Non-perturbative analysis of bifurcation and chaotic dynamics of a cantilever beam under triple time-delayed feedback control

Ahmad Almutlg; Galal M. Moatimid; T. S. Amer; Yasmeen M. Mohamed; Ahmad Almutlg; Galal M. Moatimid; T. S. Amer; Yasmeen M. Mohamed

doi:10.3934/math.2026715

AIMS Mathematics

2026, Volume 11, Issue 6: 17492-17527. doi: 10.3934/math.2026715

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Non-perturbative analysis of bifurcation and chaotic dynamics of a cantilever beam under triple time-delayed feedback control

1.
Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia
2.
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, 11566, Egypt
3.
Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt

Received: 17 April 2026 Revised: 23 May 2026 Accepted: 05 June 2026 Published: 16 June 2026
MSC : 37N35, 70K42, 74K10, 37D45, 37G35, 93D15

In this work, we aimed to improve vibration suppression in a flexible CB structure through the application of time-delays nonlinear feedback control, surpassing conventional linear control under external excitation. To improve the effectiveness and stability of vibration lessening in flexible constructions exposed to external perturbations, three time-delayed nonlinear feedback controllers provided enhanced attenuation of the fundamental CB responses compared to conventional linear or instantaneous control. The controllers employed linear/nonlinear procedures of position, velocity, and acceleration. The core methodology was based on the non-perturbative approach, mainly derived from He's frequency formula, which converted a weakly nonlinear oscillator of an ordinary differential equation into an equivalent linear one. The resulting parametric ODE was validated using the Mathematica Software. The stability/instability analyses were explored under multiple scenarios. Moreover, two objective functions were developed to optimize control gains and delay parameters for effective vibration suppression and bifurcation control. Furthermore, the method was grounded in first principles, diminished analytical complexity, and maintained high numerical accuracy in treating nonlinear parametric systems. To explore the nonlinear dynamics of the CB, several tools were employed, including bifurcation diagrams, Poincaré sections, phase portraits, and the largest Lyapunov exponents.
- cantilever beam,
- nonlinear dynamics,
- time-delayed feedback control,
- non-perturbative approach,
- He's frequency formula,
- bifurcation analysis,
- Poincaré sections,
- largest Lyapunov exponent,
- chaotic transitions,
- stability maps,
- nonlinear systems,
- differential equations
Citation: Ahmad Almutlg, Galal M. Moatimid, T. S. Amer, Yasmeen M. Mohamed. Non-perturbative analysis of bifurcation and chaotic dynamics of a cantilever beam under triple time-delayed feedback control[J]. AIMS Mathematics, 2026, 11(6): 17492-17527. doi: 10.3934/math.2026715

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Abstract

In this work, we aimed to improve vibration suppression in a flexible CB structure through the application of time-delays nonlinear feedback control, surpassing conventional linear control under external excitation. To improve the effectiveness and stability of vibration lessening in flexible constructions exposed to external perturbations, three time-delayed nonlinear feedback controllers provided enhanced attenuation of the fundamental CB responses compared to conventional linear or instantaneous control. The controllers employed linear/nonlinear procedures of position, velocity, and acceleration. The core methodology was based on the non-perturbative approach, mainly derived from He's frequency formula, which converted a weakly nonlinear oscillator of an ordinary differential equation into an equivalent linear one. The resulting parametric ODE was validated using the Mathematica Software. The stability/instability analyses were explored under multiple scenarios. Moreover, two objective functions were developed to optimize control gains and delay parameters for effective vibration suppression and bifurcation control. Furthermore, the method was grounded in first principles, diminished analytical complexity, and maintained high numerical accuracy in treating nonlinear parametric systems. To explore the nonlinear dynamics of the CB, several tools were employed, including bifurcation diagrams, Poincaré sections, phase portraits, and the largest Lyapunov exponents.

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