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A finite extensibility nonlinear oscillator. (English) Zbl 1211.65090
Summary: The dynamics of a finite extensibility nonlinear oscillator (FENO) is studied analytically by means of two different approaches: a generalized decomposition method (GDM) and a linearized harmonic balance procedure (LHB). From both approaches, analytical approximations to the frequency of oscillation and periodic solutions are obtained, which are valid for a large range of amplitudes of oscillation. Within the generalized decomposition method, two different versions are presented, which provide different kinds of approximate analytical solutions.
In the first version, it is shown that the truncation of the perturbation solution up to the third order provides a remarkable degree of accuracy for almost the whole range of amplitudes. The second version, which expands the nonlinear term in Taylor’s series around the equilibrium point, exhibits a little lower degree of accuracy, but it supplies an infinite series as the approximate solution. On the other hand, a linearized harmonic balance method is also employed, and the comparison between the approximate period and the exact one (numerically calculated) is slightly better than that obtained by both versions of the GDM.
In general, the agreement between the results obtained by the three methods and the exact solution (numerically integrated) for amplitudes $$(A)$$ between $$0 < A \leqslant 0.9$$ is very good both for the period and the amplitude of oscillation. For the rest of the amplitude range $$(0.9 < A < 1)$$, an exponentially large $$L_{2}$$ error demonstrates that all three approximations do not represent a good description for the FENO, and higher order perturbation solutions are needed instead. As a complement, very accurate asymptotic representations of the period are provided for the whole range of amplitudes of oscillation.

##### MSC:
 65L05 Numerical methods for initial value problems involving ordinary differential equations
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##### References:
 [1] Maniadis, P.; Kopidakis, G.; Aubry, S., Classical and quantum targeted energy transfer between nonlinear oscillators, Physica D, 188, 153-177, (2004) · Zbl 1047.81513 [2] Pathak, A.; Mandal, S., Classical and quantum oscillators of sextic and octic anharmonicities, Physics letters A, 298, 259-270, (2002) · Zbl 0995.70020 [3] Acebrón, J.A.; Spigler, R., Uncertainty in phase-frequency synchronization of large populations of globally coupled nonlinear oscillators, Physica D, 141, 65-79, (2000) · Zbl 0961.34025 [4] Gois, Sandra R.F.S.M.; Savi, Marcelo A., An analysis of heart rhythm dynamics using a three-coupled oscillator model, Chaos, solitons and fractals, 41, 2553-2565, (2009) · Zbl 1198.37123 [5] Ramos, J.I., Linearized Galerkin and artificial parameter techniques for the determination of periodic solutions of nonlinear oscillators, Applied mathematics and computation, 196, 483-493, (2008) · Zbl 1135.65344 [6] Guo, Z.; Leung, A.Y.T., The iterative homotopy harmonic balance method for conservative helmholtzduffing oscillators, Applied mathematics and computation, 215, 3163-3169, (2010) · Zbl 1183.65083 [7] Mickens, R.E., Quadratic non-linear oscillators, Journal of sound and vibration, 270, 427-432, (2004) · Zbl 1236.70036 [8] Scherrer, H.; Risken, H.; Leiber, T., Eigenvalues of the Schrödinger equation with rational potentials, Physical review A, 38, 3949-3959, (1988) [9] Risken, H.; Vollmer, H.D., The influence of higher order contributions to the correlation function of the intensity fluctuation in a laser near threshold, Zeitschrift fr physik A, 201, 323-330, (1967) [10] Biswas, S.N.; Datta, K.; Saxena, R.P.; Srivastava, P.K.; Varma, V.S., Eigenvalues of λx2m anharmonic oscillators, Journal of mathematical physics, 14, 1190-1195, (1973) [11] Febbo, M.; Milchev, A.; Rostiashvili, V.; Dimitrov, D.; Vilgis, T.A., Dynamics of a stretched nonlinear polymer chain, The journal of chemical physics, 129, 1549081-15490813, (2008) [12] Hatfield, J.W.; Quake, S.R., Dynamic