Hopf bifurcation calculations for functional differential equations. (English) Zbl 0592.34048

Stech, Harlan W.

J. Math. Anal. Appl. 109, 472-491 (1985).

The author presents an alternate method for analyzing the Hopf bifurcation problem in a class of functional differential equations with unbounded delay: \(\dot y(t)=\int^{0}_{\infty}d\eta (\alpha;s)y_ t(s)+H(\alpha;y_ t).\) An advantage of this method over previous ones for example by S.-N. Chow and J. Mallet-Paret [J. Differ. Equations 26, 112-159 (1977; Zbl 0367.34033)] is that it avoids all reference to the infinite-dimensional nature of the problem. Its importance lies both in its generality and its relative ease of applications. The method is based on Lyapunov-Schmidt means and is motivated by the work of J. C. De Oliveira and J. K. Hale [Tôhoku Math. J., II. Ser. 32, 577-592 (1980; Zbl 0454.34035)].

Reviewer: Jing Huang Tian

MSC:

34K99	Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C25	Periodic solutions to ordinary differential equations

Keywords:

Hopf bifurcation problem; functional differential equations; unbounded delay; Lyapunov-Schmidt means

Citations:

Zbl 0367.34033; Zbl 0454.34035

References:

[1]	Andronov, A. A.; Leontovich, E. A.; Gordon, I. I.; Maier, A. E., Theory of Bifurcations of Dynamical Systems on a Plane (1971), Israel Program for Scientific Translations: Israel Program for Scientific Translations Jerusalem
[2]	Chafee, N., A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl., 35, 312-348 (1971) · Zbl 0214.09806
[3]	Chow, S.-N; Hale, J. K., Methods of Bifurcation Theory (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0487.47039
[4]	Chow, S.-N; Mallet-Paret, J., Integral averaging and Hopf bifurcation, J. Differential Equations, 26, 112-159 (1977) · Zbl 0367.34033
[5]	Claeyssen, J. R., The integral-averaging bifurcation method and the general one-delay equation, J. Math. Anal. Appl., 78, 429-439 (1980) · Zbl 0447.34042
[6]	de Oliveira, J. C.F, Hopf bifurcation for functional differential equations, Nonlinear Anal. T. M. A., 4, 217-229 (1980) · Zbl 0456.34044
[7]	de Oliveira, J. C.F; Hale, J. K., Dynamic behavior from bifurcation equations, Tôhoku Math. J., 32, 577-592 (1980) · Zbl 0454.34035
[8]	Flockerzi, D., Existence of small periodic solutions of ordinary differential equations in \(R^2\), Arch. Math., 33, 263-278 (1979) · Zbl 0407.34037
[9]	Golobitsky, M.; Langford, W. F., Classification and unfoldings of degenerate Hopf bifurcations, J. Differential Equations, 41, 375-415 (1981) · Zbl 0442.58020
[10]	Green, D., Self-oscillations for epidemic models, Math. Biosci., 38, 91-111 (1978) · Zbl 0384.92013
[11]	Hale, J. K., Functional Differential Equations, (Applied Math. Sci., Vol. 3 (1971), Springer-Verlag: Springer-Verlag New York) · Zbl 0267.34064
[12]	Hale, J. K., Stability from the bifurcation function, (Differential Equations (1980), Academic Press: Academic Press New York), 23-30 · Zbl 0588.34043
[13]	Hale, J. K.; de Oliveira, J. C.F, Hopf bifurcation for functional equations, J. Math. Anal. Appl., 74, 41-59 (1980) · Zbl 0433.34053
[14]	Hassard, B. D.; Kazarinoff, N. D.; Wan, Y.-H, Theory and Applications of Hopf Bifurcation, (London Math. Soc. Lecture Notes, No. 41 (1981), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0474.34002
[15]	Hethcote, H. W.; Stech, H. W.; van den Driessche, P., Nonlinear oscillations in epidemic models, SIAM J. Appl. Math., 1, 1-9 (1981) · Zbl 0469.92012
[16]	Jordan, G. S.; Wheeler, R. L., Asymptotic behavior of unbounded solutions of linear Volterra integral equations, J. Math. Anal. Appl., 3, 596-615 (1976) · Zbl 0347.45022
[17]	Kaplan, J.; Yorke, J., On the stability of a periodic solution of a delay differential equation, SIAM J. Math. Anal., 6, 268-282 (1975) · Zbl 0241.34080
[18]	Kazarinoff, N. D.; van den Driessche, P.; Wan, Y.-H, Hopf bifurcation and stability of periodic solutions of differential-difference and integro-differential equations, J. Inst. Math. Appl., 21, 461-477 (1978) · Zbl 0379.45021
[19]	Kielhöfer, H., Generalized Hopf bifurcation in Hilbert space, Math. Meth. Appl. Sci., 1, 498-513 (1979) · Zbl 0451.34059
[20]	Levinger, B. W., A folk theorem in functional differential equations, J. Differential Equations, 4, 612-619 (1968) · Zbl 0174.13902
[21]	Marsden, J. E.; McCracken, M., The Hopf Bifurcation and Its Applications, (Applied Math. Sciences, Vol. 19 (1976), Springer-Verlag: Springer-Verlag New York) · Zbl 0545.58002
[22]	Naito, T., On autonomous linear functional differential equations with infinite retardations, J. Differential Equations, 21, 135-143 (1976) · Zbl 0342.34054
[23]	Negrini, P.; Tesei, A., Attractivity and Hopf bifurcation in Banach spaces, J. Math. Anal. Appl., 78, 204-221 (1980) · Zbl 0452.35050
[24]	Nussbaum, R. D., The range of periods of periodic solutions of \(x\)′ \((t)\) = −\( αf (x(t\) − 1)), J. Math. Anal. Appl., 58, 280-292 (1977) · Zbl 0359.34066
[25]	H. W. Stechin; H. W. Stechin
[26]	Stech, H. W., The Hopf bifurcation: a stability result and application, J. Math. Anal. Appl., 2, 525-546 (1979) · Zbl 0418.34073
[27]	H. W. StechSIAM J. Math. Anal.; H. W. StechSIAM J. Math. Anal. · Zbl 0586.34060
[28]	Stech, H. W., On the adjoint theory for autonomous linear functional differential equations with unbounded delays, J. Differential Equations, 27, 421-443 (1978) · Zbl 0417.34100
[29]	H. W. Stech; H. W. Stech · Zbl 0418.34073
[30]	Stech, H. W.; Williams, M., Stability in a class of cyclic epidemic models with delay, J. Math. Biol., 11, 95-103 (1981) · Zbl 0449.92022
[31]	S. A. Van Gils; S. A. Van Gils