广义Hopf-Langford系统的可积性研究
Study on the Integrability of the General Hopf-Langford System
DOI: 10.12677/aam.2025.149400, PDF,    国家自然科学基金支持
作者: 马林祺：中国地质大学(武汉)数学与物理学院，湖北 武汉
关键词: 广义Hopf-Langford系统；可积性；不变代数曲面；Darboux多项式；Puiseux级数；General Hopf-Langford System； Integrability； Invariant Algebraic Surfaces； Darboux Polynomial； Puiseux Series
摘要: 该文研究了一个三维自治广义Hopf-Langford系统的可积性和不变代数曲面。首先介绍了一种利用Puiseux级数研究二维系统不变代数曲线的方法，然后利用柱坐标变换和该方法，证明了系统在三个特殊情况下是可积的，并且得到了该系统的不变代数曲面。
Abstract: This paper investigates the integrability and invariant algebraic surfaces of a three-dimensional autonomous general Hopf-Langford system. During the study, we first introduce a recently proposed method that uses Puiseux series to study invariant algebraic curves of two-dimensional systems. Then, it is proved that the system is integrable in three special cases, and the invariant algebraic surfaces of the system are obtained by utilizing cylindrical coordinate transformations and this method.
文章引用：马林祺. 广义Hopf-Langford系统的可积性研究[J]. 应用数学进展, 2025, 14(9): 65-75. https://doi.org/10.12677/aam.2025.149400

