Primary resonance and feedback control of the fractional Duffing-van der Pol oscillator with quintic nonlinear-restoring force

Zhoujin Cui; Zhoujin Cui

doi:10.3934/math.20231271

AIMS Mathematics

2023, Volume 8, Issue 10: 24929-24946. doi: 10.3934/math.20231271

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Research article

Primary resonance and feedback control of the fractional Duffing-van der Pol oscillator with quintic nonlinear-restoring force

Zhoujin Cui ^,

School of Mathematical Sciences, Jiangsu Second Normal University, Nanjing 210013, China

Received: 21 June 2023 Revised: 10 August 2023 Accepted: 20 August 2023 Published: 28 August 2023
MSC : 34A08, 34K37, 37N99

In the present paper, the primary resonance and feedback control of the fractional Duffing-van der Pol oscillator with quintic nonlinear-restoring force is studied. The approximately analytical solution and the amplitude-frequency equation are obtained using the multiple scale method. Based on the Lyapunov theory, the stability conditions for the steady-state solution are obtained. The bifurcations of primary resonance for system parameters are analyzed, and the influence of parameters on fractional-order model is also studied. Numerical simulation shows that when the parameter values are fixed, the curve bends to the right or left, resulting in jumping phenomena and multi-valued amplitudes. As the excitation frequency changes, the typical hardening or softening characteristics of the oscillator are observed. In addition, the comparisons of approximate analytical solution and numerical solution are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution.
- fractional Duffing-van der Pol oscillator,
- primary resonance,
- multiple scale method,
- feedback control
Citation: Zhoujin Cui. Primary resonance and feedback control of the fractional Duffing-van der Pol oscillator with quintic nonlinear-restoring force[J]. AIMS Mathematics, 2023, 8(10): 24929-24946. doi: 10.3934/math.20231271

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Abstract

In the present paper, the primary resonance and feedback control of the fractional Duffing-van der Pol oscillator with quintic nonlinear-restoring force is studied. The approximately analytical solution and the amplitude-frequency equation are obtained using the multiple scale method. Based on the Lyapunov theory, the stability conditions for the steady-state solution are obtained. The bifurcations of primary resonance for system parameters are analyzed, and the influence of parameters on fractional-order model is also studied. Numerical simulation shows that when the parameter values are fixed, the curve bends to the right or left, resulting in jumping phenomena and multi-valued amplitudes. As the excitation frequency changes, the typical hardening or softening characteristics of the oscillator are observed. In addition, the comparisons of approximate analytical solution and numerical solution are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution.

References

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