article · Journal of low frequency noise, vibration and active control
An investigation into the dynamical, chaotic, and stability properties of the Duffing oscillator subjected to sinusoidal external excitation provides deeper insight into nonlinear oscillatory phenomena. The analytical approach uses the multiple-scales perturbation method to calculate third-order expansions, examine resonant conditions, and incorporate the effects of viscous damping. Numerical verification carried out through the fourth-order Runge-Kutta method validates the accuracy of these analytical solutions. Qualitative dynamics are assessed using bifurcation diagrams, Lyapunov exponent spectra, and Poincaré maps to demonstrate diverse system motions. Furthermore, stability analysis using resonance curves establishes the boundaries of stable and unstable operating regions. The findings clarify how external excitation and quadratic parameters influence transitions in system behaviour, serving as a framework to interpret nonlinear phenomena commonly encountered across physical and engineering systems.
Nonlinear oscillations can trigger sudden, unpredictable, or chaotic motions in physical and mechanical systems. By establishing clear analytical and numerical boundaries between stability and chaos under external forces, this work aids the understanding and management of complex vibrations, supporting the future design and control of systems vulnerable to structural or operational instabilities.
Although the research notes broad relevance to engineering and physics phenomena, the abstract does not indicate an application pathway.
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This paper delves into the dynamical, chaotic, and stability aspects of the Duffing oscillator (DO) under sinusoidal external excitation, underscoring its relevance in various scientific and engineering applications. The DO, with its complex behavior, poses a significant challenge to our understanding. A unique perturbation technique, the multiple-scales (MS), is harnessed to tackle this challenge and enrich our knowledge. The analysis entails determining third-order expansions, exploring resonant cases, and factoring in the influence of viscous damping. The accuracy of the analytical solution is cross-validated with numerical results using the Runge–Kutta fourth-order (RK4). The paper also employs effective methods to assess the obtained results qualitatively. As a result, Visual figures, such as bifurcation diagrams and Lyapunov exponents’ spectra (LES), have been presented to illustrate diverse system motions and demonstrate Poincaré diagrams. Moreover, stability analysis has been investigated via resonance curves showing the stable and unstable regions. These aids facilitate our comprehension of the system’s complex behavior and its variations under different conditions. The results are elucidated through displayed curves, offering insights into the dynamics of the DO under sinusoidal excitation and damping effects. The influence of excitation and the quadratic parameter on bifurcation diagrams, LES, and Poincaré maps helps us understand the intricate behavior of nonlinear oscillatory systems, such as DO. The presented oscillator is a compelling example of elucidating the nonlinear behavior observed in numerous engineering and physics phenomena.
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DOI: 10.1177/14613484241298998
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