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Global analysis of stochastic systems by the digraph cell mapping method based on short-time Gaussian approximation. (English) Zbl 1457.60109

Summary: The digraph cell mapping method is popular in the global analysis of stochastic systems. Traditionally, the Monte Carlo simulation is used in finding the image cells of one-step mapping, and it is notably costly in the computation time. In this paper, a novel short-time Gaussian approximation (STGA) scheme is incorporated into the digraph cell mapping method to study the global analysis of nonlinear dynamical systems under Gaussian white noise excitations. In order to find out all the active image cells in one-step cell mapping quickly, the STGA scheme together with a probability truncation method is introduced for systems without periodic excitation, and then in the case with periodic excitation. The global structures, such as the stochastic attractors, stochastic basins of attraction and stochastic saddles, are calculated by the digraph analysis algorithm. The proposed methodology has been applied to three typical stochastic dynamical systems. For each system, the effectiveness and superiority of the proposed STGA scheme are verified by checking the image cells of one-step mapping and comparing with the results of Monte Carlo simulation. It is found in the global analysis that the change of the amplitude of periodic excitation induces stochastic bifurcations in the stochastic Duffing system. Moreover, a stochastic bifurcation occurs in the stochastic Lorenz system with the increase of noise intensities.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
37H10 Generation, random and stochastic difference and differential equations
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[1] Arnold, L. [1995] Random Dynamical Systems (Springer-Verlag, Berlin). · Zbl 0834.58026
[2] Belardinelli, P. & Lenci, S. [2016] “ An efficient parallel implementation of cell mapping methods for MDOF systems,” Nonlin. Dyn.86, 2279-2290.
[3] Belardinelli, P., Lenci, S. & Rega, G. [2018] “ Seamless variation of isometric and anisometric dynamical integrity measures in basins’s erosion,” Commun. Nonlin. Sci. Numer. Simul.56, 499-507.
[4] Benettin, G., Galgani, L., Giorgilli, A. & Strelcyn, J.-M. [1980] “ Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Part 2: Numerical application,” Meccanica15, 21-30. · Zbl 0488.70015
[5] Chen, Z. & Liu, X. B. [2017] “ Noise induced transitions and topological study of a periodically driven system,” Commun. Nonlin. Sci. Numer. Simul.48, 454-461.
[6] 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
[7] Frey, M. & Simiu, E. [1993] “ Noise-induced chaos and phase space flux,” Physica D63, 321-340. · Zbl 0768.60039
[8] Gan, C. B. & Lei, H. [2011] “ A new procedure for exploring chaotic attractors in nonlinear dynamical systems under random excitations,” Acta Mech. Sin.27, 593-601. · Zbl 1270.70065
[9] Ge, Z. M. & Lee, S. C. [1996] “ Analysis of random dynamical systems by interpolated cell mapping,” J. Sound Vibr.194, 521-536. · Zbl 1232.70040
[10] Gong, P. L. & Xu, J. X. [2001] “ Global dynamics and stochastic resonance of the forced FitzHugh-Nagumo neuron model,” Phys. Rev. E63, 031906.
[11] Grebogi, C., Ott, E. & Yorke, J. A. [1983] “ Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D7, 181-200. · Zbl 0561.58029
[12] Gyebrószki, G. & Csernák, G. [2017] “ Clustered simple cell mapping: An extension to the simple cell mapping method,” Commun. Nonlin. Sci. Numer. Simul.42, 607-622.
[13] Han, Q., Xu, W. & Yue, X. [2014] “ Global bifurcation analysis of a Duffing-van der Pol oscillator with parametric excitation,” Int. J. Bifurcation and Chaos24, 1450051. · Zbl 1296.37043
[14] Han, Q., Xu, W. & Sun, J.-Q. [2016] “ Stochastic response and bifurcation of periodically driven nonlinear oscillators by the generalized cell mapping method,” Physica A458, 115-125. · Zbl 1400.60090
[15] Han, Q., Yue, X., Chi, H. & Chen, S. [2019] “ Stochastic response and bifurcations of a dry friction oscillator with periodic excitation based on a modified short-time Gaussian approximation scheme,” Nonlin. Dyn.96, 2001-2011.
