On existence of periodic orbits for the FitzHugh nerve system
Authors:
S. A. Treskov and E. P. Volokitin
Journal:
Quart. Appl. Math. 54 (1996), 601-607
MSC:
Primary 34C25; Secondary 34C23, 92C20
DOI:
https://doi.org/10.1090/qam/1417226
MathSciNet review:
MR1417226
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Abstract: Applying bifurcation theory, we construct various phase portraits of the FitzHugh differential system and describe the set of parameters for which this system has periodic solutions.
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445–466 (1961)
J. Sugie, Nonexistence of periodic solutions for the FitzHugh nerve system, Quart. Appl. Math. 49, 543–554 (1991)
K. P. Hadeler, U. an der Heiden, and K. Schumacher, Generation of the nervous impulse and periodic oscillations, Biol. Cybernet. 23, 211–218 (1976)
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
V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1982 (Russian original, Nauka, Moscow, 1977)
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1986, p. 459
N. N. Bautin and E. A. Leontovich, Methods and Ways (Examples) of the Qualitative Analysis of Dynamical Systems in a Plane, Nauka, Moscow, 1990, p. 480 (in Russian)
A. I. Chibnik and E. E. Shnol, Software for Qualitative Analysis of Differential Equations, Research Computer Center, USSR Academy of Sciences, Pushcino, 1982, p. 10 (in Russian)
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445–466 (1961)
J. Sugie, Nonexistence of periodic solutions for the FitzHugh nerve system, Quart. Appl. Math. 49, 543–554 (1991)
K. P. Hadeler, U. an der Heiden, and K. Schumacher, Generation of the nervous impulse and periodic oscillations, Biol. Cybernet. 23, 211–218 (1976)
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
V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1982 (Russian original, Nauka, Moscow, 1977)
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1986, p. 459
N. N. Bautin and E. A. Leontovich, Methods and Ways (Examples) of the Qualitative Analysis of Dynamical Systems in a Plane, Nauka, Moscow, 1990, p. 480 (in Russian)
A. I. Chibnik and E. E. Shnol, Software for Qualitative Analysis of Differential Equations, Research Computer Center, USSR Academy of Sciences, Pushcino, 1982, p. 10 (in Russian)
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Article copyright:
© Copyright 1996
American Mathematical Society
