×

Global invariant manifolds of dynamical systems with the compatible cell mapping method. (English) Zbl 1422.37015

Summary: An iterative compatible cell mapping (CCM) method with the digraph theory is presented in this paper to compute the global invariant manifolds of dynamical systems with high precision and high efficiency. The accurate attractors and saddles can be simultaneously obtained. The simple cell mapping (SCM) method is first used to obtain the periodic solutions. The results obtained by the generalized cell mapping (GCM) method are treated as a database. The SCM and GCM are compatible in the sense that the SCM is a subset of the GCM. The depth-first search algorithm is utilized to find the coarse coverings of global stable and unstable manifolds based on this database. The digraph GCM method is used if the saddle-like periodic solutions cannot be obtained with the SCM method. By taking this coarse covering as a new cell state space, an efficient iterative procedure of the CCM method is proposed by combining sort, search and digraph algorithms. To demonstrate the effectiveness of the proposed method, the classical Hénon map with periodic or chaotic saddles is studied in far more depth than reported in the literature. Not only the global invariant manifolds, but also the attractors and saddles are computed. The computational efficiency can be improved by up to 200 times compared to the traditional GCM method.

MSC:

37D10 Invariant manifold theory for dynamical systems
37M20 Computational methods for bifurcation problems in dynamical systems

Software:

