Research article

Harmonic and subharmonic solutions of quadratic Liénard type systems with sublinearity

  • Received: 12 July 2021 Accepted: 06 September 2021 Published: 10 September 2021
  • MSC : 34A12, 34C25

  • In this paper, we prove the existence of harmonic solutions and infinitely many subharmonic solutions of dissipative second order sublinear differential equations named quadratic Liénard type systems. The method of the proof is based on the Poincaré-Birkhoff twist theorem.

    Citation: Chunmei Song, Qihuai Liu, Guirong Jiang. Harmonic and subharmonic solutions of quadratic Liénard type systems with sublinearity[J]. AIMS Mathematics, 2021, 6(11): 12913-12928. doi: 10.3934/math.2021747

    Related Papers:

  • In this paper, we prove the existence of harmonic solutions and infinitely many subharmonic solutions of dissipative second order sublinear differential equations named quadratic Liénard type systems. The method of the proof is based on the Poincaré-Birkhoff twist theorem.



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    [1] K. Johannessen, The Duffing oscillator with damping, Eur. J. Phys., 36 (2015), 065020. doi: 10.1088/0143-0807/36/6/065020
    [2] B. Baumann, J. Schwieger, M. Wolff, F. Manders, J. Suijker, Nonlinear behavior in high-intensity discharge lamps, J. Phys. D App. Phys., 49 (2016), 255201. doi: 10.1088/0022-3727/49/25/255201
    [3] Q. Liu, L. Huang, G. Jiang, Periodic oscillations of the relativistic pendulum with friction, Electron. J. Differ. Eq., 2017 (2017), 1–10. doi: 10.1186/s13662-016-1057-2
    [4] J. A. Cid, On the existence of periodic oscillations for pendulum-type equations, Adv. Nonlinear Anal., 10 (2020), 121–130. doi: 10.1515/anona-2020-0222
    [5] L. Gao, J. He, Y. Xu, C. Zhang, Application of Duffing oscillator in fault diagnosis, Mech. Des. Manuf., 3 (2009), 77–79.
    [6] Q. Liu, Z. Wang, Periodic impact behavior of a class of Hamiltonian oscillators with obstacles, J. Math. Anal. App., 365 (2010), 67–74. doi: 10.1016/j.jmaa.2009.09.054
    [7] A. K. Tiwari, S. N. Pandey, M. Senthilvelan, M. Lakshmanan, Classification of Lie point symmetries for quadratic Lienard type equation $\ddot x+ f (x)(\dot x)^2+ g (x) = 0$, J. Math. Phy., 54 (2013), 053506. doi: 10.1063/1.4803455
    [8] V. Marinca, N. Herisanu, Nonlinear Dynamical Systems in Engineering: Some Approximate Approaches, Berlin: Springer-Verlag, 2011.
    [9] P. J. Torres, Mathematical Models with Singularities, Paris: Atlantis Press, 2015.
    [10] M. A. Delpino, R. F. Manasevich, Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity, J. Differ. Equations, 2 (1993), 260–277.
    [11] K. N. Shukla, A generalization of the Rayleigh-Plesset equation of bubble dynamics, ZAMM-Z. Angew Math. Me., 67 (2010), 470–471.
    [12] F. Hegedűs, C. Hős, L. Kullmann, Stable period 1, 2 and 3 structures of the harmonically excited Rayleigh-Plesset equation applying low ambient pressure, IMA J. Appl. Math., 78 (2013), 1179–1195. doi: 10.1093/imamat/hxs016
    [13] W. Ding, The fixed point of torsion map and periodic solution of ordinary differential equation, Acta Math. Sinica., 25 (1982), 227–235.
    [14] T. Ding, Study of the sublinear Duffing equation, Acta Math. Appl. Sin., 12 (1989), 449–455.
    [15] T. Ding, Nonlinear oscillations at a point of resonance, Sci. China Ser. A, 1 (1982), 1–13.
    [16] A. Fonda, F. Zanolin, On the use of time-maps for the solvability of nonlinear boundary value problems, Arch. Math., 59 (1992), 245–259. doi: 10.1007/BF01197322
    [17] D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach-Dancer condition, J. Differ. Equations, 171 (2001), 233–250. doi: 10.1006/jdeq.2000.3847
