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.
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. JOURNAL OF VIBRATION AND SHOCK, 2017, 36(17): 262-266.
[1] 贾 蒙. 映射动力系统一维流形并行计算方法 振动与冲击 2014,33(9):40-47.
JIA Meng, A parallel algorithm for approximating 1-D manifold of map [J].Journal of vibration and shock 2014,33(9):40-47.
[2] H. Gao, Y. Zhang, S. Liang and D.A. Li, New chaotic algorithm for image encryption[J]Chaos Solutions and Fractals (29) (2006), pp. 393–399
[3] P. Grassberger. On the fractal dimension of the Henon attractor [J] Physics Letters A 1983 97(6):224-226
[4] Palis J, Melo W D. Geometric Theory of Dynamical Systems[M].Springer, 1982.
[5]You, Z., Kostelich, E.J., Yorke, J.A.: Calculating stable and unstable manifolds. Int. J. Bifurc. Chaos Appl.Sci. Eng. 1(3), 605–623 (1991)
[6]Simó, C.: On the analytical and numerical approximation of invariant manifolds. In: Benest, D., Froeschlé,C. (eds.) Les Méthodes Modernes de la Mécanique Céleste, pp. 285–330. Goutelas (1989)
[7]Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, Berlin (1989)
[8]Hobson, D.: An efficient method for computing invariant manifolds. J. Comput. Phys. 104, 14–22 (1991)
[9]B. Krauskopf, H. M. Osinga, Growing unstable manifolds of planar maps, 1517, 1997, http://www.ima.umn.edu/preprints/OCT97/1517.ps.gz.
[10]Krauskopf, B., Osinga, H.M.: Growing 1D and quasi-2D unstable manifolds of maps. J. Comput. Phys.146, 406–419 (1998b)
[11]J. P. England, B. Krauskopf, and H. M. Osinga. Computing One-Dimensional Stable Manifolds and Stable Sets of Planar Maps without the Inverse. SIAM J. Appl. Dyn. Syst. Volume 3, Issue 2, pp. 161-190 (2004)
[12]Dellnitz, M., Hohmann, A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75, 293–317 (1997)
[13]Danny Fundinger. Toward the Calculation of Higher-Dimensional Stable Manifolds and Stable Sets for Noninvertible and Piecewise-Smooth Maps. J Nonlinear Sci (2008) 18: 391–413
[14]Krauskopf, B., Osinga, H.M.: Globalizing two-dimensional unstable manifolds of maps. Int. J. Bifurc.Chaos 8(3), 483–503 (1998a)