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Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method. (English) Zbl 1183.70045

Summary: The hyperbolic perturbation method is applied to determining the homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators of the form \(\ddot{x}+c_{1}x+c_{3}x^{3}=\varepsilon f(\mu,x,\dot{x})\) , in which the hyperbolic functions are employed instead of the periodic functions in the usual perturbation method. The generalized Liénard oscillator with \(f(\mu,x,\dot{x})=(\mu -\mu_{1}x^{2}-\mu_{2}\dot{x}^{2})\dot{x}\) is studied in detail. Comparisons with the numerical simulations obtained by using R-K method are made to show the efficacy and accuracy of the present method.

MSC:

70K44 Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics
70K60 General perturbation schemes for nonlinear problems in mechanics
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[1] Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981) · Zbl 0449.34001
[2] Mickens, R.E.: Nonlinear Oscillations. Cambridge University Press, New York (1981) · Zbl 0459.34002
[3] Cheung, Y.K., Chen, S.H., Lau, S.L.: A modified Lindstedt–Poincaré method for certain strongly nonlinear oscillators. J. Non-Linear Mech. 26(3/4), 367–378 (1991) · Zbl 0755.70021
[4] Chen, S.H., Cheung, Y.K.: A modified Lindstedt–Poincare method for a strongly non-linear two degree-of-freedom system. J. Sound Vib. 193(4), 751–762 (1996) · Zbl 1232.70019
[5] Burton, T.D., Rahman, Z.: On the multi-scale analysis of strongly non-linear forced oscillators. J. Non-Linear Mech. 21(2), 135–146 (1986) · Zbl 0583.70016
[6] Ottoy, J.P.: A perturbation method for a set of purely non-linear differential equations. Int. J. Control 30(4), 587–595 (1979) · Zbl 0422.34066
[7] Dai, S.Q.: Asymptotic analysis of strongly non-linear oscillator. Appl. Math. Mech. (Engl. Ed.) 6, 409–415 (1985) · Zbl 0585.34039
[8] Barkham, P.G.D., Souback, A.C.: An extension to the method of Kryloff and Bogoliubov. Int. J. Control 10, 337–392 (1969) · Zbl 0176.46702
[9] Yuste, S.B., Bejarano, J.D.: Extension and improvement to the Krylov–Bogoliubov methods using elliptic functions. Int. J. Control 49, 1127–1141 (1989) · Zbl 0691.34029
[10] Coppola, V.T., Rand, R.H.: Averaging using elliptic function: approximation of limit cycles. Acta Mech. 81, 125–142 (1990) · Zbl 0699.34032
[11] Roy, R.V.: Averaging method for strongly non-linear oscillators with periodic excitations. J. Non-Linear Mech. 29, 737–753 (1994) · Zbl 0813.70014
[12] Chen, S.H., Cheung, Y.K.: An elliptic Lindstedt–Poincaré method for certain strongly non-linear oscillators. Nonlinear Dyn. 12, 199–213 (1997) · Zbl 0881.70015
[13] Chen, S.H., Yang, X.M., Cheung, Y.K.: Periodic solutions of strongly quadratic non-linear oscillators by the elliptic Lindstedt–Poincaré method. J. Sound Vib. 227, 1109–1118 (1999) · Zbl 1235.70110
[14] Yang, C.H., Zhu, S.M., Chen, S.H.: A modified elliptic Lindstedt–Poincaré method for certain strongly non-linear oscillators. J. Sound Vib. 273, 921–932 (2004) · Zbl 1236.34056
[15] Chen, S.H., Cheung, Y.K.: An elliptic perturbation method for certain strongly non-linear oscillators. J. Sound Vib. 192, 453–464 (1996) · Zbl 1232.70017