[16] 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
[17] Hsu, C. S. [1987] Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems (Springer-Verlag, NY). · Zbl 0632.58002
[18] Hsu, C. S. [1995] “ Global analysis of dynamical systems using posets and digraphs,” Int. J. Bifurcation and Chaos5, 1085-1118. · Zbl 0886.58060
[19] Kong, C., Gao, X. & Liu, X. B. [2016] “ On the global analysis of a piecewise linear system that is excited by a Gaussian white noise,” J. Comput. Nonlin. Dyn.11, 051029.
[20] Li, S. B., Gong, X. J., Zhang, W. & Hao, Y. X. [2017] “ The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold,” Nonlin. Dyn.89, 939-953. · Zbl 1384.34049
[21] Li, Z. G., Jiang, J., Li, J., Hong, L. & Li, M. [2019] “ A subdomain synthesis method for global analysis of nonlinear dynamical systems based on cell mapping,” Nonlin. Dyn.95, 715-726.
[22] Liu, X. J., Hong, L. & Jiang, J. [2016] “ Global bifurcations in fractional-order chaotic systems with an extended generalized cell mapping method,” Chaos26, 084304. · Zbl 1378.34017
[23] Liu, X. M., Jiang, J., Hong, L. & Tang, D. [2018] “ Global bifurcation involving Wada boundary metamorphosis by a method of generalized cell mapping with sampling-adaptive interpolation,” Int. J. Bifurcation and Chaos28, 1830003. · Zbl 1388.34031
[24] Serdukova, L., Zheng, Y., Duan, J. & Kurths, J. [2016] “ Stochastic basins of attraction for metastable states,” Chaos26, 327-356. · Zbl 1375.37146
[25] Siewe, M. S., Tchawoua, C. & Woafo, P. [2010] “ Melnikov chaos in a periodically driven Rayleigh-Duffing oscillator,” Mech. Res. Commun.37, 363-368. · Zbl 1272.70116
[26] Sun, J.-Q. & Hsu, C. S. [1987] “ Cumulant-neglect closure method for nonlinear systems under random excitations,” J. Appl. Mech.54, 649-655. · Zbl 0618.73101
[27] Sun, J.-Q. & Hsu, C. [1988] “ A statistical study of generalized cell mapping,” J. Appl. Mech.55, 694-701. · Zbl 0659.70028
[28] 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.
[29] Sun, J.-Q. & Hsu, C. S. [1991] “ Effects of small random uncertainties on non-linear systems studied by the generalized cell mapping method,” J. Sound Vibr.147, 185-201.
[30] Sun, J.-Q. [1995] “ Random vibration analysis of a non-linear system with dry friction damping by the short-time Gaussian cell mapping method,” J. Sound Vibr.180, 785-795.
[31] Sun, J.-Q. [2006] Stochastic Dynamics and Control (Elsevier).
[32] Wiggins, S. [1990] Global Bifurcations and Chaos: Analytical Methods (Springer-Verlag, NY). · Zbl 0661.58001
[33] Wu, W. F. & Lin, Y. K. [1984] “ Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations,” Int. J. Non-Lin. Mech.19, 349-362. · Zbl 0551.70018
[34] 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, 111001.
[35] Xu, W., He, Q., Fang, T. & Rong, H. W. [2003] “ Global analysis of stochastic bifurcation in Duffing system,” Int. J. Bifurcation and Chaos13, 3115-3123. · Zbl 1078.37512
[36] Xu, W., He, Q., Fang, T. & Rong, H. W. [2004] “ Stochastic bifurcation in Duffing system subject to harmonic excitation and in presence of random noise,” Int. J. Non-Lin. Mech.39, 1473-1479. · Zbl 1348.70063
[37] Xu, J. X. [2009] “ Some advances on global analysis of nonlinear systems,” Chaos Solit. Fract.39, 1839-1848.
[38] Xu, Y., Liu, Q., Guo, G. B., Xu, C. & Liu, D. [2017] “ Dynamical responses of airfoil models with harmonic excitation under uncertain disturbance,” Nonlin. Dyn.89, 1579-1590.
[39] 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
[40] Zhang, W., Yao, M. H. & Zhang, J. H. [2009] “ Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam,” J. Sound Vibr.319, 541-569.
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