Lyapunov function as potential function：A dynamical equivalence

doi:10.1088/1674-1056/23/1/010505

Abstract For a physical system, regardless of time reversal symmetry, a potential function serves also as a Lyapunov function, providing convergence and stability information. In this paper, the converse is constructively proved that any dynamics with a Lyapunov function has a corresponding physical realization: a friction force, a Lorentz force, and a potential function. Such construction establishes a set of equations with physical meaning for Lyapunov function and suggests new approaches on the significant unsolved problem namely to construct Lyapunov functions for general nonlinear systems. In addition, a connection is found that the Lyapunov equation is a reduced form of a generalized Einstein relation for linear systems, revealing further insights of the construction.

Keywords: Lyapunov function potential function stochastic process generalized Einstein relation

Received: 20 March 2013
Revised: 16 May 2013
Accepted manuscript online:

PACS:	05.45.-a	(Nonlinear dynamics and chaos)
	05.40.-a	(Fluctuation phenomena, random processes, noise, and Brownian motion)
	89.75.-k	(Complex systems)

Fund: Project supported by the National Basic Research Program of China (Grant No. 2010CB529200) and the National Natural Science Foundation of China (Grant Nos. 61073087 and 91029738).

Corresponding Authors: Yuan Ruo-Shi
E-mail: yrs12345@sjtu.edu.cn

Cite this article:

Yuan Ruo-Shi (袁若石), Ma Yi-An (马易安), Yuan Bo (苑波), Ao Ping (敖平) Lyapunov function as potential function：A dynamical equivalence 2014 Chin. Phys. B 23 010505

[1]	Lyapunov A M 1992 Int. J. Control 55 531
[2]	Hirsch M W and Smale S 1974 Differential Equations, Dynamical Systems, and Linear Algebra (San Diego: Academic Press)
[3]	Johansson M and Rantzer A 1998 IEEE Trans. Autom. Control 43 555
[4]	Sastry S 1999 Nonlinear Systems: Analysis, Stability, and Control (New York: Springer-Verlag)
[5]	Papachristodoulou A and Prajna S 2002 Proceedings of the 41st IEEE Conference on Decision and Control, December 10–13, 2002 Las Vegas, USA, p. 3482
[6]	Haddad W M and Chellaboina V S 2008 Nonlinear Dynamical Systems and Control: A Lyapunov-based Approach (Princeton: Princeton University Press)
[7]	Strogatz S H 2000 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Reading: Perseus Books)
[8]	La Salle J P 1976 The Stability of Dynamical Systems (Philadelphia: Society for Industrial and Applied Mathematics)
[9]	Scheffer M, Carpenter S, Foley J A, Folke C and Walker B 2001 Nature 413 591
[10]	Field R J and Noyes R M 1974 J. Chem. Phys. 60 1877
[11]	Frauenfelder H, Sligar S G and Wolynes P G 1991 Science 254 1598
[12]	Zdravković S and Satarić M V 2006 Chin. Phys. Lett. 23 65
[13]	Wang J, Xu L and Wang E K 2008 Proc. Natl. Acad. Sci. USA 105 12271
[14]	Ao P 2009 J. Genet. Genomics 36 63
[15]	Qian H 2010 J. Stat. Phys. 141 990
[16]	Wang J and Verkhivker G M 2003 Phys. Rev. Lett. 90 188101
[17]	Ge H and Qian H 2009 Phys. Rev. Lett. 103 148103
[18]	Yan G, Ren J, Lai Y C, Lai C H and Li B 2012 Phys. Rev. Lett. 108 218703
[19]	Liang X S 2013 Physica D 248 1
[20]	Maschke B, Ortega R and van der Schaft A J 2000 IEEE Trans. Autom. Control 45 1498
[21]	Wang Y Z, Li C W and Cheng D Z 2003 Automatica 39 1437
[22]	Freidlin M I and Wentzell A D 2012 Random Perturbations of Dynamical Systems (3rd edn.) (New York: Springer Verlag)
[23]	Tél T and Lai Y C 2010 Phys. Rev. E 81 056208
[24]	Ao P 2004 J. Phys. A 37 25
[25]	Ao P 2008 Commun. Theor. Phys. 49 1073
[26]	Zhu X M, Yin L, Hood L and Ao P 2004 Funct. Integr. Genomics 4 188
[27]	Bechhoefer J 2005 Rev. Mod. Phys. 77 783
[28]	Gong Y W, Song Y R and Jiang G P 2013 Chin. Phys. B 22 040204
[29]	Tang Y, Yuan R S and Ma Y A 2013 Phys. Rev. E 87 012708
[30]	Wang J, Hui G T and Xie X P 2013 Chin. Phys. B 22 030206
[31]	McLachlan R I, Quispel G R W and Robidoux N 1998 Phys. Rev. Lett. 81 2399
[32]	Ao P, Chen T Q and Shi J H 2013 Chin. Phys. Lett. 30 070201
[33]	Einstein A 1905 Ann. Phys. (Leipzig) 17 549
[34]	Yuan R S and Ao P 2012 J. Stat. Mech. 2012 P07010
[35]	Yuan R S, Wang X A, Ma Y A, Yuan B and Ao P 2013 Phys. Rev. E 87 062109
[36]	Kubo R 1966 Rep. Prog. Phys. 29 255
[37]	Liu F, Luo Y P, Huang M C and Ou-Yang Z C 2009 J. Phys. A 42 332003
[38]	Zhu X M, Yin L and Ao P 2006 Int. J. Mod. Phys. B 20 817
[39]	Kwon C, Ao P and Thouless D J 2005 Proc. Natl. Acad. Sci. USA 102 13029
[40]	Xu W, Yuan B and Ao P 2011 Chin. Phys. Lett. 28 050201
[41]	Ma Y A, Tan Q J, Yuan R S, Yuan B and Ao P 2012 arXiv:1208.1654

