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February  2021, 26(2): 847-860. doi: 10.3934/dcdsb.2020144

## Chaos control in a special pendulum system for ultra-subharmonic resonance

 1 School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, China 2 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Xianwei Chen

Received  January 2019 Revised  December 2019 Published  February 2021 Early access  May 2020

Fund Project: This work is supported by the Province Natural Science Foundation of Hunan (No. 2018JJ2110)

In this paper, we study the chaos control of pendulum system with vibration of suspension axis for ultra-subharmonic resonance by using Melnikov methods, and give a necessary condition for controlling heteroclinic chaos and homoclinic chaos, respectively. We give some bifurcation diagrams by numerical simulations, which indicate that the chaos behaviors for ultra-subharmonic resonance may be inhibited to periodic orbits by adjusting phase-difference of parametric excitation, and prove that results obtained are very effective in inhibiting chaos for ultra-subharmonic resonance.

Citation: Xianwei Chen, Xiangling Fu, Zhujun Jing. Chaos control in a special pendulum system for ultra-subharmonic resonance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 847-860. doi: 10.3934/dcdsb.2020144
##### References:

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##### References:
Phase portrait of system (2) for $\alpha = 0.1$.
The chaotic attractor of system (1) for $\alpha = 0.1, \; \omega = 1.5, \; \delta = 0.38, \; f_1 = 1.381$, $\gamma = 0.01$ and $f_0 = 0$.
The bifurcation diagram of system (1) in ($\Psi$, x) plane for $\alpha = 0.1$, $f_1 = 1.381$, $f_0 = 0.2$, $\delta = 0.38$, $\Omega = 0.75$ and $\omega = 1.5$.
The bifurcation diagram of system (1) in ($\Psi$, x) plane for $\alpha = 0.1$, $f_1 = 1.381$, $f_0 = 0.2$, $\delta = 0.38$, $\Omega = 0.5$ and $\omega = 1.5$.
The bifurcation diagram of system (1) in ($\Psi$, x) plane for $\alpha = 0.1$, $f_1 = 1.381$, $f_0 = 0.4$, $\delta = 0.38$, $\Omega = 1$ and $\omega = 1.5$.
The bifurcation diagram of system (1) in ($\Psi$, x) plane for $\alpha = 0.1$, $f_1 = 1.381$, $f_0 = 2$, $\delta = 0.38$, $\Omega = 0.75$ and $\omega = 1.5$.
The bifurcation diagram of system (1) in ($\Psi$, x) plane for $\alpha = 0.1$, $f_1 = 1.381$, $\delta = 0.38$, $\Omega = 0.5$ and $\omega = 1.5$: (a) $f_0 = 1$; (b) $f_0 = 4$.
The bifurcation diagram of system (1) in ($\Psi$, x) plane for $\alpha = 0.1$, $f_1 = 1.381$, $\delta = 0.38$, $\Omega = 1$ and $\omega = 1.5$: (a) $f_0 = 2$; (b) $f_0 = 2.5$.
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