International Journal of Physics | Vol. 1, No. 9, September 2010 | pp. 69–77
DOI: 10.46882/2010/IJP/000009
Article Type: Original Research Paper
Title: Chaotic Dynamics and Bifurcation Analysis of a Modified Duffing-Van der Pol Oscillator with Non-Linear Fractional Damping
Names of Authors: F. A. Williams¹, S. T. Chon²
Authors’ Affiliations: ¹Department of Physics, Federal University of Technology, Akure, Nigeria; ²Department of Physics, Seoul National University, Seoul, South Korea
Abstract: Non-linear oscillators play a prominent role in modeling structural vibrations, biological rhythms, and secure communication networks. This study analyzes the chaotic dynamics and bifurcation sequences of a modified Duffing-Van der Pol oscillator featuring non-linear fractional-order damping. The fractional derivative is defined using the Caputo framework, with the order α varied between 0.5 and 1.0. We solved the governing non-linear differential equations using an adapted fourth-order Runge-Kutta numerical scheme. Phase space trajectories, Poincare maps, and Lyapunov exponent spectrums were constructed across a wide range of driving amplitudes and frequencies. The analysis demonstrates that reducing the fractional order α from 1.0 (standard derivative) to 0.7 shifts the onset of chaos to higher forcing amplitudes. The system transitions from periodic limit cycles to full chaotic behavior via a classic period-doubling bifurcation route. The maximum Lyapunov exponent reaches a peak value of +0.42 at an amplitude of F = 12.5 N, confirming the presence of deterministic chaos. We also identified several stability windows where the system returns to period-3 and period-5 orbits. These results indicate that adjusting the fractional damping parameter provides an effective control mechanism for stabilizing or inducing chaotic states in physical systems.
Keywords: Chaotic dynamics; Duffing-Van der Pol oscillator; Fractional damping; Bifurcation analysis; Lyapunov exponent; Phase space; Runge-Kutta method; Non-linear mechanics.
Manuscript Timeline: Received: June 22, 2010; Revised: August 01, 2010; Accepted: August 25, 2010; Published: September 18, 2010.
Citation: Williams, F. A., & Chon, S. T. (2010). Chaotic Dynamics and Bifurcation Analysis of a Modified Duffing-Van der Pol Oscillator with Non-Linear Fractional Damping. International Journal of Physics, 1(9), 69–77.
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