ varepsilon = 0 \）获得的系统被称为冻结的。如果冻结（快速）系统具有一个自由度，则在冻结哈密顿量的水平曲线闭合的区域中，存在绝热不变量。A. Neishtadt表明，在冷冻系统的分离线附近，绝热不变量表现出\（\ varepsilon \）阶的准随机跳跃。在本文中，我们将Neishtadt的结果部分扩展到多维案例。我们表明，如果冻结系统的双曲临界点具有多个横向同宿点，则对于小\（\ varepsilon \）存在轨迹掩盖同宿链。慢变量以准随机方式演化，遮盖了具有类似于绝热不变量的哈密顿量的系统的轨迹。本文扩展了V. Gelfreich和D. Turaev的工作，他们认为类似的现象远离冻结的哈密顿量的临界点。