[16] Chen, S.H., Yang, X.M., Cheung, Y.K.: Periodic solutions of strongly quadratic non-linear oscillators by the elliptic perturbation method. J. Sound Vib. 212, 771–780 (1998) · Zbl 1235.70062
[17] Otty, J.P.: Study of a non-linear perturbed oscillator. Int. J. Control 32(3), 475–487 (1980) · Zbl 0452.34033
[18] Lakrad, F., Belhaq, M.: Periodic solutions of strongly non-linear oscillators by the multiple scales method. J. Sound Vib. 258, 677–700 (2002) · Zbl 1237.34082
[19] Xu, Z., Cheung, Y.K.: Averaging method using generalized harmonic functions for strongly non-linear oscillators. J. Sound Vib. 174(4), 563–576 (1994) · Zbl 0945.70534
[20] Xu, Z.: Non-linear time transformation method for strongly nonlinear oscillation systems. Acta Mech. Sin. (Engl. Ed.) 8(3), 279–288 (1992) · Zbl 0769.34027
[21] Xu, Z., Zhang, L.: Asymptotic method for analysis of nonlinear systems with two parameters. Acta Math. Sci. (Engl. Ed.) 6(4), 453–462 (1986) · Zbl 0639.34062
[22] Xu, Z., Cheung, Y.K.: Non-linear scales method for strongly non-linear oscillators. Nonlinear Dyn. 7, 285–289 (1995)
[23] Vakakis, A.F.: Exponentially small splittings of manifolds in a rapidly forced Duffing system, Letter to the editor. J. Sound Vib. 170, 119–129 (1994) · Zbl 0925.70262
[24] Vakakis, A.F., Azeez, M.F.A.: Analytic Approximation of the homoclinic orbits of the Lorenz system at {\(\sigma\)}=10, b=8/3 and {\(\rho\)}=13.926.... Nonlinear Dyn. 15, 245–257 (1998) · Zbl 0910.34053
[25] Xu, Z., Chan, H.S.Y., Chung, K.W.: Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method. Nonlinear Dyn. 11, 213–233 (1996)
[26] Chan, H.S.Y., Chung, K.W., Xu, Z.: Stability and bifurcations of limit cycles by the perturbation-incremental method. J. Sound Vib. 206(4), 589–604 (1997) · Zbl 1235.34093
[27] Chen, S.H., Chan, J.K.H., Leung, A.Y.T.: A perturbation method for the calculation of semi-stable limit cycles of strongly nonlinear oscillators. Commun. Numer. Methods Eng. 16, 301–313 (2000) · Zbl 0964.65145
[28] Belhaq, M., Lakrad, F.: Prediction of homoclinic bifurcation: the elliptic averaging method. Chaos Solitons Fractals 11, 2251–2258 (2000) · Zbl 0953.34026
[29] Belhaq, M., Fiedler, B., Lakrad, F.: Homoclinic connections in strongly self-excited nonlinear oscillators: the Melnikov function and the elliptic Lindstedt–Poincaré method. Nonlinear Dyn. 23, 67–86 (2000) · Zbl 0967.70019
[30] Mikhlin, Yu.V.: Analytical construction of homoclinic orbits of two- and three-dimensional dynamical systems. J. Sound Vib. 230(5), 971–983 (2000) · Zbl 1235.34140
[31] Mikhlin, Yu.V., Manucharyan, G.V.: Construction of homoclinic and heteroclinic trajectories in mechanical systems with several equilibrium positions. Chaos Solitons Fractals 16, 299–309 (2003) · Zbl 1056.70011
[32] Chen, S.H., Chen, Y.Y., Sze, K.Y.: A hyperbolic perturbation method for determining homoclinic solution of certain strongly nonlinear autonomous oscillators. J. Sound Vib. 322(1–2), 381–392 (2009) · Zbl 1269.70031
[33] Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover, New York (1972) · Zbl 0543.33001
[34] Merkin, J.H., Needham, D.J.: On infinite-period bifurcations with an application to roll waves. Acta Mech. 60, 1–16 (1986) · Zbl 0588.76024
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