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Notes on a conservative nonlinear oscillator. (English) Zbl 1219.34051
Summary: The amplitude-frequency relationship is an important mathematical property for a nonlinear oscillator. He’s amplitude-frequency formulation and the Max-Min approach are used to handle the conservative nonlinear oscillator \(x''+(1+{x'}^2)x=0\) for the amplitude-frequency relationship. The obtained result is compared with those in the open literature, revealing the effectiveness of the used methods.

MSC:
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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[1] Beatty, John; Mickens, Ronald E., A qualitative study of the solutions to the differential equation \(\ddot{x} +(1 + \dot{x}^2) x = 0\), Journal of sound and vibration, 283, 1/2, 475-477, (2005) · Zbl 1237.34040
[2] Mickens, Ronald E., Investigation of the properties of the period for the nonlinear oscillator \(\ddot{x} +(1 + \dot{x}^2) x = 0\), Journal of sound and vibration, 292, 3/5, 1031-1035, (2006) · Zbl 1243.34052
[3] Xu, L., He’s parameter expanding methods for strongly nonlinear oscillators, Journal of computational and applied mathematics, 207, 1, 148-154, (2007) · Zbl 1120.65084
[4] Kalmár-Nagy, Tamás; Erneux, Thomas, Approximating small and large amplitude periodic orbits of the oscillator \(\ddot{x} +(1 + \dot{x}^2) x = 0\), Journal of sound and vibration, 313, 3/5, 806-811, (2008)
[5] Beléndez, A.; Beléndez, T.; Hernández, A.; Gallego, S.; Ortuño, M.; Neipp, C., Comments on “investigation of the properties of the period for the nonlinear oscillator \(\ddot{x} +(1 + \dot{x}^2) x = 0\)”, Journal of sound and vibration, 303, 3/5, 925-930, (2007) · Zbl 1242.34056
[6] 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, 1/2, 403-408, (2007) · Zbl 1241.70031
[7] Beléndez, A.; Hernández, A.; Beléndez, T.; Neipp, C.; Márquez, A., Erratum to “asymptotic representations of the period for the nonlinear oscillator \(\ddot{x} +(1 + \dot{x}^2) x = 0\)” [journal of sound and vibration 299 (2007) 403-408], Journal of sound and vibration, 301, 1/2, 427, (2007) · Zbl 1242.70040
[8] He, J.H., Some asymptotic methods for strongly nonlinear equations, International journal of modern physics B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039
[9] He, J.H., An improved amplitude – frequency formulation for nonlinear oscillators, International journal of nonlinear sciences and numerical simulation, 9, 2, 211-212, (2008)
[10] He, J.H., Comment on ‘he’s frequency formulation for nonlinear oscillators’, European journal of physics, 29, 4, L19-L22, (2008)
[11] He, J.H., Max – min approach to nonlinear oscillators, International journal of nonlinear sciences and numerical simulation, 9, 2, 207-210, (2008)
[12] Zhang, Y.N.; Xu, F.; Deng, L.L., Exact solution for nonlinear schrodinger equation by he’s frequency formulation, Computers and mathematics with applications, 58, 2449-2451, (2009) · Zbl 1189.81064
[13] Fan, J., He’s frequency – amplitude formulation for the Duffing harmonic oscillator, Computers and mathematics with applications, 58, 2473-2476, (2009) · Zbl 1189.65163
[14] Zhao, L., He’s frequency – amplitude formulation for nonlinear oscillators with an irrational force, Computers and mathematics with applications, 58, 2477-2479, (2009) · Zbl 1189.65185
[15] Zhang, H.L., Application of he’s frequency – amplitude formulation to an \(x(1 / 3)\) force nonlinear oscillator, International journal of nonlinear sciences and numerical simulation, 9, 297-300, (2008)
[16] He, J.H., Application of he chengtian’s interpolation to Bethe equation, Computers and mathematics with applications, 58, 11/12, 2427-2430, (2009) · Zbl 1189.65025
[17] Shen, Y.Y.; Mo, L.F., The max – min approach to a relativistic equation, Computers and mathematics with applications, 58, 2131-2133, (2009) · Zbl 1189.65174
[18] Zeng, D.Q.; Lee, Y.Y., Analysis of strongly nonlinear oscillator using the max – min approach, International journal of nonlinear sciences and numerical simulation, 10, 1361-1368, (2009)
[19] He, J.H., An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering, International journal of modern physics B, 22, 21, 3487-3578, (2008) · Zbl 1149.76607
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