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A new multistable chaotic system with memristor and memcapacitor for fractional-order: dynamical analysis, implementation, and synchronization

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Abstract

In this paper, a new multistable chaotic system with memristor and memcapacitor for fractional-order is constructed. A chaotic oscillator of parallel memcapacitor and memristor is designed, the system for fractional-order can be obtained. Then equilibrium point stability of the system is analyzed. The dynamical behavior of order and parameter is analyzed. Interestingly, special dynamical behavior state transfer is found. In addition, the multistability is found with the different initial values. The digital circuit is implemented on the DSP platform that verify the feasibility of the system. Finally, the synchronization controller is designed based on fractional Lyapunov stability theory and its synchronization is investigated. The results show that it has rich dynamical behaviors. It provides a reference for application in confidential communication.

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Data Availability Statement

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. This manuscript has associated data in a data repository [Authors’ comment: The research results in this paper provide a theoretical basis for the application of fractional-order chaotic systems.]

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Acknowledgements

This research work was funded by Institutional Fund Projects under Grant No. (IFPIP: 597-611-1443). The authors gratefully acknowledge technical and financial support provided by the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.

Author information

Authors and Affiliations

  1. School of Information Science and Engineering, Dalian Polytechnic University, Dalian, 116034, China

    Lujie Ren, Jun Mou & Yinghong Cao

  2. Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, Canada

    Hadi Jahanshahi

  3. Communication Systems and Networks Research Group, Department of Information Systems, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia

    Abdullah A. Al-Barakati

Authors
  1. Lujie Ren
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  2. Jun Mou
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  3. Hadi Jahanshahi
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  4. Abdullah A. Al-Barakati
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  5. Yinghong Cao
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Contributions

LR designed and carried out experiments, data analyzed and manuscript wrote. JM, HJ, and AAAl-B made the theoretical guidance for this paper. YC improved the algorithm. All authors reviewed the manuscript.

Corresponding authors

Correspondence to Jun Mou or Yinghong Cao.

