Chebyshev-Taylor参数化稳定/不稳定流形用于周期轨道：实现与应用
Chebyshev-Taylor parameterization of stable/unstable manifolds for periodic orbits: implementation and applications
AI总结：
本文提出一种高阶Chebyshev-Taylor级数近似方法，用于研究微分方程周期轨道的稳定/不稳定流形，通过求解边界值问题计算周期解，并应用于洛伦兹系统和环形限制三体问题。
摘要：
本文开发了一种用于研究微分方程周期轨道附加稳定/不稳定流形的计算方法。该方法结合高阶Chebyshev-Taylor级数近似与参数化方法，后者是一种通用泛函分析框架，用于不变流形。参数化方法可以跟踪嵌入中的折叠，通过简单的共轭恢复流形上的动力学，并允许自然的后验误差分析。该方法的关键在于推导描述不变流形Taylor系数的递推线性微分方程系统。通过求解耦合的边界值问题，利用Chebyshev谱方法找到这些方程的周期解。本文讨论了该方法在洛伦兹系统、环形限制三体和四体问题中的性能，并展示了该方法作为计算周期间连接轨道工具的应用。
英文摘要：
This paper develops a computational method for studying stable/unstable manifolds attached to periodic orbits of differential equations. The method uses high order Chebyshev-Taylor series approximations in conjunction with the parameterization method -- a general functional analytic framework for invariant manifolds. The parameterization method can follow folds in the embedding, recovers the dynamics on the manifold through a simple conjugacy, and admits a natural notion of a-posteriori error analysis. The key to the approach is the derivation of a recursive system of linear differential equations describing the Taylor coefficients of the invariant manifold. We find periodic solutions of these equations by solving a coupled collection of boundary value problems with Chebyshev spectral methods. We discuss the performance of the method for the Lorenz system, and for circular restricted three and four body problems. We also illustrate the use of the method as a tool for computing cycle-to-cycle connecting orbits.
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