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Suppressing chaos in crystal growth process using adaptive phase resonant perturbation

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Abstract

Chaos occurs in the flexible shaft rotating-lifting (FSRL) system of crystal growth process. Chaotic swing does harm to the quality of mono-silicon crystal production. Therefore, it must be suppressed. Previous studies have proposed impulse control method to suppress the chaos in crystal growth process. However, the impulses require sudden and intermittent changes to the rotation speed, which are difficult to implement through the soft rope connection. In this work, a small amplitude resonant perturbation to the rotation speed is being proposed to suppress chaos in the FSRL system. The system state, given by the swing angle between the rotation center on the vertical axis and the soft shaft, is observed by measuring the force on the soft shaft and by using the untraced Kalman filter. The control parameters are selected by calculating the Lyapunov exponent. As compared with the previous impulse control methods, the proposed small amplitude resonant perturbation method engenders a small continuous change instead of the sudden change in the rotation speed. In addition, the proposed method does not alter the average rotation speed, which complies with the crystal growth technique requirement. The effectiveness of the proposed chaos control method is validated by numerical simulations.

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Data Availability Statement

All data generated or analyzed during this study are included within the article.

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Funding

This work was supported in part by the Shaanxi Provincial Special Support Program for Science and Technology Innovation Leader.

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Authors and Affiliations

  1. Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing, Xi’an University of Technology, Xi’an, Shaanxi, 710048, China

    Zi-Xuan Zhou, Hai-Peng Ren & Celso Grebogi

  2. School of Computer Science and Engineering, North Minzu University, Yinchuan, Ningxia, 750021, China

    Zi-Xuan Zhou

  3. Institute for Complex System and Mathematical Biology, University of Aberdeen, Aberdeen, AB24 3UE, UK

    Celso Grebogi

Authors
  1. Zi-Xuan Zhou
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Correspondence to Hai-Peng Ren.

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Appendix

Appendix

Some function definitions are given in the following. For the elliptic integrals function of the first kind,

$$\begin{aligned} \rho =\int _{0}^{\varphi }\frac{dx}{\sqrt{1-m^2\sin ^2x}}, \end{aligned}$$
(A1)

its inverse function is written by:

$$\begin{aligned} \varphi =am(\rho ). \end{aligned}$$
(A2)

Then, Jacobian elliptic functions \(sn(\cdot )\) and \(dn(\cdot )\) are defined by:

$$\begin{aligned} sn(\cdot )=sn(\rho ,m)=\sin \varphi =\sin (am(\rho )), \end{aligned}$$
(A3)
$$\begin{aligned} dn(\cdot )=dn(\rho ,m)=\sqrt{1-m^2sn^2(\rho ,m)}. \end{aligned}$$
(A4)

The complete elliptic integral of the first kind is given by,

$$\begin{aligned} K=K(m)=\int _{0}^{\pi /2}\frac{dx}{\sqrt{1-m^2\sin ^2x}}. \end{aligned}$$
(A5)

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Zhou, ZX., Ren, HP. & Grebogi, C. Suppressing chaos in crystal growth process using adaptive phase resonant perturbation. Nonlinear Dyn 108, 2655–2669 (2022). https://doi.org/10.1007/s11071-022-07333-6

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  • DOI: https://doi.org/10.1007/s11071-022-07333-6

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