• Dynamics, Vibration and Control •

### FIXED POINT CHAOS AND FOLD/FOLD BURSTING OF A CLASS OF DUFFING SYSTEMS AND THE MECHANISM ANALYSIS 1)

Chen Yani,Meng Wenjing,Qian Youhua()

1. College of Mathematics and Computer Science,Zhejiang Normal University,Jinhua 321004,Zhejiang,China
• Received:2020-04-01 Accepted:2020-04-01 Online:2020-09-18 Published:2020-06-01
• Contact: Qian Youhua

Abstract:

In this paper, a class of bistable Duffing type system with two slow variables under new materials is explored. The system is simulated by time history diagram, phase diagram and bifurcation diagram, then the dynamic mechanism of the system under different parameters is analyzed theoretically. Firstly, this manuscript describes that when the amplitude parameter value is greater than 1, the system may exhibit fixed point chaos and explains the reason of fixed point chaos. Secondly, this manuscript introduces the phenomenon of Fold/Fold bursting in parameter space which is caused by the movement of the system from one side of the saddle-node surface to the other side. We also call it saddle-node bursting. In fact, when the system passes through the saddle-node surface, the number of equilibrium points changes. Then this manuscript uses the path of longitudinal parabolic to explain the mechanism of Fold/Fold bursting. And it is found that regardless of the value of constant coefficient term and amplitude, as long as a certain relationship is satisfied, there will always be Fold/Fold bursting. Next this manuscript uses the linear path to discuss the influence of newly added constant coefficient term. It is found that the position where the path intersects the saddle-node surface will affect the symmetry of the bursting, and the span of the path will affect the magnitude of the bursting oscillation. Finally, this manuscript uses the multiple inflection curve path to discuss the phenomenon when two incentive terms have specific relation. When $n=3$, the change of the constant coefficient term will make the system show Fold/Fold bursting with different times, and the maximum can reach triple bursting. Moreover, it is found that if you can find a path that can be divided into $n$ segments, and each segment will have an intersection with the saddle surface, then $n$ times Fold/Fold bursting will occur.

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