Analysis of period-doubling bifurcation in double-well stochastic Duffing system via Laguerre polynomial approximation. (English) Zbl 1170.70358

Summary: The Laguerre polynomial approximation method is applied to study the stochastic period-doubling bifurcation of a double-well stochastic Duffing system with a random parameter of exponential probability density function subjected to a harmonic excitation. First, the stochastic Duffing system is reduced into its equivalent deterministic one, solvable by suitable numerical methods. Then nonlinear dynamical behavior about stochastic period-doubling bifurcation can be fully explored. Numerical simulations show that similar to the conventional period-doubling phenomenon in the deterministic Duffing system, stochastic period-doubling bifurcation may also occur in the stochastic Duffing system, but with its own stochastic modifications. Also, unlike the deterministic case, in the stochastic case the intensity of the random parameter should also be taken as a new bifurcation parameter in addition to the conventional bifurcation parameters, i.e. the amplitude and the frequency of harmonic excitation.


70K50 Bifurcations and instability for nonlinear problems in mechanics
70L05 Random vibrations in mechanics of particles and systems
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[1] Zhu, W.Q., Huang, Z.L., Suzuki, Y.: Response and stability of strongly non-linear oscillators under wide-band random excitation. Int. J. Non-Linear Mech. 36, 1235–1250 (2001) · Zbl 1258.70037
[2] Rong, H.W., Meng, G., Wang, X.D., Fang, T.: Response of strongly non-linear oscillators to narrowband random excitation. J. Sound Vib. 266, 875–887 (2003) · Zbl 1236.34055
[3] Xu, W., He, Q., Fang, T., Rong, H.W.: Global analysis of stochastic in Duffing system. Int. J. Bifur. Chaos 13, 3115–3123 (2003) · Zbl 1078.37512
[4] Shinozuka, M.: Probability modeling of concrete structures. ASCE J. Eng. Mech. Div. 98, 1433–1451 (1972)
[5] Kleiber, M., Hien, T.D.: The Stochastic Finite Element Method: Basic Perturbation Technique and Computer Implementation. Wiley, New York (1992) · Zbl 0902.73004
[6] Ghamem, R., Spans, P.: Stochastic Finite Element: A Spectral Approach. Springer, Berlin (1991)
[7] Pettit, C.L., Beran, P.S.: Spectral and multiresolution Wiener expansions of oscillatory stochastic processes. J. Sound Vib. 294, 752–779 (2006)
[8] Le Maître, O.P., Najm, H.N., Ghanem, R.G., Knio, O.M.: Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197, 502–531 (2004) · Zbl 1056.65006
[9] Le Maître, O.P., Knio, O.M., Najm, H.N., Ghanem, R.G.: Uncertainty propagation using Wiener–Haar expansions. J. Comput. Phys. 197, 28–57 (2004) · Zbl 1052.65114
[10] Xiu, D.B., Karniadakis, G.E.: Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Eng. 191, 4927–4948 (2002) · Zbl 1016.65001
[11] Wan, X.L., Karniadakis, G.E.: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209, 617–642 (2005) · Zbl 1078.65008
[12] Fang, T., Leng, X.L., Song, C.Q.: Chebyshev polynomial approximation for dynamical response problem of random system. J. Sound Vib. 226, 198–206 (2003) · Zbl 1236.70045
[13] Leng, X.L., Wu, C.L., Ma, X.P., Meng, G., Fang, T.: Bifurcation and chaos analysis of stochastic Duffing system under harmonic excitations. Nonlinear Dyn. 42, 185–198 (2005) · Zbl 1105.70011
[14] Ma, S.J., Xu, W., Li, W., Fang, T.: Analysis of stochastic bifurcation and chaos in stochastic Duffing–van der Pol system via Chebyshev polynomial approximation. Chin. Phys. 15, 1231–1238 (2006)
[15] Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillators, Dynamical System and Bifurcation of Vector Fields. Springer, New York (1983) · Zbl 0515.34001
[16] Holmes, P.: A nonlinear oscillator with a strange attractor. Philos. Trans. Roy. Soc. Lond. Ser. A 292, 419–448 (1979) · Zbl 0423.34049
[17] Moon, F.C., Homles, P.J.: A megnetoelastic strange attractor. J. Sound Vib. 65, 275–296 (1979) · Zbl 0405.73082
[18] Osipov, G., Glatz, L., Troger, H.: Suppressing chaos in the Duffing oscillator by impulsive actions. Chaos Solitons Fractals 9, 307–321 (1998) · Zbl 0934.37036
[19] Litvak-Hinenzon, A., Rom-Kedar, V.: Symmetry-breaking perturbations and strange attractors. Phys. Rev. E 55, 4964–4978 (1997)
[20] Chacón, R., García-Hoz, A.M.: Bifurcation and chaos in a parametrically damped double-well Duffing oscillator subjected to symmetric periodic pulses. Phys. Rev. E 59, 6558–6568 (1999)
[21] Kim, S.Y., Kim, Y.: Dynamic stabilization in the double-well Duffing oscillator. Phys. Rev. E 61, 6517–6520 (2000)
[22] Szmplińskia-Stupnicka, W., Zubrzycki, A., Tyrkiel, E.: New phenomena in the neighborhood of the codimension-two bifurcation in the twin-well Duffing oscillator. Int. J. Bifur. Chaos 10, 1367–1381 (2000) · Zbl 1090.70510
[23] Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequality. Springer, New York (1995) · Zbl 0840.26002
[24] Kamerich, E.: A Guide to Maple. Springer, New York (1999) · Zbl 0926.68060
[25] Hu, H.Y.: Symmetry and bifurcation of the periodic response of an externally forced nonlinear oscillator. J. Vib. Eng. S 7, 26–36 (1994)
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