Dynamic analysis of the nonlinear chaotic system with multistochastic disturbances. (English) Zbl 1463.37022

Summary: The nonlinear chaotic system with multistochastic disturbances is investigated. Based on the orthogonal polynomial approximation, the method of transforming the system into an equivalent deterministic system is given. Then dynamic analysis of the nonlinear chaotic system with multistochastic disturbances can be reduced into that of its equivalent deterministic system. Especially, the Lorenz system with multistochastic disturbances is studied to demonstrate the feasibility of the given method. And its dynamic behaviors are gained including the phase portrait, the bifurcation diagram, the Poincaré section, and the maximum Lyapunov exponent.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
65C30 Numerical solutions to stochastic differential and integral equations
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
70L05 Random vibrations in mechanics of particles and systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
Full Text: DOI


[1] Sprott, J. C., Chaos and Time-Series Analysis (2004), New York, NY, USA: Oxford University Press, New York, NY, USA
[2] Chen, G.; Dong, X., From Chaos to Order: Methodologies, Perspectives and Applications (1998), Singapore: World Scientific, Singapore
[3] Chen, H. K.; Lin, T. N.; Chen, J. H., Dynamic analysis, controlling chaos and chaotification of a SMIB power system, Chaos, Solitons and Fractals, 24, 5, 1307-1315 (2005) · Zbl 1092.37510
[4] Yu, Y. G.; Zhang, S. C., Hopf bifurcation analysis of the Lü system, Chaos, Solitons and Fractals, 21, 5, 1215-1220 (2004) · Zbl 1061.37029
[5] Elabbasy, E. M.; Elsadany, A. A.; Zhang, Y., Bifurcation analysis and chaos in a discrete reduced Lorenz system, Applied Mathematics and Computation, 228, 184-194 (2014)
[6] Stratonovich, R. L., Topics in the Theory of Random Noise, 2 (1967), New York, NY, USA: Gorden and Breach Science Publishers, New York, NY, USA
[7] Crandall, S. H.; Mark, W. D., Random Vibration in Mechanical Systems (1963), New York, NY, USA: Academic Press, New York, NY, USA
[8] Li, W.; Xu, W.; Zhao, J.; Jin, Y., Stochastic stability and bifurcation in a macroeconomic model, Chaos, Solitons and Fractals, 31, 3, 702-711 (2007) · Zbl 1133.91484
[9] Jin, Y. E.; Xu, W.; Xu, M., Stochastic resonance in an asymmetric bistable system driven by correlated multiplicative and additive noise, Chaos, Solitons and Fractals, 26, 4, 1183-1187 (2005) · Zbl 1096.94509
[10] Shinozuka, M., Monte Carlo solution of structural dynamics, Computers & Structures, 2, 5-6, 855-874 (1972)
[11] Kleiber, M.; Hien, T. D., The Stochastic Finite Element Method: Basic Perturbation Technique and Computer Implementation (1992), Wiley Press
[12] Ghamem, R.; Spans, P., Stochastic Finite Element: A Spectral Approach (1991), Berlin, Germany: Springer, Berlin, Germany
[13] Jensen, H.; Iwan, W. D., Response of systems with uncertain parameters to stochastic excitation, Journal of Engineering Mechanics, 118, 5, 1012-1025 (1992)
[14] Li, J., The expanded order system method for combined random vibration analysis, Acta Mechanica Sinica. Lixue Xuebao, 28, 1, 66-75 (1996)
[15] Xiu, D.; Karniadakis, G. E., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM Journal on Scientific Computing, 24, 2, 619-644 (2002) · Zbl 1014.65004
[16] Xiu, D.; Karniadakis, G. E., Modeling uncertainty in flow simulations via generalized polynomial chaos, Journal of Computational Physics, 187, 1, 137-167 (2003) · Zbl 1047.76111
[17] le Maître, O. P.; Najm, H. N.; Ghanem, R. G.; Knio, O. M., Multi-resolution analysis of Wiener-type uncertainty propagation schemes, Journal of Computational Physics, 197, 2, 502-531 (2004) · Zbl 1056.65006
