Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse
Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse
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DOI：
10.1088/1361-6544/aa9ee5
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发表时间：
2017-06
期刊：
影响因子：
1.7
通讯作者：
C. Dettmann;V. Fain;D. Turaev
中科院分区：
文献类型：
--
作者：
C. Dettmann;V. Fain;D. Turaev
研究了轴长随时间周期性变化的椭圆内的台球动力学，并在边界上加了一个O（δ）-小四次多项式变形.在这种情况下，台球中粒子的能量不再守恒。我们证明了在这样的系统中存在一个费米加速：存在一个能量趋于无穷大的台球轨迹。该构造是基于在相空间中的动力学分析，所述相空间靠近由能量和时间参数化的通常双曲不变柱体Λ的稳定和不稳定流形的同宿交点，其对应于沿着椭圆的长轴的沿着运动。该证明依赖于将同宿道附近的台球映射约化为由定义在Λ上的沿着两个Hamilton流的移位构成的迭代函数系统。这两个流近似于所谓的内部映射和散射映射，这是研究Arnold扩散的基本工具;散射映射由稳定和不稳定不变流形Ws，u（Λ）在同宿点处的强稳定和强不稳定叶理Wss，uu的沿着投影定义。Melnikov类型的计算意味着这个问题中散射映射的行为是非常不寻常的：它只定义在Λ的一个小子集上，在大能量极限下，当δ→0时，它收缩为一组平行线t = const。
We study billiard dynamics inside an ellipse for which the axes lengths are changed periodically in time and an O(δ)-small quartic polynomial deformation is added to the boundary. In this situation the energy of the particle in the billiard is no longer conserved. We show a Fermi acceleration in such system: there exists a billiard trajectory on which the energy tends to infinity. The construction is based on the analysis of dynamics in the phase space near a homoclinic intersection of the stable and unstable manifolds of the normally hyperbolic invariant cylinder Λ, parameterised by the energy and time, that corresponds to the motion along the major axis of the ellipse. The proof depends on the reduction of the billiard map near the homoclinic channel to an iterated function system comprised by the shifts along two Hamiltonian flows defined on Λ. The two flows approximate the so-called inner and scattering maps, which are basic tools that arise in the studies of the Arnold diffusion; the scattering maps defined by the projection along the strong stable and strong unstable foliations Wss,uu of the stable and unstable invariant manifolds Ws,u(Λ) at the homoclinic points. Melnikov type calculations imply that the behaviour of the scattering map in this problem is quite unusual: it is only defined on a small subset of Λ that shrinks, in the large energy limit, to a set of parallel lines t = const as δ→0.