The number of limit cycles of the FitzHugh nerve system
Authors:
Hebai Chen and Jianhua Xie
Journal:
Quart. Appl. Math. 73 (2015), 365-378
MSC (2010):
Primary 34C05, 34C07, 34C60
DOI:
https://doi.org/10.1090/S0033-569X-2015-01384-7
Published electronically:
March 17, 2015
MathSciNet review:
3357499
Full-text PDF Free Access
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Abstract: In this paper we give a complete analysis of the number of limit cycles of the FitzHugh nerve system. First, we prove the uniqueness of the limit cycle when the unique equilibrium is a source. We then show that the system has two limit cycles if the unique equilibrium is a sink and limit cycles exist. We will also show that the mathematical study of limit cycles for FitzHugh nerve systems is related to Hilbert’s 16$^{\mbox {th}}$ problem and is therefore an important area of study.
References
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References
- W. A. Coppel, Some quadratic systems with at most one limit cycle, Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 61–88. MR 1000976 (90d:34057)
- Freddy Dumortier and Chengzhi Li, On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations, Nonlinearity 9 (1996), no. 6, 1489–1500. MR 1419457 (97k:34036), DOI https://doi.org/10.1088/0951-7715/9/6/006
- Freddy Dumortier and Chengzhi Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations 139 (1997), no. 1, 41–59. MR 1467352 (98j:34043), DOI https://doi.org/10.1006/jdeq.1997.3291
- Freddy Dumortier and Chengzhi Li, Perturbations from an elliptic Hamiltonian of degree four. I. Saddle loop and two saddle cycle, J. Differential Equations 176 (2001), no. 1, 114–157. MR 1861185 (2002h:34061), DOI https://doi.org/10.1006/jdeq.2000.3977
- Freddy Dumortier and Chengzhi Li, Perturbations from an elliptic Hamiltonian of degree four. II. Cuspidal loop, J. Differential Equations 175 (2001), no. 2, 209–243. MR 1855970 (2002h:34060), DOI https://doi.org/10.1006/jdeq.2000.3978
- Freddy Dumortier and Chengzhi Li, Perturbation from an elliptic Hamiltonian of degree four. III. Global centre, J. Differential Equations 188 (2003), no. 2, 473–511. MR 1954291 (2003j:34046a), DOI https://doi.org/10.1016/S0022-0396%2802%2900110-9
- Freddy Dumortier and Chengzhi Li, Perturbation from an elliptic Hamiltonian of degree four. IV. Figure eight-loop, J. Differential Equations 188 (2003), no. 2, 512–554. MR 1954292 (2003j:34046b), DOI https://doi.org/10.1016/S0022-0396%2802%2900111-0
- Freddy Dumortier and Christiane Rousseau, Cubic Liénard equations with linear damping, Nonlinearity 3 (1990), no. 4, 1015–1039. MR 1079280 (91m:34036)
- R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961): 445-466.
- John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990. Revised and corrected reprint of the 1983 original. MR 1139515 (93e:58046)
- A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952): 500-544.
- E. Kaumann and U. Staude, Uniqueness and nonexistence of limit cycles for the FitzHugh equation, Equadiff 82 (Würzburg, 1982) Lecture Notes in Math., vol. 1017, Springer, Berlin, 1983, pp. 313–321. MR 726594 (85a:34038), DOI https://doi.org/10.1007/BFb0103262
- Chengzhi Li and Jaume Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations 252 (2012), no. 4, 3142–3162. MR 2871796 (2012k:34058), DOI https://doi.org/10.1016/j.jde.2011.11.002
- A. Lins, W. de Melo, and C. C. Pugh, On Liénard’s equation, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 335–357. MR 0448423 (56 \#6730)
- Jitsuro Sugie and Minoru Yamamoto, On the existence of periodic solutions for the FitzHugh nerve system, Math. Japon. 35 (1990), no. 4, 759–767. MR 1067877 (91h:34063)
- Jitsuro Sugie, Nonexistence of periodic solutions for the FitzHugh nerve system, Quart. Appl. Math. 49 (1991), no. 3, 543–554. MR 1121685 (92m:92006)
- S. A. Treskov and E. P. Volokitin, On existence of periodic orbits for the FitzHugh nerve system, Quart. Appl. Math. 54 (1996), no. 4, 601–607. MR 1417226 (97h:34043)
- William C. Troy, Oscillation phenomena in the Hodgkin-Huxley equations, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 299–310 (1976). MR 0442444 (56 \#826)
- William C. Troy, Bifurcation phenomena in FitzHugh’s nerve conduction equations, J. Math. Anal. Appl. 54 (1976), no. 3, 678–690. MR 0411683 (53 \#15413)
- X. Zeng, On the uniqueness of limit cycle of Liénard’s equation, Sci. China Ser. A 25 (1982), 583-592. MR 0670882 (84a:34030)
- Zhi Fen Zhang, Tong Ren Ding, Wen Zao Huang, and Zhen Xi Dong, Qualitative theory of differential equations, Translations of Mathematical Monographs, vol. 101, American Mathematical Society, Providence, RI, 1992. Translated from the Chinese by Anthony Wing Kwok Leung. MR 1175631 (93h:34002)
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Additional Information
Hebai Chen
Affiliation:
Chengdu, Sichuan 610031, People’s Republic of China
Address at time of publication:
Department of Mechanics and Engineering, Southwest Jiaotong University
MR Author ID:
1112845
Email:
chen_hebai@sina.com
Jianhua Xie
Affiliation:
Chengdu, Sichuan 610031, People’s Republic of China
Address at time of publication:
Department of Mechanics and Engineering, Southwest Jiaotong University
Email:
jhxie2000@126.com
Keywords:
FitzHugh nerve system,
limit cycle,
Liénard system
Received by editor(s):
May 31, 2013
Published electronically:
March 17, 2015
Additional Notes:
Project supported by the National Science Foundation of China(11172246)
Article copyright:
© Copyright 2015
Brown University
The copyright for this article reverts to public domain 28 years after publication.
