Efficient and Reliable Algorithms for the Computation of Non-Twist Invariant Circles
Efficient and Reliable Algorithms for the Computation of Non-Twist Invariant Circles
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高效可靠的非扭转不变圆计算算法
DOI：
10.1007/s10208-021-09517-9
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发表时间：
2022
影响因子：
3
通讯作者：
de la Llave, Rafael
中科院分区：
文献类型：
--
作者：
González, Alejandra;Haro, Àlex;de la Llave, Rafael
本文提出了一种研究面积保留地图的非扭转不变圆及其分叉的方法，该方法得到了 Gonzalez-Enriquez 等人开发的理论框架的支持。 （Mem. Amer. Math. Soc. 227:vi+115, 2014）。我们记得，非扭曲不变圆的特征不仅是不变，而且还具有一些特定的正常行为。正常行为可能赋予它们额外的稳定性（例如，抵抗外部噪声），因此，它们在某些应用中出现为设计目标，例如在等离子体物理学、天体动力学和海洋学中。该方法导致有效的算法来计算和继续关于参数的非扭转不变圆。这些算法是二次收敛的，并且在使用 FFT 实现时，存储要求低，每步操作数也低。此外，这些算法得到严格的后验定理的支持，并在 Gonzalez-Enriquez 等人中详细证明和讨论。 (Mem. Amer. Math. Soc. 227:vi+115, 2014)，它给出了保证真实非扭转不变圆存在的充分条件，前提是近似不变圆已知。因此，即使非常接近崩溃，人们也可以自信地进行计算。通过一些额外的努力，计算可以转化为计算机辅助证明，参见 Figueras 等人。 （Found.Comput.Math.17:1123–1193, 2017）了解后者的示例。该算法还保证收敛到不变圆的崩溃，然后，它们适合计算存在非扭曲不变圆的参数区域。这些区域边界的计算所涉及的计算非常稳健，并且不需要对称性，无需连续的手动调整即可运行，极大地改进了基于计算超长周期轨道以近似不变圆的方法。本文详细描述了我们的算法、相应的实现以及通过运行计算机程序获得的一些数值结果。特别是，我们包括对存在非扭曲不变圆（具有规定频率）的二维参数区域的计算。事实上，我们在不包含对称线的系统中提出了系统结果，这似乎是以前的方法无法获得的。这些数值探索引出了一些悬而未决的问题，也包含在这里。
This paper presents a methodology to study non-twist invariant circles and their bifurcations for area preserving maps, which is supported on the theoretical framework developed in Gonzalez-Enriquez et al. (Mem. Amer. Math. Soc. 227:vi+115, 2014). We recall that non-twist invariant circles are characterized not only by being invariant, but also by having some specified normal behavior. The normal behavior may endow them with extra stability properties (e.g., against external noise), and hence, they appear as design goals in some applications, e.g., in plasma physics, astrodynamics and oceanography. The methodology leads to efficient algorithms to compute and continue, with respect to parameters, non-twist invariant circles. The algorithms are quadratically convergent and, when implemented using FFT, have low storage requirement and low operations count per step. Furthermore, the algorithms are backed up by rigorousa posterioritheorems, proved and discussed in detail in Gonzalez-Enriquez et al. (Mem. Amer. Math. Soc. 227:vi+115, 2014), which give sufficient conditions guaranteeing the existence of a true non-twist invariant circle, provided an approximate invariant circle is known. Hence, one can compute confidently even very close to breakdown. With some extra effort, the calculations could be turned into computer-assisted proofs, see Figueras et al. (Found. Comput. Math. 17:1123–1193, 2017) for examples of the latter. The algorithms are also guaranteed to converge up to the breakdown of the invariant circles, and then, they are suitable to compute regions of parameters where the non-twist invariant circles exist. The calculations involved in the computation of the boundary of these regions are very robust, and they do not require symmetries and can run without continuous manual adjustments, largely improving methods based on the computation of very long period periodic orbits to approximate invariant circles. This paper contains a detailed description of our algorithms, the corresponding implementation and some numerical results, obtained by running the computer programs. In particular, we include calculations for two-dimensional parameter regions where non-twist invariant circles (with a prescribed frequency) exist. Indeed, we present systematic results in systems that do not contain symmetry lines, which seem to be unaccessible for previous methods. These numerical explorations lead to some open questions, also included here.
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