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

Xianwei Chen; Xiangling Fu; Zhujun Jing

doi:10.3934/dcdsb.2020144

Article Contents

2021, Volume 26, Issue 2: 847-860. Doi: 10.3934/dcdsb.2020144

This issue Previous Article Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows Next Article Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields

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
^* Corresponding author: Xianwei Chen

Received: January 2019

Revised: December 2019

Early access: May 2020

Published: February 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 34G20, 34H05; Secondary: 34H10.

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Figure 1. Phase portrait of system (2) for $ \alpha = 0.1 $.

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Figure 2. 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 $.

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Figure 3. 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 $.

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Figure 4. 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 $.

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Figure 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 $.

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Figure 6. 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 $.

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Figure 7. 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 $.

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Figure 8. 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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