A note on the uniqueness of the closed orbit of the FitzHugh-Nagumo system
Author:
Makoto Hayashi
Journal:
Quart. Appl. Math. 58 (2000), 171-176
MSC:
Primary 34C25; Secondary 34C60
DOI:
https://doi.org/10.1090/qam/1739043
MathSciNet review:
MR1739043
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: A parameter range for which the ordinary differential equations governing the FitzHugh nerve system have a unique nontrivial closed orbit is given. It is wider than those already known.
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445–466 (1961)
- Makoto Hayashi, On the uniqueness of the closed orbit of FitzHugh-Nagumo system, Math. Japon. 46 (1997), no. 2, 331–336. MR 1479835
- Makoto Hayashi, On the uniqueness of the closed orbit of the Liénard system, Math. Japon. 46 (1997), no. 3, 371–376. MR 1487283
- 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, DOI https://doi.org/10.1007/BFb0103262
J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE 50, 2061–2070 (1962)
- Jitsuro Sugie, Nonexistence of periodic solutions for the FitzHugh nerve system, Quart. Appl. Math. 49 (1991), no. 3, 543–554. MR 1121685, DOI https://doi.org/10.1090/qam/1121685
- 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, DOI https://doi.org/10.1090/qam/1417226
- Xian Wu Zeng, On the uniqueness of limit cycle of Liénard’s equation, Sci. Sinica Ser. A 25 (1982), no. 6, 583–592. MR 670882
- Xian Wu Zeng, Zhi Fen Zhang, and Su Zhi Gao, On the uniqueness of the limit cycle of the generalized Liénard equation, Bull. London Math. Soc. 26 (1994), no. 3, 213–247. MR 1289041, DOI https://doi.org/10.1112/blms/26.3.213
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445–466 (1961)
M. Hayashi, On the uniqueness of the closed orbit of FitzHugh-Nagumo system, Math. Japon. 46 (2), 331–336 (1997)
M. Hayashi, On the uniqueness of the closed orbit of the Liénard system, Math. Japon. 46 (3), 371–376 (1997)
E. Kaumann and U. Staude, Uniqueness and nonexistence of limit cycles for the FitzHugh equation, Lecture Notes in Math. 1017, Springer-Verlag, New York, 1983, pp. 313–321
J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE 50, 2061–2070 (1962)
J. Sugie, Nonexistence of periodic solutions for the FitzHugh nerve system, Quart. Appl. Math. 49, 543–554 (1991)
S. A. Treskov and E. P. Volokitin, On existence of periodic orbits for the FitzHugh nerve system, Quart. Appl. Math. 54, 601–607 (1996)
X. Zeng, On the uniqueness of limit cycle of Liénard’s equation, Scientia Sinica (Series A) 25, 583–592 (1982)
X. Zeng, Z. Zhang, and S. Gao, On the uniqueness of the limit cycle of the generalized Liénard equation, Bull. London Math. Soc. 26, 213–247 (1994)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
34C25,
34C60
Retrieve articles in all journals
with MSC:
34C25,
34C60
Additional Information
Article copyright:
© Copyright 2000
American Mathematical Society