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Appendix A

Appendix A

$$\begin{aligned}{} & {} \ \left\{ \begin{array}{l} c_1^2 = - c_1^1 - c_2^1\\ c_2^2 = ac_1^1 - a\big ({k_1}\big (c_3^1\sin \big (bc_5^0\big )\mathrm{{ }} + nc_3^0c_5^1\cos \big (bc_5^0\big )\big ) + {k_2}{b^2}\big (c_3^1c_5^0c_5^0 + 2c_3^0c_5^1c_5^0\big )\big )\\ c_3^2 = c_2^1 - \big ({k_1}{k_3}\big (c_3^1\sin \big (cc_4^0\big )\sin \big (bc_5^0\big ) + cc_3^0c_4^1\cos \big (cc_4^0\big )\sin \big (bc_5^0\big ) + bc_3^0c_5^1\cos \big (bc_5^0\big )\sin \big (cc_4^0\big )\big )\\ \quad \qquad + {k_2}{k_3}{b^2}\bigg (c_3^1{ \big (c_5^0\big )^2}\sin \big (cc_4^0\big ) + cc_3^0c_4^1{ \big (c_5^0\big )^2}\cos \big (cc_4^0\big ) + 2c_3^0 \big (bc_5^0\big )c_5^1\sin \big (cc_4^0\big ) \bigg )\\ \quad \qquad + {k_1}{k_4}{c^2}\big (c_3^1{\big (c_4^0\big )^2}\sin \big (bc_5^0\big ) + bc_3^0{\big (c_4^0\big )^2}c_5^1\cos \big (bc_5^0\big ) + 2c_3^0c_4^0c_4^1\sin \big (bc_5^0\big )\big )\\ \quad \qquad + {k_2}{k_4}{b^2}{c^2}\big (c_3^1{\big (c_4^0\big )^2}{\big (c_5^0\big )^2} + 2c_3^0c_5^1{\big (c_4^0\big )^2}c_5^0 + 2c_3^0c_4^1c_4^0{\big (c_5^0\big )^2}\big )\big )\\ c_4^2 = {k_1}\big (c_3^1\sin \big (bc_5^0\big )\mathrm{{ }} + c_3^0c_5^1\cos \big (bc_5^0\big )\big ) + {k_2}{b^2}\big (c_3^1c_5^0c_5^0 + 2c_3^0c_5^1c_5^0\big )\\ c_5^2 = c_3^1 \end{array} \right. \end{aligned}$$
(36)
$$\begin{aligned}{} & {} \ \left\{ {\begin{array}{*{20}{l}} {c_1^3 = - c_1^2 - c_2^2}\\ c_2^3 = ac_1^2 - a\bigg ({k_1}\big (c_3^2\sin \big (bc_5^0\big ) + bc_3^0c_5^2\cos \big (bc_5^0\big ) + \bigg (bc_3^1c_5^1\cos \big (bc_5^0\big ) - \big ({b^2}c_3^0{\big (c_5^1\big )^2}sin\big (bc_5^0\big )\big )/2\bigg ) \frac{{\Gamma \big (2q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^2}}}\bigg )\\ \quad \qquad + {k_2}{b^2}\bigg (c_3^2{\big (c_5^0\big )^2} + 2c_3^0c_5^2c_5^0 + \big (2c_3^1c_5^0c_5^1 + c_3^0{\big (c_5^1\big )^2}\big )\frac{{\Gamma \big (2q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^2}}}\bigg )\bigg )\\ c_3^3 = ac_1^2 - a\big ({k_1}{k_3}\big (c_3^2\sin \big (cc_4^0\big )\sin \big (bc_5^0\big ) + cc_3^0c_4^2\cos \big (cc_4^0\big )\sin \big (bc_5^0\big ) + bc_3^0c_5^2\cos \big (bc_5^0\big )\sin \big (cc_4^0\big )\\ \quad \qquad + \bigg (cc_3^1c_4^1\cos \big (cc_4^0\big )\sin \big (bc_5^0\big ) + nc_3^1c_5^1\cos \big (bc_5^0\big )\sin \big (cc_4^0\big ) - \big ({c^2}c_3^0{\big (c_4^1\big )^2}\sin \big (cc_4^0\big )\sin \big (bc_5^0\big )\big )/2\\ \quad \qquad - \big ({b^2}c_3^0{\big (c_5^1\big )^2}\sin \big (cc_4^0\big )\sin \big (bc_5^0\big )\big )/2 + cbc_3^0c_4^1c_5^1\cos \big (cc_4^0\big )\cos \big (bc_5^0\big )\big )\frac{{\Gamma \big (2q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^2}}}\bigg )\\ \quad \qquad + {k_2}{k_3}{b^2}\big (c_3^2{\big (c_5^0\big )^2}\sin \big (cc_4^0\big ) + cc_3^0c_4^2{\big (c_5^0\big )^2}\cos \big (cc_4^0\big ) + 2c_3^0c_5^0c_5^2\sin \big (cc_4^0\big )\\ \quad \qquad + \bigg (c_3^0{\big (c_5^1\big )^2}\sin \big (cc_4^0\big ) + cc_3^1c_4^1{\big (c_5^0\big )^2}\cos \big (cc_4^0\big ) - \big ({c^2}c_3^0{\big (c_4^1\big )^2}{\big (c_5^0\big )^2}\sin \big (cc_4^0\big )\big )/2\\ \quad \qquad + 2c_3^1c_5^0c_5^1\sin \big (cc_4^0\big ) + 2cc_3^0c_4^1c_5^0c_5^1\cos \big (cc_4^0\big )\big )\frac{{\Gamma \big (2q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^2}}}\bigg )+ {k_1}{k_4}{c^2}\big (c_3^2{\big (c_4^0\big )^2}\sin \big (bc_5^0\big )\\ \quad \qquad + bc_3^0{\big (c_4^0\big )^2}c_5^2\cos \big (bc_5^0\big ) + 2c_3^0c_4^0c_4^2\sin \big (bc_5^0\big )\\ \quad \qquad + \big (c_3^0{\big (c_4^1\big )^2}\sin \big (bc_5^0\big ) + bc_3^1{\big (c_4^0\big )^2}c_5^1\cos \big (bc_5^0\big ) - \big ({b^2}c_3^0{\big (c_4^0\big )^2}{\big (c_5^1\big )^2}\sin \big (bc_5^0\big )\big )/2\\ \quad \qquad + 2c_3^1c_4^0c_4^1\sin \big (bc_5^0\big ) + 2bc_3^0c_4^0c_4^1c_5^1\cos \big (bc_5^0\big )\big )\frac{{\Gamma \big (2q + 1\big )}}{{\big (\Gamma {{\big (q + 1\big )}^2}\big )}}\big )\\ \quad \qquad + {k_2}{k_4}{c^2}{b^2}\big (c_3^2{\big (c_4^0\big )^2}{\big (c_5^0\big )^2} + 2c_3^0c_5^2{\big (c_4^0\big )^2}c_5^0 + 2c_3^0c_4^2c_4^0{\big (c_5^0\big )^2}\\ \quad \qquad + \big (2c_3^1{\big (c_4^0\big )^2}c_5^0c_5^1 + c_3^0{\big (c_4^0\big )^2}{\big (c_5^1\big )^2} + 2c_3^1c_4^0c_4^1{\big (c_5^0\big )^2} + 4c_3^0c_4^0c_4^1c_5^0c_5^1 + c_3^0{\big (c_4^1\big )^2}{\big (c_5^0\big )^2}\big )\frac{{\Gamma \big (2q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^2}}}\bigg )\bigg )\\ c_4^3 = {k_1}\bigg (c_3^2\sin \big (bc_5^0\big ) + bc_3^0c_5^2\cos \big (bc_5^0\big ) + \big (bc_3^1c_5^1\cos \big (bc_5^0\big ) - \bigg ({b^2}c_3^0{ \big (c_5^1\big )^2} \sin \big (bc_5^0\big ) \bigg )/2\bigg ) \frac{{\Gamma \big (2q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^2}}}\bigg )\\ \quad \qquad + {k_2}{b^2}\bigg (c_3^2{\big (c_5^0\big )^2} + 2c_3^0c_5^2c_5^0 + \bigg (2c_3^1c_5^0c_5^1 + c_3^0{\big (c_5^1\big )^2}\bigg ) \frac{{\Gamma \big (2q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^2}}}\bigg )\\ {c_5^3 = c_3^2} \end{array}} \right. \end{aligned}$$
