We consider a Hamiltonian system depending on a parameter which slowly changes with rate ε ≪ 1. If trajectories of the frozen autonomous system are periodic, then the system has adiabatic invariant which changes much slower than energy. For a system with 1 degree of freedom and a figure 8 separatrix, Anatoly Neishtadt [18] showed that for trajectories crossing the separatrix, the adiabatic invariant, and hence the energy, have quasirandom jumps of order ε. We prove a partial analog of Neishtadt’s result for a system with n degrees of freedom such that the frozen system has a hyperbolic equilibrium possessing several homoclinic orbits. We construct trajectories staying near the homoclinic set with energy having jumps of order ε at time intervals of order ∣ln ε∣, so the energy may grow with rate ε/∣ln ε∣. Away from the homoclinic set faster energy growth is possible: if the frozen system has chaotic behavior, Gelfreich and Turaev [16] constructed trajectories with energy growth rate of order ε.

中文翻译：

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慢时变哈密顿系统的同宿集合附近的能量跳跃

我们根据一个随速度ε≪ 1缓慢变化的参数来考虑哈密顿系统。如果冻结的自治系统的轨迹是周期性的，则该系统具有绝热不变性，其变化远比能量慢。对于一个自由度为1，数字为8的分离线的系统，Anatoly Neishtadt [18]指出，对于穿过分离线的轨迹，绝热不变量和能量具有ε阶的准随机跳跃。对于具有n个自由度的系统，我们证明了Neishtadt结果的部分模拟，从而使冻结的系统具有一个拥有几个同斜轨道的双曲线平衡。我们构造的轨迹停留在同宿点附近，其能量具有ε阶跃变在订单|ln的时间间隔ε |，因此能量可以用速度增长ε / |ln ε |。远离同斜率集，可能会有更快的能量增长：如果冻结系统具有混沌行为，则Gelfreich和Turaev [16]会构建能量增长率为ε的轨迹。