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Oscillators with symmetric and asymmetric quadratic nonlinearity

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Abstract

In this paper, oscillators with asymmetric and symmetric quadratic nonlinearity are compared. Both oscillators are modeled as ordinary second-order differential equations with strong quadratic nonlinearities: one with positive quadratic term and the second with a quadratic term which changes the sign. Solutions for both equations are obtained in the form of Jacobi elliptic functions. For the asymmetric oscillator, conditions for the periodic motion are determined, while for the symmetric oscillator a new approximate solution procedure based on averaging is developed. Obtained results are tested on an optomechanical system where the motion of a cantilever in the intracavity field is oscillatory. Two types of quadratic nonlinearities in the system are investigated: symmetric and asymmetric. The advantage and disadvantage of both models is discussed. The analytical procedure suggested in the paper is applied. The obtained solution agrees well with a numerical one.

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References

  1. Kovacic I., Brennan M.J.: The Duffing Equation: Nonlinear Oscillators and their Behaviour. Wiley, Hoboken (2011)

    Book  MATH  Google Scholar 

  2. Duffing G.: Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung. Vieweg & Sohn, Braunschweig (1918)

    MATH  Google Scholar 

  3. Nayfeh A.H.: Quenching of primary resonance by a superharmonic resonance. J. Sound Vib. 92(3), 363–377 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Nayfeh A.H.: Combination tones in the response of single degree of freedom systems with quadratic and cubic non-linearities. J. Sound Vib. 92(3), 379–386 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Atadan A.S., Huseyin K.: An intrinsic method of harmonic analysis for non-linear oscillations (a perturbation technique). J. Sound Vib. 95(4), 525–530 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Nayfeh A.H.: Interaction of fundamental parametric resonances with subharmonic resonances. J. Sound Vib. 96(3), 333–340 (1984)

    Article  MathSciNet  Google Scholar 

  7. Bajkowski J., Szemplinska-Stupnicka W.: Internal resonances effects-simulation versus analytical methods results. J. Sound Vib. 104(2), 259–275 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Szemplinska-Stupnicka W., Bajkowski J.: The 1/2 subharmonic resonance and its transition to chaotic motion in a non-linear oscillator. Int. J. Non-Linear Mech. 21(5), 401–419 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. HaQuang N., Mook D.T.: A non-linear analysis of the interactions between parametric and external excitations. J. Sound Vib. 118(3), 425–439 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rudowski J., Szemplinska-Stupnicka W.: On an approximate criterion for chaotic motion in a model of a buckled beam. Ing. Arch. 57, 243–255 (1987)

    Article  MATH  Google Scholar 

  11. Szemplinska-Stupnicka W.: Secondary resonances and approximate models of routes to chaotic motion in non-linear oscillators. J. Sound Vib. 113(1), 155–172 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zavodney L.D., Nayfeh A.H., Sanchez N.E.: The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a principal parametric resonance. J. Sound Vib. 129(3), 417–442 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Szemplinska-Stupnicka W., Niezgodski P.: The approximate approach to chaos phenomena in oscillators having single equilibrium position. J. Sound Vib. 141(2), 181–192 (1990)

    Article  MathSciNet  Google Scholar 

  14. Sarma M.S., Beena A.P., Rao B.N.: Applicability of the perturbation technique to the periodic solution of \({\ddot{x}+\alpha x+\beta x^{2}+\gamma x^{3}=0}\). J. Sound Vib. 180(1), 177–184 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rega G., Salvatori A.: Bifurcation structur at 1/3-subharmonic resonance in an asymmetric nonlinear elastic oscillator. Int. J. Bifurc. Chaos 6(8), 1529–1546 (1996)

    Article  MATH  Google Scholar 

  16. Szemplinska-Stupnicka W., Tyrkiel E.: Effects of multi global bifurcations on basin organization, catastrophes and final outcomes in a driven nonlinear oscillator at the 2T-subharmonic resonance. Nonlinear Dyn. 17, 41–59 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Maccari A.: Modulated motion and infinite-period homoclinic bifurcation for parametrically excited Lienard systems. Int. J. Non-Linear Mech. 35, 239–262 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hu H.: Solution of a mixed parity nonlinear oscillator: harmonic balance. J. Sound Vib. 299, 331–338 (2007)

    Article  Google Scholar 

  19. He Q., Daqaq M.F.: Influence of potential function asymmetries on the performance of nonlinear energy harvesters under white noise. J. Sound Vib. 333, 3479–3489 (2014)

    Article  Google Scholar 

  20. Szemplinska-Stupnicka W., Rudowski J.: On minimum safe impulsive velocity in the driven escape oscillator. Int. J. Non-Linear Mech. 31(3), 255–266 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rodriguez-Lozano E.D.R., Velarde M.G.: Note on stability of limit cycles of an asymmetric (Helmholtz–Thompson) non-linear oscillator. J. Sound Vib. 172(2), 283–288 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mickens R.E.: A uniformly valid asymptotic solution for \({d^{2}y/dt^{2}+y=a + \varepsilon y^{2}}\). J. Sound Vib. 76(1), 150–152 (1981)