properties of an extended polymer in solution, Physical review letters, 82, 3548-3551, (1999) [13] Koplik, J.; Banavar, J.R., Extensional rupture of model non-Newtonian fluid filaments, Physical review E, 67, 115021-1150212, (2003) [14] Kremer, N., Computational soft matter: from synthetic polymers to proteins, NIC series, vol. 23, (2004), John von Neumann Institute for Computing [15] Gatz, S.; Herrmann, J., Soliton propagation and soliton collision in double-doped fibers with a non-Kerr-like nonlinear refractive-index change, Optics letters, 17, 84-486, (1992) [16] Melvin, T.R.O.; Champneys, A.R.; Kevrekidis, P.G.; Cuevas, J., Travelling solitary waves in the discrete schrodinger equation with saturable nonlinearity: existence, stability and dynamics, Physica D, 237, 551-567, (2008) · Zbl 1167.35448 [17] Khare, A.; Rasmussen, K.; Samuelsen, M.R.; Saxena, Avadh, Exact solutions of the saturable discrete nonlinear Schrödinger equation, Journal of physics A, 38, 807-814, (2005) · Zbl 1069.81016 [18] Mickens, R.E., Construction of a perturbation solution for a non-linear, singular oscillator equation, Journal of sound and vibration, 130, 513-515, (1989) · Zbl 1235.70153 [19] He, J-H., Some asymptotic methods for strongly nonlinear equations, International journal of modern physics, 20, 1141-1199, (2006) · Zbl 1102.34039 [20] He, J-H., Addendum: new interpretarion of homotoy perturbation methods, International journal of modern physics, 20, 2561-2568, (2006) [21] Hu, H.; Tang, J.H., Solution of a Duffing-harmonic oscillator by the method of harmonic balance, Journal of sound and vibration, 294, 637-639, (2006) · Zbl 1243.34049 [22] Gottlieb, H.P.W., Harmonic balance approach to periodic solutions of non-linear Jerk equations, Journal of sound and vibration, 271, 671-683, (2004) · Zbl 1236.34049 [23] Wu, B.S.; Lim, C.W., Large amplitude nonlinear oscillations of a general conservative system, International journal of non-linear mechanics, 39, 859-870, (2004) · Zbl 1348.34074 [24] Beléndez, A.; Méndez, D.I.; Beléndez, T.; Hernández, A.; Álvarez, M.L., Harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable, Journal of sound and vibration, 314, 775-782, (2008) · Zbl 1145.70012 [25] Nayfeh, A.H.; Mook, D.T., Nonlinear oscillations, (1979), John Wiley and Sons [26] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Boston, MA · Zbl 0802.65122 [27] Hosseini, M.M., Adomian decomposition method with Chebyshev polynomials, Applied mathematics and computation, 175, 1685-1693, (2006) · Zbl 1093.65073 [28] Mickens, R.E., Harmonic balance and iteration calculations of periodic solutions to $$\ddot{y} + y^{- 1} = 0$$, Journal of sound and vibration, 306, 968-972, (2007) [29] Mickens, R.E., A generalized iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillators, Journal of sound and vibration, 287, 1045-1051, (2005) · Zbl 1243.65079 [30] Ramos, J.I., Generalized decomposition method for singular oscillators, Chaos, solitons and fractals, 42, 1149-1155, (2009) · Zbl 1198.65152 [31] Ramos, J.I., Generalized decomposition method for nonlinear oscillators, Chaos, solitons and fractals, 41, 1078-1084, (2009) · Zbl 1198.65151 [32] Beléndez, A.; Pascual, C.; Mendez, D.I.; Neipp, C., Solution of the relativistic (an)harmonic oscillator using the harmonic balance method, Journal of sound and vibration, 311, 1447-1456, (2008) [33] Meirovitch, L., Methods of analytical dynamics, (2003), Dover Mineola, NY · Zbl 1115.70002 [34] Gradshteyn, I.S.; Ryzhik, I.M., Table of integrals, series, and products, (1996), Academic Press, Inc. · Zbl 0918.65001 [35] Apostol, T.M., Mathematical analysis, (1974), Addison-Wesley · Zbl 0126.28202 [36] Beléndez, A.; Hernández, A.; Beléndez, T.; Neipp, C.; Márquez, A., Asymptotic representations of the period for the nonlinear oscillator, Journal of sound and vibration, 299, 403-408, (2007) · Zbl 1241.70031 [37] Rektorys, K., Variational methods in mathematics, science and engineering, (1980), D. Reidel Dordrecht, Holland · Zbl 0481.49002
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