Dynamical Analysis of a New Chaotic System and Its Sliding Mode Control
ZHANG Hong, ZHANG Fuchen
Chongqing Key Laboratory of Statistical Intelligent Computing and Monitoring, School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
Abstract：This paper proposes a novel three-dimensional autonomous chaotic system in order to discover more chaotic behaviors in electrons and circuits, which includes four parameters and three nonlinear coupling terms. Based on the Lagrangian stability theory, the quantitative estimate expression of the ultimate bound of this system was strictly derived by constructing the generalized Lyapunov function. A new sliding mode control strategy was designed based on the Vaidyanathan theorem, achieving global asymptotic synchronization of the system. The research results show that the system has a global exponential attractive set. Numerical simulation verifies the strong robustness and fast convergence of the sliding mode control, and its performance is superior to the adaptive control. This achievement not only enriches the theory of chaotic systems but also provides an effective control method for engineering applications.
张洪, 张付臣. 一个新混沌系统动力学分析及其滑模控制[J]. 复杂系统与复杂性科学, 2026, 23(2): 86-93.
ZHANG Hong, ZHANG Fuchen. Dynamical Analysis of a New Chaotic System and Its Sliding Mode Control[J]. Complex Systems and Complexity Science, 2026, 23(2): 86-93.
[1] LORENZ E N. Deterministic non-periods flows[J]. Journal of the Atmospheric Science,1963, 20:130-141. [2] ZHANG F C, XU F, ZHANG X. Qualitative behaviors of a four-dimensional Lorenz system[J]. Journal of Physics A: Mathematical and Theoretical, 2024, 57(9): 095201. [3] ZHANG F C, CHEN S, CHEN X S, et al. Qualitative behaviors and control of a new four-dimensional Lorenz system[J]. Journal of Applied Analysis and Computation, 2025, 15(5): 3025-3044. [4] ZHANG F C, ZHOU P, XU F. Qualitative properties of a physically extended six-dimensional Lorenz system[J]. International Journal of Bifurcation and Chaos, 2024, 34(7): 2450083. [5] 周雯静, 张付臣. 新非线性混沌系统动力学分析及仿真[J]. 复杂系统与复杂性科学, 2025, 22(1): 77-82. ZHOU W J, ZHANG F C. Analysis and simulations of a new nonlinear chaotic system[J]. Complex Systems and Complexity Science, 2025, 22(1): 77-82. [6] ZHANG F C, LIAO X F, ZHANG G Y, et al. Dynamical behaviors of a generalized Lorenz family[J]. Discrete and Continuous Dynamical Systems-Series B, 2017, 22(10): 3707-3720. [7] 廖晓昕, 罗琦. Lorenz混沌系统Lyapunov稳定性简洁的代数充要条件及其应用[J]. 中国科学: 信息科学, 2010, 8: 1086-1095. LIAO X X, LUO Q. Simple algebraic necessary and sufficient conditions for the stability of Lorenz chaotic system lyapunov and their applications[J]. Science China: Information Science, 2010 (8): 1086-1095. [8] 陈关荣, 吕金虎. Lorenz 系统族的动力学分析、控制与同步[M]. 北京: 科学出版社, 2003. CHEN G R, LU J H. Dynamic analysis, control and synchronization of the Lorenz system family[M]. Beijing: Science Press, 2003. [9] PECORA L M, CARRLOLL T L. Synchronization in chaotic systems[J]. Physical review letters, 1990, 64(8): 821-825. [10] YASSEN M T. Chaos synchronization between two different chaotic systems using active control[J]. Chaos, Solitons & Fractals, 2005, 23(1): 131-140. [11] ADLOO H, ROOPAEI M. Review article on adaptive synchronization of chaotic systems with unknown parameters[J]. Nonlinear Dynamics, 2011, 65: 141-159. [12] 陈松, 张付臣, 肖敏. 一个复杂混沌系统的分析与同步控制[J]. 系统科学与数学, 2024, 44 (5): 1311-1323. CHEN S, ZHANG F C, XIAO M. Analysis and synchronization control of a complex chaotic system[J]. Journal of Systems Science and Mathematical Sciences, 2024, 44 (5): 1311-1323. [13] KHATTAR D, AGRAWAL N, SINGHi G. Chaos synchronization of a new chaotic system having exponential term via adaptive and sliding mode control[J]. Differential Equations and Dynamical Systems, 2025, 33(2):475-493. [14] 廖晓昕, 罗海庚, 傅予力, 等. 论Lorenz系统族的全局指数吸引集和正向不变集[J]. 中国科学(E辑:信息科学), 2007, 6: 757-769. LIAO X X, LUO H G, FU Y L, et al. Analysis on the global exponential set and positive invariant set of the Lorenz family[J]. Science China (Technology Science), 2007, 37: 757-769. [15] 廖晓昕. Lorenz混沌族中若干数学问题新研究[M]. 武汉: 华中科技大学出版社, 2017. LIAO X X. New Research on Some Mathematical Problems in Lorenz Chaotic Family[M]. Wuhan: Huazhong University of Science and Technology Press, 2017. [16] ZHANG F C, ZHANG G Y. Further results on ultimate bound on the trajectories of the Lorenz system[J]. Qualitative Theory of Dynamical Systems, 2016, 15: 221-235. [17] VAIDYANATHAN S, SAMPATH S, AZAR A T. Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system[J]. International Journal of Modelling, Identification and Control, 2015, 23(1): 92-100. [18] 秦铭宏, 赖强, 吴永红. 具有无穷共存吸引子的简单忆阻混沌系统的分析与实现[J]. 物理学报, 2022, 71(16): 160502. QIN M H, LAI Q, WU Y H. Analysis and implementation of simple memristor chaotic systems with infinite coexisting attractors[J]. Acta Physica Sinica, 2022, 71(16): 160502. [19] 赖强, 王君. 基于滑模趋近律的忆阻混沌系统有限和固定时间同步[J]. 物理学报, 2024, 73(18): 75-83. LAI Q, WANG J. Finite and fixed-time synchronization of memristive chaotic systems based on sliding mode reaching law[J]. Acta Physica Sinica, 2024, 73(18): 75-83. [20] 赖强, 秦铭宏. 基于忆阻电磁辐射的极简环形HNN 动力学行为增强研究[J]. 贵州师范大学学报(自然科学版), 2025, 43(3): 1-11. LAI Q, QIN M H. Enhanced dynamics of an extremely simple cyclic HNN based on memristive electromagnetic radiation[J]. Journal of Guizhou Normal University (Natural Sciences), 2025, 43(3): 1-11. [21] QI G Y, CHEN G R, DU S Z, et al. Analysis of a new chaotic system[J]. Physica A: Statistical Mechanics and its Applications, 2005, 352(2-4): 295-308.
