On Truly Nonlinear Oscillator Equations of Ermakov-Pinney Type

Main Article Content

Marcellin Nonti
Kolawole Kegnide Damien Adjai
Jean Akande
Marc Delphin Monsia

Abstract

In this paper we present a general class of differential equations of Ermakov-Pinney type which may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations.

Article Details

References

  1. F. Gungor, P.J. Torres, Lie point symmetry analysis of a second order differential equation with singularity, (2017). http://arxiv.org/abs/1612.07080.
  2. M. D Monsia, Analysis of a purely nonlinear generalized isotonic oscillator equation, (2020). https://vixra.org/pdf/ 2010.0195v1.pdf.
  3. M. D. Monsia, On a nonlinear differential equation of Lienard type, (2020). https://vixra.org/pdf/2011.0050v3.pdf.
  4. E. Pinney, The nonlinear differential equation y 00 + p(x)y + cy−3 = 0, Proc. Amer. Math. Soc. 1 (1950), 681.
  5. M. Euler, N. Euler and P. Euler, The Riccati and Ermakov-Pinney hierarchies, J. Nonlinear Math. Phys. 14(2) (2007), 290 – 310.
  6. M. C. Nucci and P.G. L. Leach, Jacobi’s Last multiplier and the complete symmetry group of the Ermakov-Pinney equation, J. Nonlinear Math. Phys. 12 (2) (2005), 305 – 320.
  7. R. E. Michens, Truly Nonlinear Oscillators, World Scientific, Singapore, (2010).
  8. M. Gadella and L. P. Lara, On the solutions of a nonlinear pseudo-oscillator equation, Phys. Scripta, 89 (2014), 105205.
  9. V. R. Gorder, Continuous periodic of a nonlinear pseudo-oscillator equation in which the restoring force is inversely proportional to the dependent variable,Phys. Scripta, 90 (2015), 085208.
  10. E. A. Doutetien, A. R. Yehossou, P. Mallick, B. Rath and M. D. Monsia, On the general solutions of a nonlinear pseudooscillator equation and related quadratic lienard systems, PINSA. 86 (2020). https://doi.org/10.16943/ptinsa/2020/ 154987.
  11. L. Cveticanin, Oscillator with fraction order restoring force, J. Sound Vibration, 320 (2009), 1064 – 1077.
  12. Y. Wang, Unboundedness in a Duffing Equation with Polynomial Potentials, J. Differ. Equations, 160 (2) (2000), 467 – 479.
  13. M. D. Monsia, The non-periodic solution of a truly nonlinear oscillator with power nonlinearity, (2020). https://vixra. org/pdf/2009.0174v1.pdf.
  14. S. C. Mancas and H. Rosu, Existence of periodic orbits in nonlinear oscillators of Emden-Fowler form, Phys. Lett. A, 380 (3) (2016), 422–428.
  15. J. Akande, D. K. K. Adjaı and M. D. Monsia, Theory of exact trigonometric periodic solutions to quadratic Lienard type equations, J. Math. Stat. 14 (1) (2018), 232 – 240.
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, ed. Elsevier, California, 2007.
  17. P.F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer, Berlin, 1954.
  18. W. A. Schwalm, Lectures on Selected Topics in Mathematical Physics: Elliptic Functions and Elliptic Integrals, IOP Publishing, 2015.