AccScience Publishing / NSCE / Online First / DOI: 10.36922/NSCE026140012
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RESEARCH ARTICLE

A simple autonomous Jerk system with coexisting multiple attractors and its cryptographic application

Song Liu1,2 ,  Zihan Li1 ,  Min Luo1 ,  Da Qiu1 ,  Xu Quan3 ,  Sheng Xu4 ,  Guoping Zhang2*
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1 College of Intelligence Science and Engineering, Hubei Minzu University, Enshi, Hubei, China
2 College of Physical Science and Technology, Central China Normal University, Wuhan, Hubei, China
3 Guilin Power Supply Bureau, Guangxi Power Grid Co. Ltd, Guilin, Guangxi, China
4 Radiation Oncology Center, The Central Hospital of Enshi Tujia and Miao Autonomous Prefecture, Enshi, Hubei, China
NSCE, 026140012 https://doi.org/10.36922/NSCE026140012
Received: 31 March 2026 | Revised: 1 May 2026 | Accepted: 20 May 2026 | Published online: 12 June 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

In nonlinear dynamical systems, multiple attractors emerge, with the system converging to different attractors depending on the initial conditions, leading to complex dynamic behavior. This paper introduces a simple autonomous jerk system, which combines hyperbolic sine and absolute value functions. By adjusting the system’s parameters, we explore its dynamic behavior, uncovering various complex phenomena such as period-doubling bifurcations, antimonotonicity, transient chaos, periodic windows, and multiple attractors, which collectively illustrate the system’s rich dynamics. Furthermore, we examine the basins of attraction for each attractor, providing insights into the system’s behavior under various initial conditions. Through this analysis of the basins, we deepen our understanding of the system’s diversity and complexity, highlighting its sensitivity to initial conditions. This comprehensive analysis provides essential clues for future investigations. Additionally, the system is implemented on a field-programmable gate array platform, demonstrating its practicality for real-world use. Finally, the chaotic sequences generated by the system are applied to image encryption, and the influence of multistability on encryption performance is assessed. The results show that chaotic-state encryption outperforms periodic-state encryption in terms of cryptographic security. These findings emphasize the considerable impact of multiple attractors on the system’s dynamic behavior and encryption performance, offering new theoretical insights and approaches for further research in this area.

Keywords
Multistability
Hopf bifurcation
Antimonotonicity
Hardware implementation
Image encryption
Funding
This work was supported by the China University- Industry-Research Innovation Fund (Grant No. 2024IT118), and the Hubei Provincial Natural Science Joint Fund (Grant No. 2026AFC0242).
Conflict of interest
The authors declare no conflicts of interest.
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