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Effective computation of periodic orbits and bifurcation diagrams in delay equations. (English) Zbl 0419.34070

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C25 Periodic solutions to ordinary differential equations
34A99 General theory for ordinary differential equations
Full Text: DOI EuDML
[1] Chow, S.N., Hale, J.K.: Periodic solutions of autonomous functional differential equations. J. Differential Equations15, 350-378 (1978) · Zbl 0295.34055 · doi:10.1016/0022-0396(74)90084-9
[2] Grafton, R.B.: A periodicity theorem for autonomous functional difference equations. J. Differential Equations6, 87-109 (1969) · Zbl 0175.38503 · doi:10.1016/0022-0396(69)90119-3
[3] Hadeler, K.P., Tomiuk, J.: Periodic solutions of difference-differential equations. Arch. Rational Mech. Anal.65, 87-95 (1977) · Zbl 0426.34058 · doi:10.1007/BF00289359
[4] Hadeler, K.P.: Periodic solutions ofx(t)=?f(x(t), x(t?1)) Math. Meth. in the Appl. Sciences1, 62-59 (1979) · Zbl 0461.34052 · doi:10.1002/mma.1670010106
[5] Hadeler, K.P.: Delay equations in Biology in [17], pp. 136-156 · Zbl 0412.92006
[6] Hale, J.K.: Theory of functional differential equations. Appl. Math. Sci. Vol. 3. Berlin Heidelberg New York: Springer 1977 · Zbl 0352.34001
[7] Hassard, B., Wan, Y.H.: Bifurcation formulae derived from center manifold theory. J. Math. Anal. Appl.63, 297-312 (1978) · Zbl 0435.34034 · doi:10.1016/0022-247X(78)90120-8
[8] an der Heiden, U.: Periodic solutions of non-linear second order differential equation with delay. J. Math. Anal. Appl.70, 599-609 (1979) · Zbl 0426.34059 · doi:10.1016/0022-247X(79)90068-4
[9] Jones, G.S.: The existence of periodic solutions off?(x)=??f(x?1) [1+f(x)]. J. Math. Anal. Appl.5, 435-450 (1962) · Zbl 0106.29504 · doi:10.1016/0022-247X(62)90017-3
[10] J?rgens, H., Peitgen, H.O., Saupe, D.: Topological perturbations in the numerical study of nonlinear eigenvalue- and bifurcation problems, in: Proceedings Symposium on analysis and computation of fixed points, Madison 1979, S.M. Robinson (ed.). New York: Academic Press 1979
[11] Kaplan, J., Yorke, J.: Ordinary differential equations which yield periodic solutions of differential-delay equations. J. Math. Anal. Appl.48, 317-324 (1974) · Zbl 0293.34102 · doi:10.1016/0022-247X(74)90162-0
[12] Kaplan, J., Yorke, J.: On the nonlinear differential delay equationx?(t)=?f(x(t), x(t?1)). J. Differential Equations23, 293-314 (1977) · Zbl 0337.34067 · doi:10.1016/0022-0396(77)90132-2
[13] Kazarinoff, N., 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 · doi:10.1093/imamat/21.4.461
[14] Nussbaum, R.D.: Periodic solutions of some nonlinear autonomous functional diferential equations I. Ann. Mat. Pura Appl.101, 236-306 (1974); II, J. Differential Equations14, 360-394 (1973)
[15] Nussbaum, R.D.: A global bifurcation theorem with applications to functional differential equations. J. Functional Analysis19, 319-338 (1975) · Zbl 0314.47041 · doi:10.1016/0022-1236(75)90061-0
[16] Nussbaum, R.D.: Periodic solutions of nonlinear autonomous functional equations, in [17] pp. 283-325
[17] Peitgen, H.O., Walter, H.O. (eds.): Functional differential equations and approximation of fixed points. Proceedings Bonn 1978, Lecture Notes in Mathematics 730. Berlin Heidelberg New York: Springer 1979
[18] Peitgen, H.O., Pr?fer, M.: The Leray-Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems, in [17] pp. 326-409
[19] Walther, H.O.: A theorem on the amplitudes of periodic solutions of differential delay equations with application to bifurcation. J. Differential Equations39, 396-404 (1978) · Zbl 0378.34054 · doi:10.1016/0022-0396(78)90049-9
[20] Wright, E.M.: A nonlinear difference-differential equation. J. Reine Angew. Math.194, 66-87 (1955) · Zbl 0064.34203 · doi:10.1515/crll.1955.194.66
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