[1]	Ratchet transport of self-propelled chimeras in an asymmetric periodic structure Wei-Jing Zhu(朱薇静) and Bao-Quan Ai(艾保全). Chin. Phys. B, 2022, 31(4): 040503.
[2]	Dynamical behavior and optimal impulse control analysis of a stochastic rumor spreading model Liang'an Huo(霍良安) and Xiaomin Chen(陈晓敏). Chin. Phys. B, 2022, 31(11): 110204.
[3]	Dynamics of a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise Liang-An Huo(霍良安), Ya-Fang Dong(董雅芳), and Ting-Ting Lin(林婷婷). Chin. Phys. B, 2021, 30(8): 080201.
[4]	Near-optimal control of a stochastic rumor spreading model with Holling II functional response function and imprecise parameters Liang'an Huo(霍良安) and Xiaomin Chen(陈晓敏). Chin. Phys. B, 2021, 30(12): 120205.
[5]	Phonon dispersion relations of crystalline solids based on LAMMPS package Zhiyong Wei(魏志勇), Tianhang Qi(戚天航), Weiyu Chen(陈伟宇), and Yunfei Chen(陈云飞). Chin. Phys. B, 2021, 30(11): 114301.
[6]	Transport of velocity alignment particles in random obstacles Wei-jing Zhu(朱薇静), Xiao-qun Huang(黄小群), Bao-quan Ai(艾保全). Chin. Phys. B, 2018, 27(8): 080504.
[7]	Stability analysis of a simple rheonomic nonholonomic constrained system Chang Liu(刘畅), Shi-Xing Liu(刘世兴), Feng-Xing Mei(梅凤翔). Chin. Phys. B, 2016, 25(12): 124501.
[8]	Two kinds of generalized gradient representationsfor holonomic mechanical systems Feng-Xiang Mei(梅凤翔) and Hui-Bin Wu(吴惠彬). Chin. Phys. B, 2016, 25(1): 014502.
[9]	Stochastic stability of the derivative unscented Kalman filter Hu Gao-Ge (胡高歌), Gao She-Sheng (高社生), Zhong Yong-Min (种永民), Gao Bing-Bing (高兵兵). Chin. Phys. B, 2015, 24(7): 070202.
[10]	Stability and performance analysis of a jump linear control system subject to digital upsets Wang Rui (王蕊), Sun Hui (孙辉), Ma Zhen-Yang (马振洋). Chin. Phys. B, 2015, 24(4): 040201.
[11]	Current and efficiency of Brownian particles under oscillating forces in entropic barriers Ferhat Nutku, Ekrem Aydıner. Chin. Phys. B, 2015, 24(4): 040501.
[12]	Skew-gradient representation of generalized Birkhoffian system Mei Feng-Xiang (梅凤翔), Wu Hui-Bin (吴惠彬). Chin. Phys. B, 2015, 24(10): 104502.
[13]	Stability analysis of Markovian jumping stochastic Cohen–Grossberg neural networks with discrete and distributed time varying delays M. Syed Ali. Chin. Phys. B, 2014, 23(6): 060702.
[14]	Modified 2CLJDQP model and the second virial coefficients for linear molecules Zhang Ying (张颖), Wang Sheng (王升), He Mao-Gang (何茂刚). Chin. Phys. B, 2014, 23(12): 125101.
[15]	Global dynamics of a novel multi-group model for computer worms Gong Yong-Wang (巩永旺), Song Yu-Rong (宋玉蓉), Jiang Guo-Ping (蒋国平). Chin. Phys. B, 2013, 22(4): 040204.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Lyapunov function as potential function：A dynamical equivalence

Cite this article:

Online attention