    [18] Z. Wang, Lazer–Leach type conditions on periodic solutions of semilinear resonant Duffing equations with singularities, Z. Angew Math. Phys., 65 (2014), 69–89. doi: 10.1007/s00033-013-0323-3
    [19] A. Fonda, A. Sfecci, Periodic solutions of weakly coupled superlinear systems, J. Differ. Equations, 260 (2016), 2150–2162. doi: 10.1016/j.jde.2015.09.056
    [20] A. Fonda, A. C. Lazer, Subharmonic solutions of conservative systems with nonconvex potentials, Proc. Amer. Math. Soc., 115 (1992), 183–190. doi: 10.1090/S0002-9939-1992-1087462-X
    [21] E. Serra, M. Tarallo, S. Terracini, Subharmonic solutions to second-order differential equations with periodic nonlinearities, Nonlinear Anal. Theor., 41 (2000), 649–667. doi: 10.1016/S0362-546X(98)00302-2
    [22] A. Boscaggin, F. Zanolin, Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions, Discrete Cont. Dyn. Sys., 33 (2013), 89–110. doi: 10.3934/dcds.2013.33.89
    [23] A. Boscaggin, G. Feltrin, Positive subharmonic solutions to nonlinear ODEs with indefinite weight, Commun. Contemp. Math., 20 (2018), 1750021. doi: 10.1142/S0219199717500213
    [24] A. Fonda, R. Toader, Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal., 8 (2019), 583–602.
    [25] A. Fonda, R. Manásevich, F. Zanolin, Subharmonic solutions for some second-order differential equations with singularities, SIAM J. Math. Anal., 24 (1993), 1294–1311. doi: 10.1137/0524074
    [26] Y. Chen, Q. Liu, H. Su, Generalized Hamiltonian forms of dissipative mechanical systems via a unified approach, J. Geom. Phys., 160 (2021), 103976. doi: 10.1016/j.geomphys.2020.103976
    [27] P. Mathews, M. Lakshmanan, On a unique nonlinear oscillator, Quart. Appl. Math., 32 (1974), 215–218. doi: 10.1090/qam/430422
    [28] M. Lakshmanan, V. K. Chandrasekar, Generating finite dimensional integrable nonlinear dynamical systems, Eur. Phys. J. Spec. Top., 222 (2013), 665–688. doi: 10.1140/epjst/e2013-01871-6
    [29] A. Schulze-Halberg, Closed-form solutions and supersymmetric partners of the inverted Mathews-Lakshmanan oscillator, Eur. Phys. J. Plus., 130 (2015), 1–10. doi: 10.1140/epjp/i2015-15001-1
    [30] M. Sabatini, On the period function of $x''+ f (x) x'^2+ g (x) = 0$, J. Differ. Equations, 196 (2004), 151–168. doi: 10.1016/S0022-0396(03)00067-6
    [31] A. R. Chouikha, Isochronous centers of Lienard type equations and applications, J. Math. Anal. Appl., 331 (2004), 358–376.
    [32] I. Boussaada, A. R. Chouikha, J. M. Strelcyn, Isochronicity conditions for some planar polynomial systems, Bull. Sci. Math., 135 (2011), 89–112. doi: 10.1016/j.bulsci.2010.01.004
    [33] M. Bardet, I. Boussaada, A. R. Chouikha, J. M. Strelcyn, Isochronicity conditions for some planar polynomial systems Ⅱ, Bull. Sci. Math., 135 (2011), 230–249. doi: 10.1016/j.bulsci.2010.12.003
    [34] S. Atslega, On solutions of Neumann boundary value problem for the Liénard type equation, Math. Model. Anal., 13 (2008), 161–169. doi: 10.3846/1392-6292.2008.13.161-169
    [35] T. Ding, Application of Qualitative Methods of Ordinary Differential Equations, Higher Education Press, 2004.
    [36] Z. Zhang, K. Zhang, Q. Liu, W. Zhang, The existence of harmonic and subharmonic solutions for superlinear Liénard type equation with forced term, J. Nonlinear Funct. Anal., 2021.
    [37] D. W. Jordan, P. Smith, Nonlinear Odinary Dfferential Euations: An Introduction for Scientists and Engineers, Oxford University Press on Demand, 2007.
    [38] F. Nakajima, Even and periodic solutions of the equation $\ddot u + g(u) = p(t)$, J. Differ. Equations, 83 (1990), 277–299. doi: 10.1016/0022-0396(90)90059-X
    [39] T. Ding, F. Zanolin, Subharmonic solutions of second order nonlinear equations: A time-map approach, Nonlinear Anal., 20 (1993), 509–532. doi: 10.1016/0362-546X(93)90036-R
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