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Variational approach for nonlinear oscillators. (English) Zbl 1152.34327
Summary: We propose a novel variational approach for limit cycles of a kind of nonlinear oscillators. Some examples are given to illustrate the effectiveness and convenience of the method. The obtained results are valid for the whole solution domain with high accuracy.

34C05Location of integral curves, singular points, limit cycles (ODE)
49M15Newton-type methods in calculus of variations
65L10Boundary value problems for ODE (numerical methods)
Full Text: DOI
[1] He, J. H.: Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos, solitons & fractals 19, No. 4, 847-851 (2004) · Zbl 1135.35303
[2] He, J. H.: Variational theory for one-dimensional longitudinal beam dynamics. Phys lett A 352, No. 4 -- 5, 276-277 (2006) · Zbl 1187.74108
[3] He, J. H.: A generalized variational principle in micromorphic thermoelasticity. Mech res commun 32, No. 1, 93-98 (2005) · Zbl 1091.74012
[4] Liu, H. M.: Generalized variational principles for ion acoustic plasma waves by he’s semi-inverse method. Chaos, solitons & fractals 23, No. 2, 573-576 (2005) · Zbl 1135.76597
[5] Wu, Y.: Variational approach to higher-order water-wave equations. Chaos, solitons & fractals 32, No. 1, 195-198 (2007) · Zbl 1131.76015
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[8] He JH. Non-perturbative methods for strongly nonlinear problems. Dissertation, de-Verlag im Internet GmbH, Berlin; 2006.
[9] He, J. H.: Preliminary report on the energy balance for nonlinear oscillations. Mech res commun 29, No. 2 -- 3, 107-111 (2002) · Zbl 1048.70011
[10] He, J. H.: Determination of limit cycles for strongly nonlinear oscillators. Phys rev lett 90, No. 17 (2003)
[11] D’acunto, M.: Determination of limit cycles for a modified van der Pol oscillator. Mech res commun 33, No. 1, 93-98 (2006)
[12] D’acunto, M.: Self-excited systems: analytical determination of limit cycles. Chaos, solitons & fractals 30, No. 3, 719-724 (2006) · Zbl 1142.70010
[13] He, J. H.: Variational iteration method -- a kind of non-linear analytical technique: some examples. Int J nonlinear mech 34, No. 4, 699-708 (1999) · Zbl 05137891
[14] He, J. H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comp meth appl mech eng 167, No. 1 -- 2, 57-68 (1998) · Zbl 0942.76077
[15] He, J. H.: Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comp meth appl mech eng 167, No. 1 -- 2, 69-73 (1998) · Zbl 0932.65143
[16] He, J. H.: Variational iteration method for autonomous ordinary differential systems. Appl math comput 114, No. 2 -- 3, 115-123 (2000) · Zbl 1027.34009
[17] He, J. H.; Wu, X. H.: Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, solitons & fractals 29, No. 1, 108-113 (2006) · Zbl 1147.35338
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[21] Bildik, N.; Konuralp, A.: The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int J nonlinear sci numer simulat 7, No. 1, 65-70 (2006) · Zbl 1115.65365
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