一类具有忆阻器的Lorenz 型混沌系统稳定性及余维一分岔分析
Stability and Co-Dimension One Bifurcation Analysis of a Class of Lorenz-Type Chaotic System with Memristor
DOI: 10.12677/AAM.2019.84096, PDF,  被引量 下载: 1,180  浏览: 1,459  国家自然科学基金支持
作者: 黄 俊, 陈玉明：广东技术师范大学，数学与系统科学学院，广东 广州
关键词: Lorenz 系统；Pitchfork 分岔；Hopf 分岔；Lorenz-Type System； Pitchfork Bifurcation； Hopf Bifurcation
摘要: 基于经典的Lorenz 系统，本文通过反馈控制的方式得到了一类具有忆阻器的三维混沌电路系统，并对该系统的局部动力学行为进行了分析。首先，通过分析线性化系统，得到了原点平衡点的局部稳定性性质；其次，基于中心流形及Hopf 分岔理论，对原点平衡点处的余维一Pitchfork 分岔及Hopf 分岔进行了分析，并通过数值仿真进行了验证。
Abstract: Based on the classical Lorenz system, this paper obtains a class of 3D memristive chaotic circuit system through feedback control, and analyzes the local dynamics of this system. Firstly, the local stability  at the origin  of this system  is investigated through analyzing linearized system. Secondly, based on the center manifold theorem and Hopf bifurcation theory, the co-dimension one Pitchfork bifurcation and Hopf bifurcation at the origin of this system are investigated, and the results are verified by numerical simulation.
文章引用：黄俊, 陈玉明. 一类具有忆阻器的Lorenz 型混沌系统稳定性及余维一分岔分析[J]. 应用数学进展, 2019, 8(4): 858-867. https://doi.org/10.12677/AAM.2019.84096

参考文献

[1] Lorenz, E. (1963) Deterministic Non-Periodic Flow. Journal of the Atmospheric Sciences, 20, 130-141.
https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[2] [2] Hirsh, M.W., Smale, S. and Devaney, R.L. (2007) Differential Equations, Dynamical Systems, and an Introduction to Chaos. Elsevier Academic Press, New York.
[3] [3] Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V., et al. (2001) Methods of Qualitative Theory in Nonlinear Dynamics. World Scientific, Singapore.
https://doi.org/10.1142/4221
[4] [4] Wiggins, S. (1990) Introduction to Applied Nonlinear Dynamical Systems and Chaos. 2nd Edition, Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4757-4067-7
[5] [5] Hastings, S. and Troy, W. (1996) A Shooting Approach to Chaos in the Lorenz Equations.
[6] Journal of Differential Equations, 127, 41-53.
https://doi.org/10.1006/jdeq.1996.0060
[7] [6] Chen, A.M., Lu, J.A., Lu, J.H. and Yu, S.M. (2006) Generating Hyperchaotic Lü Attractor via State Feedback Control. Physica A, 364, 103-110.
https://doi.org/10.1016/j.physa.2005.09.039
[8] [7] Zhong, L., Rahman, M.F., Hu, Y.W., Lim, K.W. and Rahman, M.A. (1999) A Direct Torque Controller for Permanent Magnet Synchronous Motor Drives. IEEE Transactions on Energy Conversion, 14, 637-642.
https://doi.org/10.1109/60.790928
[9] [8] Chen, Y.M. and Yang, Q.G. (2014) Complex Dynamics in the Unified Lorenz-Type System.
[10] International Journal of Bifurcation and Chaos, 24, Article ID: 1450055.
[11] [9] 陈玉明, 陈春涛. 一类 Lorenz 型超混沌系统的 Zero-Zero-Hopf 分岔及共存吸引子研究 [J]. 动 力学与控制学报, 2018, 16(3): 227-232.
[12] [10] 张锦炎, 冯贝叶. 常微分方程几何理论与分支问题 [M]. 北京: 北京大学出版社, 2000.
[13] [11] 马知恩, 周义仓, 李承志. 常微分方程定性与稳定性方法 [M]. 北京: 科学出版社, 2015.
[14] [12] Llibre, J. and Zhang, X. (2002) Invariant Algebraic Surfaces of the Lorenz System. Journal of Mathematical Physics, 43, 1622-1645.
https://doi.org/10.1063/1.1435078

为你推荐