Effective computation of periodic orbits and bifurcation diagrams in delay equations

SummaryBy employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.

[1]  E. M. Wright.  Linear difference-differential equations , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  H. Peitgen,et al.  Topological Perturbations in the Numerical Study of Nonlinear Eigenvalue and Bifurcation Problems , 1980 .

[3]  K. P. Hadeler,et al.  Delay equations in biology , 1979 .

[4]  N. D. Kazarinoff,et al.  Hopf Bifurcation and Stability of Periodic Solutions of Differential-difference and Integro-differential Equations , 1978 .

[5]  Roger D. Nussbaum,et al.  A global bifurcation theorem with applications to functional differential equations , 1975 .

[6]  K. P. Hadeler,et al.  Periodic solutions of difference-differential equations , 1977 .

[7]  James A. Yorke,et al.  On the nonlinear differential delay equation x′(t) = −f(x(t), x(t − 1)) , 1977 .

[8]  M. Prüfer,et al.  The Leray-Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems , 1979 .

[9]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[10]  Shui-Nee Chow,et al.  Existence of periodic solutions of autonomous functional differential equations , 1974 .

[11]  Hans-Otto Walther,et al.  A theorem on the amplitudes of periodic solutions of differential delay equations with applications to bifurcation , 1978 .

[12]  E. M. Wright.  A non-linear difference-differential equation. , 1946 .

[13]  G.Stephen Jones,et al.  The existence of periodic solutions of f′(x) = − αf(x − 1){1 + f(x)} , 1962 .

[14]  R. B. Grafton,et al.  A periodicity theorem for autonomous functional differential equations , 1969 .

[15]  Roger D. Nussbaum,et al.  Periodic solutions of some nonlinear autonomous functional differential equations , 1974 .

[16]  K. Hadeler.  Periodic solutions of ẋ(t) = –f(x(t), x(t – 1)) , 1979 .

[17]  James A. Yorke,et al.  Ordinary differential equations which yield periodic solutions of differential delay equations , 1974 .

[18]  U. Heiden.  Periodic solutions of a nonlinear second order differential equation with delay , 1979 .

[19]  H. Peitgen,et al.  Functional Differential Equations and Approximation of Fixed Points , 1979 .

[20]  B. Hassard,et al.  Bifurcation formulae derived from center manifold theory , 1978 .

[21]  R. Nussbaum.  Periodic solutions of nonlinear autonomous functional differential equations , 1979 .

[22]  R. Nussbaum.  Periodic solutions of autonomous functional differential equations , 1973 .