Nonexistence of periodic solutions for the FitzHugh nerve system
Author:
Jitsuro Sugie
Journal:
Quart. Appl. Math. 49 (1991), 543-554
MSC:
Primary 92C20; Secondary 34C25
DOI:
https://doi.org/10.1090/qam/1121685
MathSciNet review:
MR1121685
Full-text PDF Free Access
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Additional Information
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338
R. FitzHugh, Thresholds and plateaus in the Hodgkin-Huxley nerve equations, J. Gen. Phys. 43, 867–896 (1960)
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445–466 (1961)
- K. P. Hadeler, U. an der Heiden, and K. Schumacher, Generation of the nervous impulse and periodic oscillations, Biol. Cybernet. 23 (1976), no. 4, 211–218. MR 496806, DOI https://doi.org/10.1007/BF00340337
- In Ding Hsü and N. D. Kazarinoff, An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model, J. Math. Anal. Appl. 55 (1976), no. 1, 61–89. MR 466758, DOI https://doi.org/10.1016/0022-247X%2876%2990278-X
- In Ding Hsü, A higher-order Hopf bifurcation formula and its application to Fitzhugh’s nerve conduction equations, J. Math. Anal. Appl. 60 (1977), no. 1, 47–57. MR 470350, DOI https://doi.org/10.1016/0022-247X%2877%2990046-4
- 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
- Jitsuro Sugie and Tadayuki Hara, Nonexistence of periodic solutions of the Liénard system, J. Math. Anal. Appl. 159 (1991), no. 1, 224–236. MR 1119432, DOI https://doi.org/10.1016/0022-247X%2891%2990232-O
- William C. Troy, Bifurcation phenomena in FitzHugh’s nerve conduction equations, J. Math. Anal. Appl. 54 (1976), no. 3, 678–690. MR 411683, DOI https://doi.org/10.1016/0022-247X%2876%2990187-6
E. A. Coddington and N. Levinson, Theory and Ordinary Differential Equations, McGraw-Hill, New York, 1955
R. FitzHugh, Thresholds and plateaus in the Hodgkin-Huxley nerve equations, J. Gen. Phys. 43, 867–896 (1960)
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445–466 (1961)
K. P. Hadeler, U. an der Heiden, and K. Schumacher, Generation of the nervous impulse and periodic oscillations, Biol. Cybernet. 23, 211–218 (1976)
I. D. Hs̈u and N. D. Kazarinoff, An applicable Hopf bifurcation formula and instability of small periodic solutions of Field-Noyes model, J. Math. Anal. Appl. 55, 61–89 (1976)
I. D. Hs̈u, A higher-order Hopf bifurcation formula and its application to FitzHugh’s nerve conduction equations, J. Math. Anal. Appl. 60, 47–57 (1977)
E. Kaumann and U. Staude, Uniqueness and nonexistence of limit cycles for the FitzHugh equation, Equadiff 82 (H. W. Knobloch and K. Schmitt, eds.), Lecture Notes in Math., vol. 1017, Springer-Verlag, 1983, pp. 313–321
J. Sugie and T. Hara, Non-existence of periodic solutions of the Liénard system, scheduled for J. Math. Anal. Appl. 159, No. 1 (1991)
W. C. Troy, Bifurcation phenomena in FitzHugh’s nerve conduction equations, J. Math. Anal. Appl. 54, 678–690 (1976)
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Article copyright:
© Copyright 1991
American Mathematical Society
