Unusual Nonpolynomial Van der Pol Oscillator Equations With Exact Harmonic and Isochronous Solutions

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DOI: https://doi.org/10.28924/2291-8639-20-2022-73
Published: Dec 26, 2022
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Kolawolé Kêgnidé Damien Adjaï, Marcellin Nonti, Jean Akande, Marc Delphin Monsia, Unusual Nonpolynomial Van der Pol Oscillator Equations With Exact Harmonic and Isochronous Solutions, Int. J. Anal. Appl., 20 (2022), 73.

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Kolawolé Kêgnidé Damien Adjaï, Marcellin Nonti, Jean Akande, Marc Delphin Monsia

Abstract

We do not know Van der Pol-type equations with nonlinear restoring force having explicitly an exact periodic solution. We present, for the first time, nonpolynomial Van der Pol oscillator equations that do not satisfy the classical existence theorems. We exhibit their exact harmonic and isochronous solutions and prove the existence of limit cycles by using averaging theory. We also present first integrals and exact solutions of polynomial Van der Pol-Duffing equations to show that they do not have any limit cycle. Additionally, we prove that the damped Duffing-type equations are equivalent to the conservative Duffing equations exhibiting nonoscillatory solutions.

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References

  1. D.W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations, 4 th ed, Oxford University Press, New York, 2007
  2. R.E. Mickens, Oscillations in Planar Dynamic Systems, World Scientific, Singapore, 1996. https://doi.org/10.1142/2778.
  3. M. Cioni, G. Villari, An Extension of Dragilev?s Theorem for the Existence of Periodic Solutions of the Liénard Equation, Nonlinear Anal.: Theory Meth. Appl. 127 (2015), 55-70. https://doi.org/10.1016/j.na.2015.06.026.
  4. M. D. Monsia, J. Akande, K.K.D. Adjaï, On the Non-Existence of Limit Cycles of the Duffing-Van Der Pol Type Equations, (2021). https://doi.org/10.6084/m9.figshare.14547501.v1.
  5. F.E. Udwadia, H. Cho, First Integrals and Solutions of Duffing-van Der Pol Type Equations, J. Appl. Mech. 81 (2013), 034501. https://doi.org/10.1115/1.4024673.
  6. T. Stachowiak, Hypergeometric First Integrals of the Duffing and Van Der Pol Oscillators, J. Differ. Equ. 266 (2019), 5895-5911. https://doi.org/10.1016/j.jde.2018.10.049.
  7. J. Akande, K.K.D. Adjaï, A.B. Yessoufou, et al. Exact and Sinusoidal Periodic Solutions of Lienard Equation Without Restoring Force, (2021). https://doi.org/10.6084/M9.FIGSHARE.14546019.
  8. A.R.O. Akplogan, K.K.D. Adjaï, J. Akande, et al. Modified Van Der Pol-Helmohltz Oscillator Equation With Exact Harmonic Solutions, (2021). https://doi.org/10.21203/rs.3.rs-730159/v1.
  9. S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, Cambridge, 2001.
  10. V.K. Chandrasekar, S.N. Pandey, M. Senthilvelan, et al. A Simple and Unified Approach to Identify Integrable Nonlinear Oscillators and Systems, J. Math. Phys. 47 (2006), 023508. https://doi.org/10.1063/1.2171520.
  11. E.E. Obinwanne, A.R. Okeke, On Application of Lyapunov and Yoshizawa’s Theorems on Stability, Asymptotic Stability, Boundaries and Periodicity of Solution of Duffing’s Equation, Asian J. Appl. Sci. 2 (2014), 970-975.
  12. M. Sabatini, On the Period Function of Liénard Systems, J. Differ. Equ. 152 (1999), 467-487. https://doi.org/10.1006/jdeq.1998.3520.