[18] Pettit, C. L.; Beran, P. S., Spectral and multiresolution Wiener expansions of oscillatory stochastic processes, Journal of Sound and Vibration, 294, 4, 752-779 (2006)
[19] Kim, H.; Kim, Y.; Yoon, D., Dependence of polynomial chaos on random types of forces of KdV equations, Applied Mathematical Modelling, 36, 7, 3080-3093 (2012) · Zbl 1252.60063
[20] Fang, T.; Leng, X. L.; Song, C. Q., Chebyshev polynomial approximation for dynamical response problem of random system, Journal of Sound and Vibration, 266, 1, 198-206 (2003) · Zbl 1236.70045
[21] Fang, T.; Leng, X.; Ma, X.; Meng, G., λ-PDF and Gegenbauer polynomial approximation for dynamic response problems of random structures, Acta Mechanica Sinica, 20, 3, 292-298 (2004)
[22] Ma, S. J.; Xu, W.; Fang, T., Analysis of period-doubling bifurcation in double-well stochastic Duffing system via Laguerre polynomial approximation, Nonlinear Dynamics, 52, 3, 289-299 (2008) · Zbl 1170.70358
[23] Ma, S. J.; Xu, W., Period-doubling bifurcation in an extended van der Pol system with bounded random parameter, Communications in Nonlinear Science and Numerical Simulation, 13, 10, 2256-2265 (2008)
[24] Li, W.; Xu, W.; Zhao, J.; Ma, S., Stochastic optimal control of first-passage failure for coupled Duffing-van der Pol system under Gaussian white noise excitations, Chaos, Solitons and Fractals, 25, 5, 1221-1228 (2005) · Zbl 1142.93443
[25] Ma, S. J., The orthogonal polynomial approximation analysis of the stochastic bifurcation [M.S. thesis] (2005), Northwestern Polytechnical University
[26] Wang, X. Z.; Guo, D. R., Special Functions (1989), Singapore: World Scientific Publishing, Singapore
[27] Li, X.; Chlouverakis, K. E.; Xu, D., Nonlinear dynamics and circuit realization of a new chaotic flow: a variant of Lorenz, Chen and Lü, Nonlinear Analysis: Real World Applications, 10, 4, 2357-2368 (2009) · Zbl 1163.34306
[28] Dadras, S.; Momeni, H. R.; Qi, G., Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos, Nonlinear Dynamics, 62, 1-2, 391-405 (2010) · Zbl 1242.37025
[29] Jafari, S.; Sprott, J. C.; Golpayegani, S. M. R. H., Elementary quadratic chaotic flows with no equilibria, Physics Letters A, 377, 9, 699-702 (2013)
[30] Algaba, A.; Fernández-Sánchez, F.; Merino, M.; Rodríguez-Luis, A. J., Chens attractor exists if Lorenz repulsor exists: the Chen system is a special case of the Lorenz system, Chaos, 23 (2013)
[31] Algaba, A.; Fernández-Sánchez, F.; Merino, M.; Rodríguez-Luis, A. J., The Lü system is a particular case of the Lorenz system, Physics Letters A, 377, 39, 2771-2776 (2013) · Zbl 1301.37020
[32] Lorenz, E. N., Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20, 2, 130-141 (1963) · Zbl 1417.37129
[33] Moghtadaei, M.; Hashemi Golpayegani, M. R., Complex dynamic behaviors of the complex Lorenz system, Scientia Iranica, 19, 3, 733-738 (2012)
[34] Tee, L. S.; Salleh, Z., Dynamical analysis of a modified Lorenz system, Journal of Mathematics, 2013 (2013) · Zbl 1297.37016
[35] Knobloch, E., On the statistical dynamics of the Lorenz model, Journal of Statistical Physics, 20, 6, 695-709 (1979)
[36] Song, T.; Cattani, C., Fast detection of weak singularities in a chaotic signal using Lorenz system and the bisection algorithm, Mathematical Problems in Engineering, 2012 (2012) · Zbl 1264.94069
[37] Weiland, J.; Mondt, J. P.; Gerwin, R. A., Complex Lorenz system for collisional inhomogeneous plasmas, Physical Review A, 34, 1, 647-650 (1986)
[38] Lin, X. S.; Yu, Y. G.; Wang, H., Dynamics analysis of the stochastic Lorenz system, Proceedings of the 4th International Workshop on Chaos-Fractals Theories and Applications (IWCFTA ’11)
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