GAIO
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Belardinelli, P. & Lenci, S. [2016] “ An efficient parallel implementation of cell mapping methods for MDOF systems,” Nonlin. Dyn.86, 2279-2290. · Zbl 1448.65271
[2] Chiu, H. M. & Hsu, C. S. [1986] “ A cell mapping method for nonlinear deterministic and stochastic systems — Part II: Examples of application,” J. Appl. Mech.53, 702-710. · Zbl 0678.70026
[3] Crespo, L. G. & Sun, J. Q. [2002] “ Stochastic optimal control of nonlinear systems via short-time Gaussian approximation and cell mapping,” Nonlin. Dyn.28, 323-342. · Zbl 1018.93034
[4] Dellnitz, M. & Hohmann, A. [1997] “ A subdivision algorithm for the computation of unstable manifolds and global attractors,” Numer. Math.75, 293-317. · Zbl 0883.65060
[5] Dellnitz, M. & Junge, O. [1998] “ An adaptive subdivision technique for the approximation of attractors and invariant measures,” Comput. Vis. Sci.1, 63-68. · Zbl 0970.65130
[6] Dellnitz, M., Froyland, G. & Junge, O. [2001] “ The algorithms behind gaio — Set oriented numerical methods for dynamical systems,” Proc. Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 145-174. · Zbl 0998.65126
[7] Doedel, E., Keller, H. B. & Kernevez, J. P. [1991a] “ Numerical analysis and control of bifurcation problems (I): Bifurcation in finite dimensions,” Int. J. Bifurcation and Chaos1, 493-520. · Zbl 0876.65032
[8] Doedel, E., Keller, H. B. & Kernevez, J. P. [1991b] “ Numerical analysis and control of bifurcation problems (II): Bifurcation in infinite dimensions,” Int. J. Bifurcation and Chaos1, 745-772. · Zbl 0876.65060
[9] Eason, R. P. & Dick, A. J. [2014] “ A parallelized multi-degrees-of-freedom cell mapping method,” Nonlin. Dyn.77, 467-479.
[10] Golat, M. & Flashner, H. [2002] “ A new methodology for the analysis of periodic systems,” Nonlin. Dyn.28, 29-51. · Zbl 1065.34029
[11] Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer Science & Business Media). · Zbl 0515.34001
[12] Guckenheimer, J. & Vladimirsky, A. [2004] “ A fast method for approximating invariant manifolds,” SIAM J. Appl. Dyn. Syst.3, 232-260. · Zbl 1059.37019
[13] Guder, R., Dellnitz, M. & Kreuzer, E. [1997] “ An adaptive method for the approximation of the generalized cell mapping,” Chaos Solit. Fract.8, 525-534. · Zbl 0935.37055
[14] Guder, R. & Kreuzer, E. [1999] “ Control of an adaptive refinement technique of generalized cell mapping by system dynamics,” Nonlin. Dyn.20, 21-32. · Zbl 0980.70019
[15] Han, Q., Xu, W., Yue, X. L. & Zhang, Y. [2015] “ First-passage time statistics in a bistable system subject to Poisson white noise by the generalized cell mapping method,” Commun. Nonlin. Sci. Numer. Simul.23, 220-228. · Zbl 1351.60117
[16] He, Q., Xu, W., Rong, H. W. & Fang, T. [2004] “ Stochastic bifurcation in Duffing – van der Pol oscillators,” Physica A338, 319-334.
[17] Henderson, M. E. [2005] “ Computing invariant manifolds by integrating fat trajectories,” SIAM J. Appl. Dyn. Syst.4, 832-882. · Zbl 1090.37012
[18] Hong, L. & Xu, J. X. [1999] “ Crises and chaotic transients studied by the generalized cell mapping digraph method,” Phys. Lett. A262, 361-375. · Zbl 0940.37012
[19] Hong, L. & Sun, J. Q. [2006] “ Codimension two bifurcations of nonlinear systems driven by fuzzy noise,” Physica D213, 181-189. · Zbl 1104.34030
[20] Hsu, C. S. [1980] “ A theory of cell-to-cell mapping dynamical systems,” J. Appl. Mech.47, 931-939. · Zbl 0452.58019
[21] Hsu, C. S. [1981] “ A generalized theory of cell-to-cell mapping for nonlinear dynamical systems,” J. Appl. Mech.48, 634-642. · Zbl 0482.70017
[22] Hsu, C. S. & Chiu, H. M. [1986] “ A cell mapping method for nonlinear deterministic and stochastic systems — Part I: The method of analysis,” J. Appl. Mech.53, 695-701. · Zbl 0623.70016
[23] Hsu, C. S. [1992] “ Global analysis by cell mapping,” Int. J. Bifurcation and Chaos2, 727-771. · Zbl 0870.58034
[24] Hsu, C. S. [1995] “ Global analysis of dynamical systems using posets and digraphs,” Int. J. Bifurcation and Chaos5, 1085-1118. · Zbl 0886.58060
[25] Jiang, J. & Xu, J. X. [1994] “ A method of point mapping under cell reference for global analysis of nonlinear dynamical systems,” Phys. Lett. A188, 137-145. · Zbl 0941.37502
[26] Junge, O. & Kevrekidis, I. G. [2017] “ On the sighting of unicorns: A variational approach to computing invariant sets in dynamical systems,” Chaos27, 063102. · Zbl 1390.37140
[27] Krauskopf, B. & Osinga, H. M. [2003] “ Computing geodesic level sets on global (un)stable manifolds of vector fields,” SIAM J. Appl. Dyn. Syst.2, 546-569. · Zbl 1089.37014
[28] Krauskopf, B., Osinga, H. M., Doedel, E. J., Henderson, M. E., Guckenheimer, J., Vladimirsky, A., Dellnitz, M. & Junge, O. [2005] “ A survey of methods for computing (un)stable manifolds of vector fields,” Int. J. Bifurcation and Chaos15, 763-791. · Zbl 1086.34002
[29] Kreuzer, E. & Lagemann, B. [1996] “ Cell mappings for multi-degree-of-freedom-systems — Parallel computing in nonlinear dynamics,” Chaos Solit. Fract.7, 1683-1691. · Zbl 1080.37504
[30] Li, Z. G., Jiang, J. & Hong, L. [2015] “ Transient behaviors in noise-induced bifurcations captured by generalized cell mapping method with an evolving probabilistic vector,” Int. J. Bifurcation and Chaos25, 1550109-1-11. · Zbl 1321.60016
[31] Liu, X. M., Jiang, J., Hong, L. & Tang, D. F. [2018] “ Studying the global bifurcation involving Wada boundary metamorphosis by a method of generalized cell mapping with sampling-adaptive interpolation,” Int. J. Bifurcation and Chaos28, 1830003-1-19. · Zbl 1388.34031
[32] Sun, J. Q. & Hsu, C. S. [1988] “ First-passage time probability of non-linear stochastic systems by generalized cell mapping method,” J. Sound Vibr.124, 233-248. · Zbl 1235.70200
[33] Sun, J. Q. & Hsu, C. S. [1990] “ The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation,” J. Appl. Mech.57, 1018-1025.
[34] Sun, J. Q. & Luo, A. C. J. [2012] Global Analysis of Nonlinear Dynamics (Springer Science & Business Media, NY).
[35] Tongue, B. H. [1987] “ On obtaining global nonlinear system characteristics through interpolated cell mapping,” Physica D28, 401-408.
[36] Tongue, B. H. & Gu, K. [1988] “ Interpolated cell mapping of dynamical systems,” J. Appl. Mech.55, 461-466. · Zbl 0666.70019
[37] Xiong, F. R., Qin, Z. C., Ding, Q., Hernández, C., Fernandez, J., Schütze, O. & Sun, J. Q. [2015] “ Parallel cell mapping method for global analysis of high-dimensional nonlinear dynamical systems,” J. Appl. Mech.82, 111010.
[38] Xiong, F. R., Schütze, O., Ding, Q. & Sun, J. Q. [2016] “ Finding zeros of nonlinear functions using the hybrid parallel cell mapping method,” Commun. Nonlin. Sci. Numer. Simul.34, 23-37. · Zbl 1510.65099
[39] Xu, W., He, Q. & Li, S. [2007] “ The cell mapping method for approximating the invariant manifolds,” Proc. IUTAM Symp. Dynamics and Control of Nonlinear Systems with Uncertainty, pp. 117-126. · Zbl 1188.37030
[40] Yue, X. L., Xu, W., Wang, L. & Zhou, B. C. [2012a] “ Transient and steady-state responses in a self-sustained oscillator with harmonic and bounded noise excitations,” Probab. Eng. Mech.30, 70-76.
[41] Yue, X. L., Xu, W. & Zhang, Y. [2012b] “ Global bifurcation analysis of Rayleigh-Duffing oscillator through the composite cell coordinate system method,” Nonlin. Dyn.69, 437-457.
[42] Yue, X. L., Xu, W., Zhang, Y. & Du, L. [2018] “ Analysis of global properties for dynamical systems by a modified digraph cell mapping method,” Chaos Solit. Fract.111, 206-212. · Zbl 1398.65342
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.