    Article  MATH  Google Scholar 

  23. Atadan A.S., Huseyin K.: A note on “a uniformly valid asymptotic solution for \({d^{2}y/dt^{2}+y=a + \varepsilon y^{2}}\)”. J. Sound Vib. 85(1), 129–131 (1982)

    Article  MATH  Google Scholar 

  24. Doelman A., Koenderink A.F., Maas L.R.M.: Quasi-periodically forced nonlinear Helmholtz oscillators. Phys. D Nonlinear Phenom. 164(1–2), 1–27 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Almendral J.A., Sanjuán M.A.F: Integrability and symmetries for the Helmholtz oscillator with friction. J. Phys. A Math. Gen. 36(3), 695–710 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lenci S., Rega G.: Optimal control of homoclinic bifurcation: theoretical treatment and practical reduction of safe basin erosion in the Helmholtz oscillator. JVC J. Vib. Control 9(3-4), 281–315 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Balibrea F., Chacón R., López M.A.: Reshaping-induced order-chaos routes in a damped driven Helmholtz oscillator. Chaos Solitons Fractals 24(2), 459–470 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rasband S.N.: Marginal stability boundaries for some driven, damped, non-linear oscillators. International Journal of Non-Linear Mechanics 22(6), 477–495 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pehlivani I., Wei Z.: Analysis, nonlinear control, and chaos generator circuit of another strange chaotic system. Turk. J. Electr. Eng. Comput. Sci. 20(2), 1229–1239 (2012)

    Google Scholar 

  30. Thylwe K.E.: Exact quenching phenomenon of undamped driven Helmholtz and Duffing oscillators. J. Sound Vib. 161(2), 203–211 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Szemplinska-Stupnicka W., Rudowski J.: Bifurcation phenomena in a nonlinear oscillator: approximate analytical studies versus computer simulation results. Phys. D 66, 368–380 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. Bhushan A., Inamdar M.M., Pawaskar D.N.: Dynamic analysis of a double-sided actuated MEMS oscillator using second-order averaging. Lect. Notes. Eng. Comput. Sci. LNECS 3, 1640–1645 (2013)

    Google Scholar 

  33. Isar A., Scheid W.: Deformed quantum harmonic oscillator with diffusion and dissipation. Phys. A 310, 364–376 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pham T.T., Lamarque C.H., Pernot S.: Passive control of one degree of freedom nonlinear quadratic oscillator under combination resonance. Commun. Nonlinear Sci. Numer. Simul. 16, 2279–2288 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lai S.K., Chow K.W.: Exact solutions for oscillators with quadratic damping and mixed-parity nonlinearity. Phys. Scr. 85(045006), 1–10 (2012)

    MATH  Google Scholar 

  36. Ramananarivo S., Godoy-Diana R., Thiria B.: Rather than resonance, flapping wing flyers may play on aerodynamics to improve performance. PNAS 108(15), 5964–5969 (2011)

    Article  Google Scholar 

  37. Zurkinden A.S., Ferri F., Beatty S., Kofoed J.P., Kramer M.M.: Non-linear numerical modeling and experimental testing of a point absorber wave energy converter. Ocean Eng. 78(2014), 11–21 (2014)

    Article  Google Scholar 

  38. Zhang L., Kong H.Y.: Self-sustained oscillation and harmonic generation in optomechanical systems with quadratic couplings. Phys. Rev. A 89(023847), 1–12 (2014)

    MathSciNet  Google Scholar 

  39. Sharma A.K., Patidar R.K., Raghuramaiah M., Joshi A.S., Naik P.A., Gupta P.D.: A study on wavelength dependence and dynamic range of the quadratic response of commercial grade light emitting diodes. Opt. Commun. 285, 3300–3305 (2012)

    Article  Google Scholar 

  40. Zhang H.L.: Application of He’s amplitude-frequency formulation to a nonlinear oscillator with discontinuity. Comput. Math. Appl. 58, 2197–2198 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Bejarano J.D.: Anharmonic asymmetric oscillator: a classical and quantm treatment. J. Chem. Phys. 85(9), 5128–5131 (1986)

    Article  Google Scholar 

  42. Chen C.H., Yang X.M., Cheung Y.K.: Periodic solutions of strongly quadratic nonlinear oscillators by the elliptic perturbation method. J. Sound Vib. 212, 771–780 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  43. Byrd P.F.: Handbook of Elliptic Integrals for Engineers and Physicists. Springer, Berlin (1954)

    Book  MATH  Google Scholar 

  44. Cveticanin L.: Oscillator with fraction order restoring force. J Sound Vib. 320, 1064–1077 (2009)

    Article  Google Scholar 

  45. Cveticanin L., Pogany T.: Oscillator with a sum of non-integer order non-linearities. J. Appl. Math. 2012(649050), 1–20 (2012)

