Lyapunov function as potential function：A dynamical equivalence

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

Chin. Phys. B ›› 2014, Vol. 23 ›› Issue (1): 10505-010505.doi: 10.1088/1674-1056/23/1/010505

Lyapunov function as potential function：A dynamical equivalence

袁若石^{a b}, 马易安^c, 苑波^d, 敖平^{b e}

a School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;
b Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, China;
c Department of Applied Mathematics, University of Washington, Seattle 98195, USA;
d Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;
e Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China

收稿日期:2013-03-20 修回日期:2013-05-16 出版日期:2013-11-12 发布日期:2013-11-12
基金资助:
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).

Lyapunov function as potential function：A dynamical equivalence

Yuan Ruo-Shi (袁若石)^{a b}, Ma Yi-An (马易安)^c, Yuan Bo (苑波)^d, Ao Ping (敖平)^{b e}

a School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;
b Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, China;
c Department of Applied Mathematics, University of Washington, Seattle 98195, USA;
d Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;
e Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China

Received:2013-03-20 Revised:2013-05-16 Online:2013-11-12 Published:2013-11-12
Contact: Yuan Ruo-Shi E-mail:yrs12345@sjtu.edu.cn
Supported by:
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).

摘要/Abstract

摘要： 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.

关键词: Lyapunov function, potential function, stochastic process, generalized Einstein relation

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.

Key words: Lyapunov function, potential function, stochastic process, generalized Einstein relation

中图分类号: (Nonlinear dynamics and chaos)

05.45.-a

05.40.-a (Fluctuation phenomena, random processes, noise, and Brownian motion) 89.75.-k (Complex systems)

袁若石, 马易安, 苑波, 敖平. Lyapunov function as potential function：A dynamical equivalence[J]. Chin. Phys. B, 2014, 23(1): 10505-010505.

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[J]. Chin. Phys. B, 2014, 23(1): 10505-010505.

参考文献 41

[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

Lyapunov function as potential function：A dynamical equivalence

Lyapunov function as potential function：A dynamical equivalence

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