×

zbMATH — the first resource for mathematics

Determination of periodic solutions for nonlinear oscillators with fractional powers by He’s modified Lindstedt-Poincaré method. (English) Zbl 1184.70018
Summary: He’s modified Lindstedt-Poincaré method is applied to nonlinear oscillatiors with fractional powers. Comparison of the obtained results with exact solutions provides confirmation for the validity of He’s modified Lindstedt-Poincaré method.

MSC:
70K42 Equilibria and periodic trajectories for nonlinear problems in mechanics
70K60 General perturbation schemes for nonlinear problems in mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Brogliato R (1988) Nonsmooth mechanics. Springer, Berlin · Zbl 0646.00014
[2] Oyedeji KO (2005) An analysis of a nonlinear elastic force van der Pol oscillator equation. J Sound Vib 281:417–422 · Zbl 1236.70021 · doi:10.1016/j.jsv.2004.03.040
[3] Mickens RE, Oyedeji KO, Rucker SA (2003) Analysis of the simple harmonic oscillator with fractional damping. J Sound Vib 268:839–842 · Zbl 1236.70040 · doi:10.1016/S0022-460X(03)00371-7
[4] Mickens RE (1994) Nonstandard finite difference models of differential equations. World Scientific, Singapore · Zbl 0810.65083
[5] Hogan SJ (2003) Relaxation oscillations in a system with a piecewise smooth drag coefficient. J Sound Vib 263:467–471 · Zbl 1237.34061 · doi:10.1016/S0022-460X(02)01431-1
[6] Mickens RE (2003) A combined equivalent linearization and averaging perturbation method for non-linear oscillator equations. J Sound Vib 264:1195–1200 · Zbl 1236.34053 · doi:10.1016/S0022-460X(02)01510-9
[7] Lim CW, Wu BS (2003) Accurate higher-order approximations to frequencies of nonlinear oscillators with fractional powers. J Sound Vib 281:1157–1162 · Zbl 1236.34052 · doi:10.1016/j.jsv.2004.04.030
[8] He JH (2003) Linearized perturbation technique and its applications to strongly nonlinear oscillators. Comput Math Appl 45:1–8 · Zbl 1035.65070 · doi:10.1016/S0898-1221(03)80002-0
[9] He JH (2006) Non-perturbative methods for strongly nonlinear problems. dissertation.de-verlag im Internet GmbH, Berlin
[10] He JH (2005) Limit cycle and bifurcation of nonlinear problems. Chaos Solitons Fractals 26:827–833 · Zbl 1093.34520 · doi:10.1016/j.chaos.2005.03.007
[11] He JH (2003) Determination of limit cycles for strongly nonlinear oscillators. Phys Rev Lett 90:174301 · doi:10.1103/PhysRevLett.90.174301
[12] Öziş T, Yıldırım A (2007) A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity. Chaos Solitons Fractals 34:989–991 · doi:10.1016/j.chaos.2006.04.013
[13] Öziş T, Yıldırım A (2007) Determination of limitcycles by a modified straightforward expansion for nonlinear oscillators. Chaos Solitons Fractals 32:445–448 · doi:10.1016/j.chaos.2006.06.034
[14] Öziş T, Yıldırım A (2007) Determination of frequency-amplitude relation for duffing-harmonic oscillator by the energy balance method. Comput Math Appl 54:1184–1187 · Zbl 1147.34321 · doi:10.1016/j.camwa.2006.12.064
[15] Öziş T, Yıldırım A (2007) Determination of periodic solution for a u1/3 force by He’s modified Lindstedt-Poincaré method. J Sound Vib 301:415–419 · Zbl 1242.70044 · doi:10.1016/j.jsv.2006.10.001
[16] Öziş T, Yıldırım A (2007) A study of nonlinear oscillators with u1/3 force by He’s variational iteration method. J Sound Vib 306:372–376 · Zbl 1242.74214 · doi:10.1016/j.jsv.2007.05.021
[17] He JH (1999) Analytical solution of a nonlinear oscillator by the linearized perturbation technique. Commun Nonlinear Sci Numer Simul 4:109–113 · Zbl 0928.34013 · doi:10.1016/S1007-5704(99)90021-7