(37)
$$\begin{aligned}{} & {} \ \left\{ \begin{array}{l} c_1^4 = - c_1^3 - c_2^3\\ c_2^4 = ac_1^3 - a\big ({k_1}\big (c_3^3\sin \big (bc_5^0\big ) + bc_3^0c_5^3\cos \big (bc_5^0\big )+ \big (bc_3^1c_5^2\cos \big (bc_5^0\big ) + bc_3^2c_5^1\cos \big (bc_5^0\big ) \\ \quad \qquad - bc_3^0c_5^1c_5^2\sin \big (bc_5^0\big )\big )\frac{{\Gamma \big (3q + 1\big )}}{{\Gamma \big (q + 1\big )\Gamma \big (2q + 1\big )}} - \big ({b^3}c_3^0{\big (c_5^1\big )^3}\cos \big (bc_5^0\big )/6\\ \quad \qquad + {b^2}c_3^1{\big (c_5^1\big )^2}\sin \big (bc_5^0\big )/2\big )\frac{{\Gamma \big (3q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^3}}}\big ) + {k_2}\big (c_3^3{\big (c_5^0\big )^2} + 2c_3^0c_5^0c_5^3 + c_3^1{\big (c_5^1\big )^2}\frac{{\Gamma \big (3q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^3}}} \\ \quad \qquad + \big (2c_3^0c_5^1c_5^2 + 2c_3^1c_5^0c_5^2 + 2c_3^2c_5^0c_5^1\big )\frac{{\Gamma \big (3q + 1\big )}}{{\Gamma \big (q + 1\big )\Gamma \big (2q + 1\big )}}\big )\big )\\ c_3^4 = \big (c_2^3 - \big ({k_1}{k_3}\big (c_3^3\sin \big (cc_4^0\big )\sin \big (bc_5^0\big ) + mc_3^0c_4^3\cos \big (cc_4^0\big )\sin \big (bc_5^0\big ) + cc_3^0c_5^3\cos \big (cc_4^0\big )\sin \big (bc_5^0\big )\\ \quad \qquad + \big (cc_3^1c_4^2\cos \big (cc_4^0\big )\sin \big (bc_5^0\big ) + c_3^2c_4^1\cos \big (cc_4^0\big )\sin \big (bc_5^0\big ) + bc_3^1c_5^2\cos \big (bc_5^0\big )\sin \big (cc_4^0\big )\\ \quad \qquad + bc_3^2c_5^1\cos \big (bc_5^0\big )\sin \big (cc_4^0\big ) + bcc_3^0c_4^1c_5^2\cos \big (cc_4^0\big )\cos \big (bc_5^0\big ) + bcc_3^0c_4^2c_5^1\cos \big (cc_4^0\big )\cos \big (bc_5^0\big )\\ \quad \qquad - {c^2}c_3^0c_4^1c_4^2\sin \big (cc_4^0\big )\sin \big (bc_5^0\big ) - {b^2}c_3^0c_5^1c_5^2\sin \big (cx_4^0\big )\sin \big (bx_5^0\big )\big )\frac{{\Gamma \left( {3q + 1} \right) }}{{\Gamma \left( {q + 1} \right) \Gamma \left( {2q + 1} \right) }}\mathrm{{ }}\\ \quad \qquad + \mathrm{{ }}\big (bcc_3^1c_4^1c_5^1\cos \big (cc_4^0\big )\cos \big (bc_5^0\big ) - \big (b{c^2}c_3^0{\big (c_4^1\big )^2}c_5^1\cos \big (bc_5^0\big )\sin \big (cc_4^0\big )\big )/2 - \big ({b^2}cc_3^0c_4^1{\big (c_5^1\big )^2}\cos \big (cc_4^0\big )\sin \big (bc_5^0\big )\big )/2\\ \quad \qquad - \big ({c^3}c_3^0{\big (c_4^1\big )^3}\cos \big (cx_4^0\big )\sin \big (bx_5^0\big )\big )/6 - \big ({b^3}c_3^0{\big (c_5^1\big )^3}\cos \big (bc_5^0\big )\sin \big (cc_4^0\big )\big )/6 - \big ({c^2}c_3^1{\big (c_4^1\big )^2}\sin \big (cc_4^0\big )\sin \big (bc_5^0\big )\big )/2\\ \quad \qquad - \big ({b^2}c_3^1{\big (c_5^1\big )^2}\sin \big (cc_4^0\big )\sin \big (bc_5^0\big )\big )/2\big )\frac{{\Gamma \big (3q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^3}}}\big )\big )+ {k_2}{k_3}{b^2}\big (c_3^3{\big (c_5^0\big )^2}\sin \big (cc_4^0\big )\\ \quad \qquad + cc_3^0c_4^3{\big (c_5^0\big )^2}\cos \big (cc_4^0\big ) + 2c_3^0c_5^0c_5^3\sin \big (cc_4^0\big )\\ \quad \qquad + \big (cc_3^1c_4^2{\big (c_5^0\big )^2}\cos \big (cc_4^0\big ) + cc_3^2c_4^1{\big (c_5^0\big )^2}\cos \big (cc_4^0\big ) + 2c_3^0c_5^1c_5^2\sin \big (cc_4^0\big ) + 2c_3^1c_5^0c_5^2\sin \big (cc_4^0\big )\\ \quad \qquad + 2c_3^2c_5^0c_5^1\sin \big (cc_4^0\big ) + 2cc_3^0c_4^1c_5^0c_5^2\cos \big (cc_4^0\big ) + 2cc_3^0c_4^2c_5^0c_5^1\cos \big (cc_4^0\big ) - {c^2}c_3^0c_4^1c_4^2{\big (c_5^0\big )^2}\sin \big (cc_4^0\big )\big )\frac{{\Gamma \left( {3q + 1} \right) }}{{\Gamma \left( {q + 1} \right) \Gamma \left( {2q + 1} \right) }}\\ \quad \qquad + \big (c_3^1{\big (c_5^1\big )^2}\sin \big (cc_4^0\big ) + cc_3^0c_4^1{\big (c_5^1\big )^2}\cos \big (cc_4^0\big ) - \big ({c^3}c_3^0{\big (c_4^1\big )^3}{\big (c_5^0\big )^2}\cos \big (cc_4^0\big )\big )/6 - \big ({m^2}c_3^1{\big (c_4^1\big )^2}{\big (c_5^0\big )^2}\sin \big (cc_4^0\big )\big )/2\\ \quad \qquad + 2cx_3^1x_4^1x_5^0x_5^1\cos \big (cx_4^0\big ) - {c^2}x_3^0{\big (x_4^1\big )^2}x_5^0x_5^1\sin \big (cx_4^0\big )\big )\frac{{\Gamma \big (3q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^3}}}\big )\\ \quad \qquad + {k_1}{k_4}{c^2}\big (c_3^3{\big (c_4^0\big )^2}\sin \big (bc_5^0\big ) + bc_3^0{\big (c_4^0\big )^2}c_5^3\cos \big (bc_5^0\big ) + 2c_3^0c_4^0c_4^3\sin \big (bc_5^0\big )\\ \quad \qquad + \big (bc_3^1{\big (c_4^0\big )^2}c_5^2\cos \big (bc_5^0\big ) + bc_3^2{\big (c_4^0\big )^2}c_5^1\cos \big (bc_5^0\big ) + 2c_3^0c_4^1c_4^2\sin \big (bc_5^0\big ) + 2c_3^1c_4^0c_4^2\sin \big (bc_5^0\big ) + 2c_3^2c_4^0c_4^1\sin \big (bc_5^0\big )\\ \quad \qquad + 2bc_3^0c_4^0c_4^1c_5^2\cos \big (bc_5^0\big ) + 2bc_3^0c_4^0c_4^2c_5^1\cos \big (bc_5^0\big ) - {b^2}c_3^0{\big (c_4^0\big )^2}c_5^1c_5^2\sin \big (bc_5^0\big )\big )\frac{{\Gamma \big (3q + 1\big )}}{{\Gamma \big (q + 1\big )\Gamma \big (2q + 1\big )}}\\ \quad \qquad + \big (c_3^1{\big (c_4^1\big )^2}\sin \big (bc_5^0\big ) + bc_3^0{\big (c_4^1\big )^2}c_5^1\cos \big (bc_5^0\big ) - \big ({b^3}c_3^0{\big (c_4^0\big )^2}{\big (c_5^1\big )^3}\cos \big (bc_5^0\big )\big )/6 - \big ({b^2}c_3^1{\big (c_4^0\big )^2}{\big (c_5^1\big )^2}\sin \big (bc_5^0\big )\big )/2\\ \quad \qquad + 2bc_3^1c_4^0c_4^1c_5^1\cos \big (bc_5^0\big ) - {b^2}c_3^0c_4^0c_4^1{\big (c_5^1\big )^2}\sin \big (bc_5^0\big )\big )\frac{{\Gamma \big (3q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^3}}}\big )\\ \quad \qquad + {k_2}{k_4}{b^2}{c^2}\big (c_3^3{\big (c_4^0\big )^2}{\big (c_5^0\big )^2} + 2c_3^0c_4^0c_4^3{\big (c_5^0\big )^2} + 2c_3^0{\big (c_4^0\big )^2}c_5^0c_5^3\\ \quad \qquad + \big (2c_3^0c_4^1c_4^2{\big (c_5^0\big )^2} + 2c_3^1c_4^0c_4^2{\big (c_5^0\big )^2} + 2c_3^2c_4^0c_4^1{\big (c_5^0\big )^2} + 2c_3^0{\big (c_4^0\big )^2}c_5^1c_5^2 + 2c_3^1{\big (c_4^0\big )^2}c_5^0c_5^2\\ \quad \qquad + 2c_3^2{\big (c_4^0\big )^2}c_5^0c_5^1 + 4c_3^0c_4^0c_4^1c_5^0c_5^2 + 4c_3^0c_4^0c_4^2c_5^0c_5^1\big )\frac{{\Gamma \left( {3q + 1} \right) }}{{\Gamma \left( {q + 1} \right) \Gamma \left( {2q + 1} \right) }}\\ \quad \qquad + \big (c_3^1{\big (c_4^0\big )^2}{\big (c_5^1\big )^2} + c_3^1{\big (c_4^1\big )^2}{\big (c_5^0\big )^2} + 2c_3^0c_4^0c_4^1{\big (c_5^1\big )^2} + 2c_3^0{\big (c_4^1\big )^2}c_5^0c_5^1 + 4c_3^1c_4^0c_4^1c_5^0c_5^1\big )\frac{{\Gamma \big (3q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^3}}}\big )\big )\\ c_4^4 = {k_1}\big (c_3^3\sin \big (bc_5^0\big ) + bc_3^0c_5^3\cos \big (bc_5^0\big ) + \big (bc_3^1c_5^2\cos \big (bc_5^0\big ) + bc_3^2c_5^1\cos \big (bc_5^0\big ) - {b^2}c_3^0c_5^1c_5^2\sin \big (bc_5^0\big )\big )\frac{{\Gamma \big (3q + 1\big )}}{{\Gamma \big (q + 1\big )\Gamma \big (2q + 1\big )}}\\ \quad \qquad - \big ({b^3}c_3^0{\big (c_5^1\big )^3}\cos \big (bc_5^0\big )/6 + {b^2}c_3^1{\big (c_5^1\big )^2}\sin \big (bc_5^0\big )/2\big )\frac{{\Gamma \big (3q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^3}}}\big )+ {k_2}\big (c_3^3{\big (c_5^0\big )^2} \\ \quad \qquad + 2c_3^0c_5^0c_5^3 + c_3^1{\big (c_5^1\big )^2}\frac{{\Gamma \big (3q + 1\big )}}{{{{\big (\Gamma \big (q + 1\big )\big )}^3}}} + \big (2c_3^0c_5^1c_5^2 + 2c_3^1c_5^0c_5^2 + 2_3^2c_5^0c_5^1\big )\frac{{\Gamma \big (3q + 1\big )}}{{\Gamma \big (q + 1\big )\Gamma \big (2q + 1\big )}}\big )\\ c_5^4 = c_3^3\end{array} \right. \end{aligned}$$
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Ren, L., Mou, J., Jahanshahi, H. et al. A new multistable chaotic system with memristor and memcapacitor for fractional-order: dynamical analysis, implementation, and synchronization. Eur. Phys. J. Plus 138, 748 (2023). https://doi.org/10.1140/epjp/s13360-023-04379-2

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