不变流形演进方向对非双曲型非线性时间序列重影算法的影响分析
The Effects of Homoclinic Tangencies and Homoclinic Intersectionons on Shadowing Algorithm for Series of Non-Hyperbolic Nonlinear Systems

DOI: 10.12677/AIRR.2016.53007, PP. 63-72

Keywords: 非双曲型非线性系统，轨迹重影，Newton-Raphson算法，梯度下降，同宿切面，同宿横截点
Non-Hyperbolic Nonlinear System, Shadowing, Newton-Raphson Algorithm, Gradient Descent, Homoclinic Tangencies, Homoclinic Intersections

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Abstract:

非双曲型非线性系统同宿切面点和同宿横截点的存在，使得在机器精度内实现其时间序列轨迹重影变得十分困难。本文从原理上分析了同宿切面点对轨迹重影算法的影响，并给出可降低甚或避免同宿切面点对算法性能影响的措施。不同于现有文献认为轨迹重影算法仅受同宿切面点的影响而忽视同宿横截点对算法性能影响的做法，研究得出了同宿横截点间的最小距离和干扰噪声均方差二者间的关系，首次定量地估计了同宿横截点可能对算法造成的影响。
The presence of homoclinic tangencies and homoclinic intersections makes it very difficult, sometimes even impossible, to shadow the trajectory of the non-hyperbolic nonlinear system. Different from former methods, this paper analyzed the effects of the homoclinic tangencies on the algorithm, and proposed methods that can decrease, or even avoid the effects. Different from those methods which take it for granted that the failure of denoising algorithms is related with the homoclinic tangencies only, experiments in this paper demonstrate a quantitative correlation between the minimal distance of homoclinic intersections and the standard variance of noise. Thus the probability that the algorithm converges to the true trajectory could be boosted efficiently, and without any doubts, this strategy would be a heuristic approach to other similar methods.

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