参考文献

[1] Zhao, M., Yang, Q. and Zhang, X. (2024) Dynamical Analysis of a Class of Generalized Chua’s Systems with Infinitely Many Attractors. Computational and Applied Mathematics, 43, Article No. 367. [Google Scholar] [CrossRef] 
[2] Lu, X. and Yang, Q. (2024) Jacobi Stability, Hamilton Energy and the Route to Hidden Attractors in the 3D Jerk Systems with Unique Lyapunov Stable Equilibrium. Physica D: Nonlinear Phenomena, 470, Article ID: 134423. [Google Scholar] [CrossRef] 
[3] Zhang, L. and Ding, Y. (2024) A Novel Hyperchaotic System with Corresponding Multi-Scroll System. International Journal of Dynamics and Control, 12, 1-15.
[4] Li, C., Ma, Z. and Zhou, Y. (2016) Periodic Orbits in 3-Dimensional Systems and Application to a Perturbed Volterra System. Journal of Differential Equations, 260, 2750-2762. [Google Scholar] [CrossRef] 
[5] Belozyorov, V.Y. (2015) Exponential-Algebraic Maps and Chaos in 3D Autonomous Quadratic Systems. International Journal of Bifurcation and Chaos, 25, Article 1550048. [Google Scholar] [CrossRef] 
[6] Zhang, Y., Zhang, W. and Yue, Y. (2024) Hopf Bifurcation Analysis of a Nose Landing Gear System of Aircraft. International Journal of Bifurcation and Chaos, 34, Article ID: 2450184. [Google Scholar] [CrossRef] 
[7] Goriely, A. (2001) Integrability and Nonintegrability of Dynamical Systems. World Scientific. [Google Scholar] [CrossRef] 
[8] Lakshmanan, M. and Rajaseekar, S. (2012) Nonlinear Dynamics: Integrability, Chaos and Patterns. Springer Science & Business Media.
[9] 马文秀. 一族Liouville可积的有限维Hamilton系统[J]. 应用数学和力学, 1992, 13(4): 349-357.
[10] Ollagnier, J.M. (1999) Rational Integration of the Lotka-Volterra System. Bulletin des Sciences Mathématiques, 123, 437-466. [Google Scholar] [CrossRef] 
[11] Llibre, J. and Zhang, X. (2000) Invariant Algebraic Surfaces of the Rikitake System. Journal of Physics A: Mathematical and General, 33, 7613-7635. [Google Scholar] [CrossRef] 
[12] Llibre, J. and Zhang, X. (2002) Invariant Algebraic Surfaces of the Lorenz System. Journal of Mathematical Physics, 43, 1622-1645. [Google Scholar] [CrossRef] 
[13] Cao, J. and Zhang, X. (2007) Dynamics of the Lorenz System Having an Invariant Algebraic Surface. Journal of Mathematical Physics, 48, 4370-4371. [Google Scholar] [CrossRef] 
[14] Deng, X. and Chen, A. (2011) Invariant Algebraic Surfaces of the Chen System. International Journal of Bifurcation and Chaos, 21, 1645-1651. [Google Scholar] [CrossRef] 
[15] Cândido, M.R., Llibre, J. and Valls, C. (2018) Invariant Algebraic Surfaces and Hopf Bifurcation of a Finance Model. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 28, Article ID: 1850150. [Google Scholar] [CrossRef] 
[16] Dias, F.S. and Valls, C. (2020) Global Dynamics of the Maxwell-Bloch System with Invariant Algebraic Surfaces. Dynamical Systems, 35, 668-681. [Google Scholar] [CrossRef] 
[17] 杨静, 谈文慧, 魏周超. Vallis系统的不变代数曲面研究[J]. 应用数学和力学, 2022, 43(1): 84-93.
[18] Karim, A., Amen, A. and Aziz, W. (2023) Integrability of a Family of Lotka-Volterra Three Species Biological System.
[19] Chen, Y. and Yang, Q. (2023) Limit Cycles from Perturbed Center on the Invariant Algebraic Surface of Unified Lorenz-Type System. International Journal of Bifurcation and Chaos, 33, Article ID: 2350172. [Google Scholar] [CrossRef] 
[20] Xu, Y., Huang, J., Lenci, S. and Luo, A.C.J. (2023) Constructed Complex Motions and Chaos. Chaos: An Interdisciplinary Journal of Nonlinear Science, 33, 1263-1273. [Google Scholar] [CrossRef] [PubMed]
[21] Hopf, E. (1948) A Mathematical Example Displaying Features of Turbulence. Communications on Pure and Applied Mathematics, 1, 303-322. [Google Scholar] [CrossRef] 
[22] Langford, W.F. (1979) Periodic and Steady-State Mode Interactions Lead to Tori. SIAM Journal on Applied Mathematics, 37, 22-48. [Google Scholar] [CrossRef] 
[23] 刘素华, 唐驾时. Langford系统Hopf分叉的线性反馈控制[J]. 物理学报, 2007, 56(6): 3145-3151.
[24] Nikolov, S. and Bozhkov, B. (2004) Bifurcations and Chaotic Behavior on the Lanford System. Chaos, Solitons & Fractals, 21, 803-808. [Google Scholar] [CrossRef] 
[25] Krishchenko, A.P. and Starkov, K.E. (2006) Localization of Compact Invariant Sets of Nonlinear Systems with Applications to the Lanford System. International Journal of Bifurcation and Chaos, 16, 3249-3256. [Google Scholar] [CrossRef] 
[26] Guo, G., Wang, X., Lin, X. and Wei, M. (2017) Steady-State and Hopf Bifurcations in the Langford ODE and PDE Systems. Nonlinear Analysis: Real World Applications, 34, 343-362. [Google Scholar] [CrossRef] 
[27] Yang, Q. and Yang, T. (2017) Complex Dynamics in a Generalized Langford System. Nonlinear Dynamics, 91, 2241-2270. [Google Scholar] [CrossRef] 
[28] Nikolov, S.G. and Vassilev, V.M. (2021) Completely Integrable Dynamical Systems of Hopf-Langford Type. Communications in Nonlinear Science and Numerical Simulation, 92, Article ID: 105464. [Google Scholar] [CrossRef] 
[29] Demina, M.V. (2018) Novel Algebraic Aspects of Liouvillian Integrability for Two-Dimensional Polynomial Dynamical Systems. Physics Letters A, 382, 1353-1360. [Google Scholar] [CrossRef] 
[30] Walker, R.J. (1978) Algebraic Curves. Springer-Verlag.
[31] Conte, R. (1999) The Painlevé Approach to Nonlinear Ordinary Differential Equations. In: Conte, R., Ed., The Painlevé Property, Springer, 77-180. [Google Scholar] [CrossRef] 

为你推荐