A solution method for decomposing vector fields in Hamilton energy

doi:10.1088/1674-1056/ad5a74

Abstract Hamilton energy, which reflects the energy variation of systems, is one of the crucial instruments used to analyze the characteristics of dynamical systems. Here we propose a method to deduce Hamilton energy based on the existing systems. This derivation process consists of three steps: step 1, decomposing the vector field; step 2, solving the Hamilton energy function; and step 3, verifying uniqueness. In order to easily choose an appropriate decomposition method, we propose a classification criterion based on the form of system state variables, i.e., type-I vector fields that can be directly decomposed and type-I$\!$I vector fields decomposed via exterior differentiation. Moreover, exterior differentiation is used to represent the curl of low-high dimension vector fields in the process of decomposition. Finally, we exemplify the Hamilton energy function of six classical systems and analyze the relationship between Hamilton energy and dynamic behavior. This solution provides a new approach for deducing the Hamilton energy function, especially in high-dimensional systems.

Keywords: Hamilton energy dynamical systems vector field exterior differentiation

Received: 13 May 2024
Revised: 14 June 2024
Accepted manuscript online: 21 June 2024

PACS:	87.19.L-	(Neuroscience)
	87.19.lj	(Neuronal network dynamics)
	05.45.-a	(Nonlinear dynamics and chaos)
	05.45.Pq	(Numerical simulations of chaotic systems)

Fund: This work was supported by the National Natural Science Foundation of China (Grant Nos. 12305054, 12172340, and 12371506)

Corresponding Authors: Lu-Lu Lu
E-mail: lululu@cug.edu.cn

Cite this article:

Xin Zhao(赵昕), Ming Yi(易鸣), Zhou-Chao Wei(魏周超), Yuan Zhu(朱媛), and Lu-Lu Lu(鹿露露) A solution method for decomposing vector fields in Hamilton energy 2024 Chin. Phys. B 33 098702

[1] Lu L L, Jia Y, Xu Y, Ge M Y, Yang L J and Zhan X 2018 Sci. China Technol. Sci. 62 427
[2] Yang Y M, Ma J, Xu Y and Jia Y 2020 Cogn. Neurodyn. 15 265
[3] Lin H R, Wang C H, Yao W and Tan Y M 2020 Commun. Nonlinear Sci. Numer. Simulat. 90 105390
[4] He Z W and Yao C G 2020 Sci. China Technol. Sci. 63 2339
[5] Lu L L, Gao Z H, Wei Z C and Yi M 2023 Chaos 33 013127
[6] Yang F F, Wang Y and Ma J 2023 Commun. Nonlinear Sci. Numer. Simulat. 119 107127
[7] Torrealdea F J, d’Anjou A, Graña M and Sarasola C 2006 Phys. Rev. E 74 011905
[8] Yao C G, Yao Y G, Qian Y and Xu X F 2022 Chaos, Solitons Fractals 164 112667
[9] Xu Y and Ma J 2022 Commun. Nonlinear Sci. Numer. Simulat. 111 106426
[10] Lu L L, Yi M, Gao Z H, Wu Y and Zhao X 2023 Nonlinear Dyn. 111 16557
[11] Torrealdea F J, Sarasola C and d’Anjou A 2009 Chaos, Solitons Fractals 40 60
[12] Lv M, Ma J, Yao Y G and Alzahrani F 2018 Sci. China Technol. Sci. 62 448
[13] Wang G W, Xu Y, Ge M Y, Lu L L and Jia Y 2020 AEU Int. J. Electron. Commun. 120 153209
[14] Song X L, Jin W Y and Ma J 2015 Chin. Phys. B 24 128710
[15] Sarasola C, D’Anjou A, Torrealdea F J and Moujahid A 2005 Int. J. Bifur. Chaos 15 2507
[16] Ma J, Wu F Q, Jin W Y, Zhou P and Hayat T 2017 Chaos 27 053108
[17] Cang S J, Wu A G, Wang Z H and Chen Z Q 2017 Int. J. Bifur. Chaos 27 1750024
[18] Huang L L, Ma Y H and Li C 2024 Chin. Phys. B 33 010503
[19] Guo Y L, Qi G Y and Hamam Y 2016 Nonlinear Dyn. 85 2765
[20] Yamakou M E 2020 Nonlinear Dyn. 101 487
[21] Ortigueira M D, Rivero M and Trujillo J J 2015 Commun. Nonlinear Sci. Numer. Simulat. 22 1036
[22] Sarasola C, Torrealdea F J, d’Anjou A, Moujahid A and Graña M 2004 Phys. Rev. E 69 011606
[23] An X L and Zhang L 2018 Nonlinear Dyn. 94 2995
[24] Aqeel M, Azam A and Ahmad S 2018 Chin. J. Phys. 56 1220
[25] Yao Z, Zhou P, Alsaedi A and Ma J 2020 Appl. Math. Comput. 374 124998
[26] Azam A, Aqeel M and Hussain Z 2020 Soft Comput. 25 2521
[27] Ezhilarasu P M, Inbavalli M, Murali K and Thamilmaran K 2018 Pramana 91 4
[28] Njitacke Z T, Koumetio B N, Ramakrishnan B, Leutcho G D, Fozin T F, Tsafack N, Rajagopal K and Kengne J 2021 Cogn. Neurodyn. 16 899
[29] Li J H, Wu H B and Mei F X 2017 Nonlinear Dyn. 90 2557
[30] Azam A, Aqeel M, Ahmad S and Ahmad F 2017 Nonlinear Dyn. 90 1
[31] Tirandaz H, Aminabadi S S and Tavakoli H 2018 Alexandria Eng. J. 57 1519
[32] Sprott J C 1994 Phys. Rev. E 50 R647
[33] Torrealdea F J, Sarasola C, d’Anjou A, Moujahid A and de Mendizábal N V 2009 Biosystems 97 60
[34] Bao H, Yu X H, Zhang Y Z, Liu X F and Chen M 2023 Chaos, Solitons Fractals 177 114167
[35] Njitacke Z T, Takembo C N, Awrejcewicz J, Fouda H P E and Kengne J 2022 Chaos, Solitons Fractals 160 112211
[36] Thottil S K and Ignatius R P 2018 Nonlinear Dyn. 95 239
[37] Wang C N, Sun G P, Yang F F and Ma J 2022 AEU Int. J. Electron. Commun. 153 154280
[38] Xu L, Qi G Y and Ma J 2022 Appl. Math. Model. 101 503

