×

zbMATH — the first resource for mathematics

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

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.