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
Full Text:

References:

 [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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.