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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Angheluta-Bauer, Luiza / Chen, Xi / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi / Yang, Xu

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Volume 20, Issue 2


Effect of Fractional Damping in Double-Well Duffing–Vander Pol Oscillator Driven by Different Sinusoidal Forces

M. V. Sethu Meenakshi
  • Department of Mathematics, St. Xavier’s College, Manonmaniam Sundaranar University, Tirunelveli-627 002, Tamilnadu, India
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ S. Athisayanathan
  • Department of Mathematics, St. Xavier’s College, Manonmaniam Sundaranar University, Tirunelveli-627 002, Tamilnadu, India
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ V. Chinnathambi
  • Corresponding author
  • Department of Physics, Sadakathullah Appa College, Manonmaniam Sundaranar University, Tirunelveli 627 011, Tamilnadu, India
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ S. Rajasekar
Published Online: 2019-01-29 | DOI: https://doi.org/10.1515/ijnsns-2016-0165


The effect of nonlinear damping including fractional damping on the onset of horseshoe chaos is studied both analytically and numerically in the double-well Duffing–Vander Pol (DVP) oscillator driven by various sinusoidal forces. The sinusoidal type periodic forces of our interest are sine wave, rectified sine wave, and modulus of sine wave. Using the Melnikov analytical method, the threshold condition for the onset of horseshoe chaos is obtained for each sinusoidal force. Melnikov threshold curves are drawn in (f,\;ω) parameters space for each force. When the damping component (p) increases from a small value, the Melnikov threshold value (fM) is decreased for each force. Suppression of horseshoe chaos is predicted due to the effect of weak periodic perturbation and nonlinear fractional damping. Analytical predictions are demonstrated through direct numerical simulations.

Keywords: Duffing–Vander Pol oscillator; fractional damping; sinusoidal force; homoclinic bifurcation; Melnikov method; chaos

JEL Classification: primary 37D45; secontary 34C37; 34D10; 34A08; 37J20; 37C29


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About the article

Received: 2016-11-10

Accepted: 2019-01-12

Published Online: 2019-01-29

Published in Print: 2019-04-26

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 20, Issue 2, Pages 115–124, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2016-0165.

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