Confusion threshold study of the Duffing oscillator with a nonlinear fractional damping term

Wang Mei-Qi,Ma Wen-Li,Chang Yu-Jian,Wanjie Zhang,Chen En-Li,Yang Shao-Pu

doi:10.1177/1461348420922686

Wang Mei-Qi, Ma Wen-Li + Show 4 more

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https://doi.org/10.1177/1461348420922686

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Abstract

In this study, the critical conditions for generating chaos in a Duffing oscillator with nonlinear damping and fractional derivative are investigated. The Melnikov function of the Duffing oscillator is established based on Melnikov theory. The necessary analytical conditions and critical value curves of chaotic motion in the sense of Smale horseshoe are obtained. The numerical solutions of chaotic motion, including time history diagram, frequency spectrum diagram, phase diagram, and Poincare map, are studied. The correctness of the analytical solution is verified through a comparison of numerical and analytical calculations. The effects of linear and nonlinear parameters on chaotic motion are also analyzed. These results are relevant to the study of system dynamics.

Highlights

Nonlinear damping systems are widely used in natural science and engineering technology fields, such as in constructing wire ropes, wire meshes, metal rubber nonlinear dry friction dampers, and other nonlinear dry friction dampers, and in addressing gaps and dead zone characteristics in a system structure
The approximate solution obtained by the traditional perturbation method is influenced by parameters and long-term items
Different from the traditional perturbation method, the new perturbation method assumes that the approximate solution can be expressed as a series of small parameters

Summary

Introduction

Nonlinear damping systems are widely used in natural science and engineering technology fields, such as in constructing wire ropes, wire meshes, metal rubber nonlinear dry friction dampers, and other nonlinear dry friction dampers, and in addressing gaps and dead zone characteristics in a system structure. It has advantages in analyzing the chaos threshold of multi-parameter nonlinear systems.Qin and Xie used the Melnikov method to study the critical value of chaos in a single-degree-of-freedom dry friction oscillator system with linear damping and periodic excitation.[35] Zhu et al used Melnikov’s theory to deduce the condition of chaotic motion in a nonlinear dynamic system under bounded noise and harmonic excitation.[36] nonlinear dry friction dampers and systems with nonlinear damping,[37] such as gap and dead zone characteristics in a system structure, still lack research.

Results

Conclusion