    MathSciNet  MATH  Google Scholar 

  46. Cveticanin L.: Strongly nonlinear oscillators—analytical solutions. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  47. Cveticanin, L.: An analytical method for truly nonlinear oscillators, In: Gumel, A.B. (ed.) Mathematics of Continuous and Discrete Dynamicsl Systems. Ser. Contemporary Mathematics, 618, pp. 229–245. AMS (2014)

  48. Eichenfield M., Chan J., Camacho R.M., Vahal K.J., Painter O.: Optomechanical cristals. Nature 462, 78–82 (2009)

    Article  Google Scholar 

  49. Peano V., Thorwart M.: Macroscopic quantum effects in a strongly driven nanomechanical resonator. Phys. Rev. B Condens. Matter Mater. Phys. 70(23), 1–5 (2004)

    Article  Google Scholar 

  50. Chang D.E., Regal C.A., Papp S.B., Wilsonb D.J., Ye J., Painter O., Kimbleb H.J., Zoller P.: Cavoty opto-mechanics using an optically levitated nanosphere. Proc. Natl. Acad. Sci. USA 107(3), 1005–1010 (2010)

    Article  Google Scholar 

  51. Borkje K., Nunnenkamp A., Teufel J.D., Girvin S.M.: Signatures of nonlinear cavity optomechanics in the weak coupling regime. Phys. Rev. Lett. 111(5), 053603 (2013)

    Article  Google Scholar 

  52. Eichenfeld M., Chan J., Camacho R.M., Vahala K.J., Painter O.: Optomechanical cristals. Nature 462, 78–82 (2009)

    Article  Google Scholar 

  53. Purdy T.P., Brooks D.W.C., Botter T., Brahms N., Ma Z.-Y., Stamper-Kurn D.M.: Turnable cavity optomechanics with ultracold atoms. Phys. Rev. Lett. 105(13), 133602 (2010)

    Article  Google Scholar 

  54. Sankey J.C., Yang C., Zwickl B.M., Jayich A.M., Harris J.G.E.: Strong and tunable nonlinear optomechanical coupling in a low-loss system. Nat. Phys. 6(9), 707–712 (2010)

    Article  Google Scholar 

  55. Safavi-Naeini A.H., Alegre T.P.M., Chan J., Chang D.E., Painter O.: Electromagnetically induced transparency and slow light with optomechanics. Nature 472(7341), 69–73 (2011)

    Article  Google Scholar 

  56. Bagheri M., Poot M., Li M., Pernice W.P.H., Tang H.X.: Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation. Nat. Nanotechnol. 6(11), 726–732 (2011)

    Article  Google Scholar 

  57. Brennecke F., Ritter S., Donner T., Esslinger T.: Cavity optomechanics with a Bose–Einstein condensate. Science 322, 235–238 (2008)

    Article  Google Scholar 

  58. Xiong H., Si L.-G., Lu X.-Y., Yang X., Wu Y.: Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system in sideband generation. Opt. Lett. 38(3), 353–355 (2013)

    Article  Google Scholar 

  59. Liu Y.-C., Hu Y.-W., Wong C.-W., Xiao Y.-F.: Review of cavity optomechanical cooling. Chin. Phys. B 22(11), 114213 (2013)

    Article  Google Scholar 

  60. Xiong H., Si L.-G., Lu X.-Y.: Nanosecond-pulse-controlled higher order side band comb in a GaAs optomechanical disk resonator in the perturbative regime. Ann. Phys. 349, 43–53 (2014)

    Article  Google Scholar 

  61. Wang G., Fang Z., Wu F.: Control of two-dimensional electron population in semiconductor quantum well. Phys. E Low-Dimens. Syst. Nanostruct. 75(12136), 241–245 (2016)

    Article  Google Scholar 

  62. Suzuki H., Brown E., Sterling R.: Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second order sideband generation. Phys. Rev. A At. Mol. Opt. Phys. 92(3), 033823 (2015)

    Article  Google Scholar 

  63. Ludwig M., Kubala B., Marquardt F.: The optomechanical instability in the quantum regime. New J. Phys. 10(095013), 1–12 (2008)

    Google Scholar 

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Authors and Affiliations

  1. University of Novi Sad, Trg D. Obradovica 6, Novi Sad, 21000, Serbia

    L. Cveticanin & M. Zukovic

  2. Obuda University, Nepszinhaz u.8, Budapest, 1081, Hungary

    L. Cveticanin & Gy. Mester

  3. University of Szeged, Mars tér 7, Szeged, 6724, Hungary

    I. Biro & J. Sarosi

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  1. L. Cveticanin
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  2. M. Zukovic
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Cveticanin, L., Zukovic, M., Mester, G. et al. Oscillators with symmetric and asymmetric quadratic nonlinearity. Acta Mech 227, 1727–1742 (2016). https://doi.org/10.1007/s00707-016-1582-9

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  • DOI: https://doi.org/10.1007/s00707-016-1582-9

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