[18] He JH (1999) Modified straightforward expansion. Meccanica 34:287–289 · Zbl 1002.70019 · doi:10.1023/A:1004730415955
[19] Mickens RE (2001) Oscillations in an x 4/3 potential. J Sound Vib 246:375–378 · Zbl 1237.34044 · doi:10.1006/jsvi.2000.3583
[20] He JH (2002) Modified Lindstedt-Poincaré methods for some strongly non-linear oscillations, part I: expansion of a constant. Int J Non-Linear Mech 37:309–314 · Zbl 1116.34320 · doi:10.1016/S0020-7462(00)00116-5
[21] Wang SQ, He JH (2008) Nonlinear oscillator with discontinuity by parameter-expansion method. Chaos Solitons Fractals 35:688–691 · Zbl 1210.70023 · doi:10.1016/j.chaos.2007.07.055
[22] Shou DH, He JH (2007) Application of parameter-expanding method to strongly nonlinear oscillators. Int J Nonlinear Sci Numer Simul 8:121–124 · Zbl 06942250
[23] Zengin FO, Kaya MO, Demirbag SA (2008) Application of parameter-expansion method to nonlinear oscillators with discontinuities. Int J Nonlinear Sci Numer Simul 9:267–270
[24] Zhang LN, Xu L (2007) Determination of the limit cycle by He’s parameter-expansion for oscillators in a u(3)/(1+u(2)) potential. Z Naturforsch A, J Phys Sci 62:396–398 · Zbl 1203.34053
[25] Xu L (2007) Application of He’s parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire. Phys Lett A 368:259–262 · doi:10.1016/j.physleta.2007.04.004
[26] Xu L (2007) Determination of limit cycle by He’s parameter-expanding method for strongly nonlinear oscillators. J Sound Vib 302:178–184 · Zbl 1242.70038 · doi:10.1016/j.jsv.2006.11.011
[27] Nayfeh AH (1973) Perturbation methods. Wiley-Interscience, New York · Zbl 0265.35002
[28] Mickens RE (1996) Oscillations in planar dynamics systems. World Scientific, Singapore
[29] Jordan DW, Smith P (1987) Nonlinear ordinary differential equations. Clarendon, Oxford
[30] Hagedorn P (1981) Nonlinear oscillations (translated by Wolfram Stadler). Clarendon, Oxford · Zbl 0629.70016
[31] He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 20:1141–1199 · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[32] He JH (2002) Recent developments in asymptotic methods for nonlinear ordinary equations. Int J Comput Numer Anal Appl 2:127–190 · Zbl 1046.34001
[33] Gottlieb HPW (2003) Frequencies of oscillators with fractional-power nonlinearities. J Sound Vib 261:557–566 · Zbl 1237.34060 · doi:10.1016/S0022-460X(02)01003-9
[34] Waluya SB, van Horssen WT (2003) On the periodic solutions of a generalized non-linear Van der Pol oscillator. J Sound Vib 268:209–215 · Zbl 1236.70029 · doi:10.1016/S0022-460X(03)00251-7
[35] Wu B, Li P (2001) A method for obtaining approximate analytic periods for a class of nonlinear oscillators. Meccanica 36:167–176 · Zbl 1008.70016 · doi:10.1023/A:1013067311749
[36] He JH (2000) A modified perturbation technique depending upon an artificial parameter. Meccanica 35:299–311 · Zbl 0986.70016 · doi:10.1023/A:1010349221054
[37] Marinca V, Herişanu N, Bota C (2008) Application of the variational iteration method to some nonlinear one-dimensional oscillations. Meccanica 43:75–79 · Zbl 1238.70018 · doi:10.1007/s11012-007-9086-2
[38] Villaggio P (2008) Small perturbations and large self-quotations. Meccanica 43:81–83 · Zbl 1165.01333 · doi:10.1007/s11012-007-9088-0
[39] Spanos PD, Di Paola M, Failla G (2002) A Galerkin approach for power spectrum determination of nonlinear oscillators. Meccanica 37:51–65 · Zbl 1060.70031 · doi:10.1023/A:1019610512675
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.