[1]	Computing large deviation prefactors of stochastic dynamical systems based on machine learning Yang Li(李扬), Shenglan Yuan(袁胜兰), Linghongzhi Lu(陆凌宏志), and Xianbin Liu(刘先斌). Chin. Phys. B, 2024, 33(4): 040501.
[2]	Memristors-coupled neuron models with multiple firing patterns and homogeneous and heterogeneous multistability Xuan Wang(王暄), Santo Banerjee, Yinghong Cao(曹颖鸿), and Jun Mou(牟俊). Chin. Phys. B, 2024, 33(10): 100501.
[3]	Dynamical analysis, geometric control and digital hardware implementation of a complex-valued laser system with a locally active memristor Yi-Qun Li(李逸群), Jian Liu(刘坚), Chun-Biao Li(李春彪), Zhi-Feng Hao(郝志峰), and Xiao-Tong Zhang(张晓彤). Chin. Phys. B, 2023, 32(8): 080503.
[4]	Diffusive field coupling-induced synchronization between neural circuits under energy balance Ya Wang(王亚), Guoping Sun(孙国平), and Guodong Ren(任国栋). Chin. Phys. B, 2023, 32(4): 040504.
[5]	Surface lattice resonance of circular nano-array integrated on optical fiber tips Jian Wu(吴坚), Gao-Jie Ye(叶高杰), Xiu-Yang Pang(庞修洋), Xuefen Kan(阚雪芬), Yan Lu(陆炎), Jian Shi(史健), Qiang Yu(俞强), Cheng Yin(殷澄), and Xianping Wang(王贤平). Chin. Phys. B, 2023, 32(12): 120701.
[6]	Novel energy dissipative method on the adaptive spatial discretization for the Allen-Cahn equation Jing-Wei Sun(孙竟巍), Xu Qian(钱旭), Hong Zhang(张弘), and Song-He Song(宋松和). Chin. Phys. B, 2021, 30(7): 070201.
[7]	Energy feedback and synchronous dynamics of Hindmarsh-Rose neuron model with memristor K Usha, P A Subha. Chin. Phys. B, 2019, 28(2): 020502.
[8]	Three-dimensional modulations on the states of polarization of light fields Peng Li(李鹏), Dongjing Wu(吴东京), Sheng Liu(刘圣), Yi Zhang(章毅), Xuyue Guo(郭旭岳), Shuxia Qi(齐淑霞), Yu Li(李渝), Jianlin Zhao(赵建林). Chin. Phys. B, 2018, 27(11): 114201.
[9]	Study on a new chaotic bitwise dynamical system and its FPGA implementation Wang Qian-Xue (王倩雪), Yu Si-Min (禹思敏), C. Guyeux, J. Bahi, Fang Xiao-Le (方晓乐). Chin. Phys. B, 2015, 24(6): 060503.
[10]	Average vector field methods for the coupled Schrödinger–KdV equations Zhang Hong (张弘), Song Song-He (宋松和), Chen Xu-Dong (陈绪栋), Zhou Wei-En (周炜恩). Chin. Phys. B, 2014, 23(7): 070208.
[11]	Three-dimensional spiral structure of tropical cyclone under four-force balance Liu Shi-Kuo (刘式适), Fu Zun-Tao (付遵涛), Liu Shi-Da (刘式达). Chin. Phys. B, 2014, 23(6): 069201.
[12]	A high order energy preserving scheme for the strongly coupled nonlinear Schrödinger system Jiang Chao-Long (蒋朝龙), Sun Jian-Qiang (孙建强). Chin. Phys. B, 2014, 23(5): 050202.
[13]	Tangent response in coupled dynamical systems Yan Hua(闫华), Wei Ping(魏平), and Xiao Xian-Ci(肖先赐). Chin. Phys. B, 2010, 19(9): 090501.
[14]	A fuzzy crisis in a Duffing－van der Pol system Hong Ling(洪灵). Chin. Phys. B, 2010, 19(3): 030513.
[15]	Hybrid control of bifurcation and chaos in stroboscopic model of Internet congestion control system Ding Da-Wei (丁大为), Zhu Jie(朱杰), and Luo Xiao-Shu(罗晓曙). Chin. Phys. B, 2008, 17(1): 105-110.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

A solution method for decomposing vector fields in Hamilton energy

Cite this article:

Online attention