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

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
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[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 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 New York
[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, () · Zbl 0267.34064
[12] Hale, J.K, Stability from the bifurcation function, (), 23-30
[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, () · 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, () · 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] {\scH. W. Stech}, Hopf bifurcation in higher order scalar functional differential equations, in “Physical Mathematics and Nonlinear Partial Differential Equations,” Dekker, New York, to appear.
[26] Stech, H.W, The Hopf bifurcation: a stability result and application, J. math. anal. appl., 2, 525-546, (1979) · Zbl 0418.34073
[27] {\scH. W. Stech}, Nongeneric Hopf bifurcations in functional differential equations, SIAM J. Math. Anal., in press. · 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] {\scH. W. Stech}, On the computation of the stability of the Hopf bifurcation, unpublished manuscript. · 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] {\scS. A. Van Gils}, On a formula for the direction of Hopf bifurcation, preprint.
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