论文标题
在空间等边限制的四个身体问题中的计算机辅助证明同层混乱
Computer assisted proof of homoclinic chaos in the spatial equilateral restricted four body problem
论文作者
论文摘要
我们开发了计算机辅助论点，以证明存在横向同型连接轨道的存在，并将这些论点应用于空间等边圆形限制的四个身体问题中的许多非扰动参数和能量值。这个想法是要制定所需的连接轨道作为轨道段的某些两个点边界值问题的解决方案，这些轨道片段源于周期性轨道附加的局部稳定/不稳定流形。这些边界价值问题是通过牛顿 - 坎多维奇的参数研究的，该论点是在Chebyshev系数快速衰减序列的Banach代数的适当笛卡尔产物中研究。问题中最微妙的部分也许是控制边界条件，这必须位于周期轨道的局部稳定/不稳定的歧管上。对于问题的这一部分，我们使用一种参数化方法来开发配备有A-posteriori误差边界的傅里叶塔勒近似值。这需要在适当的笛卡尔产物中通过牛顿 - 坎多维奇参数进行有限数量的傅里叶泰勒系数的验证计算，以快速衰减的傅立叶系数序列，然后进行固定点参数，以绑定泰勒膨胀的尾巴项。横向性是牛顿 - 坎多维奇论点的结果。
We develop computer assisted arguments for proving the existence of transverse homoclinic connecting orbits, and apply these arguments for a number of non-perturbative parameter and energy values in the spatial equilateral circular restricted four body problem. The idea is to formulate the desired connecting orbits as solutions of certain two point boundary value problems for orbit segments which originate and terminate on the local stable/unstable manifolds attached to a periodic orbit. These boundary value problems are studied via a Newton-Kantorovich argument in an appropriate Cartesian product of Banach algebras of rapidly decaying sequences of Chebyshev coefficients. Perhaps the most delicate part of the problem is controlling the boundary conditions, which must lie on the local stable/unstable manifolds of the periodic orbit. For this portion of the problem we use a parameterization method to develop Fourier-Taylor approximations equipped with a-posteriori error bounds. This requires validated computation of a finite number of Fourier-Taylor coefficients via Newton-Kantorovich arguments in appropriate Cartesian product of rapidly decaying sequences of Fourier coefficients, followed by a fixed point argument to bound the tail terms of the Taylor expansion. Transversality follows as a consequence of the Newton-Kantorovich argument.
