高维非线性映射系统的不稳定流形计算方法研究

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振动与冲击 ›› 2017, Vol. 36 ›› Issue (17) : 262-266.

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贾蒙

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The computation of Unstable Manifold for Hénon Map

Jia Meng

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摘要

提出了一种基于曲率约束条件的计算离散动力系统鞍型不动点一维不稳定流形的新算法，并以Hénon映射为例进行了计算。新算法以增长流形为基本思想，通过曲率约束和距离控制来确定离散点间的距离；提出流形的偏转角度可以通过流形上的已知点来预测，解决了流形上新点的原像位置快速确定的困难。仿真发现：Hénon映射的一维不稳定流形在标准参数下与Hénon映射产生的散点图分布一致，在其它几组参数下，一维不稳定流形的两个分支之间保持着某种程度的对称性，本研究对Hénon映射的进一步研究打下基础。

Abstract

A new algorithm is presented for computing one dimensional unstable manifold of a map and Hénon map is taken as an example to test the performance of the algorithm. The unstable manifold is grown with new point added at each step and the distance between consecutive points is adjusted according to the local curvature. It is proved that the gradient of the manifold at the new point can be predicted by the known points on the manifold and in this way the preimage of the new point could be located immediately. During the simulation, it is found that the unstable manifold of Hénon map coincides with its direct iteration when canonical parameters are chosen which means order is obtained out of chaos. In the other several groups of parameters the two branches of the unstable manifolds are nearly symmetric, and they serve as the borderline of the Hénon map iteration sequence. We hope that this would contribute to the further exploration of Hénon map.

导出引用

贾蒙. 高维非线性映射系统的不稳定流形计算方法研究[J]. 振动与冲击, 2017, 36(17): 262-266

Jia Meng. The computation of Unstable Manifold for Hénon Map[J]. Journal of Vibration and Shock, 2017, 36(17): 262-266

参考文献

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