New periodic-chaotic attractors in slow-fast Duffing system with periodic parametric excitation

Xianghong Li,Yongjun Shen,Shaopu Yang,Jian-Qiao Sun

doi:10.1038/s41598-019-46768-7

Xianghong Li, Yongjun Shen + Show 2 more

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https://doi.org/10.1038/s41598-019-46768-7

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Abstract

A new type of responses called as periodic-chaotic motion is found by numerical simulations in a Duffing oscillator with a slowly periodically parametric excitation. The periodic-chaotic motion is an attractor, and simultaneously possesses the feature of periodic and chaotic oscillations, which is a new addition to the rich nonlinear motions of the Duffing system including equlibria, periodic responses, quasi-periodic oscillations and chaos. In the current slow-fast Duffing system, we find three new attractors in the form of periodic-chaotic motions. These are called the fixed-point chaotic attractor, the fixed-point strange nonchaotic attractor, and the critical behavior with the maximum Lyapunov exponent fluctuating around zero. The system periodically switches between one attractor with a fixed single-well potential and the other with time-varying two-well potentials in every period of excitation. This behavior is apparently the mechanism to generate the periodic-chaotic motion.

Highlights

Chaos is a typical motion in nonlinear systems, which is characterized by the unpredictable behavior and extreme sensitivity to initial conditions[1]
We find that the maximum Lyapunov exponent of the critical behavior between fixed-point chaos and nonchaos always oscillates around zero
The periodic and chaotic motions have been found to coexist in the response of the Duffing system with time-varying linear terms over one period

Summary

Consider the Duffing system with periodical parametric excitation

Where b is the damping coefficient, and a and ω are the amplitude and frequency of the excitation. When ω = 0.7 and ω = 0.1355, chaotic attractors exist as shown in the phase plane plots of Fig. 2(a,b) This is the reason that we call this phenomenon as the fixed-point chaos. The sensitivity to initial conditions shown in this system is clearly a property of chaos This phenomenon is common in slow-fast systems with switches between different attractors of the fast subsystem. As the excitation frequency decreases, the fixed-point chaos may turn into another attractor.

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Conclusions

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