A Hopf global bifurcation theorem for retarded functional differential equations

Nussbaum, Roger D.

doi:10.1090/S0002-9947-1978-0482913-0

A Hopf global bifurcation theorem for retarded functional differential equations
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by Roger D. Nussbaum

Trans. Amer. Math. Soc. 238 (1978), 139-164

DOI: https://doi.org/10.1090/S0002-9947-1978-0482913-0

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Abstract:

We prove a result concerning the global nature of the set of periodic solutions of certain retarded functional differential equations. Our main theorem is an analogue, for retarded F.D.E.’s, of a result by J. Alexander and J. Yorke for ordinary differential equations.

References

J. C. Alexander and James A. Yorke, Global bifurcations of periodic orbits, Amer. J. Math. 100 (1978), no. 2, 263–292. MR 474406, DOI 10.2307/2373851
Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
Shui Nee Chow and John Mallet-Paret, The Fuller index and global Hopf bifurcation, J. Differential Equations 29 (1978), no. 1, 66–85. MR 492560, DOI 10.1016/0022-0396(78)90041-4
Kenneth L. Cooke and James A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosci. 16 (1973), 75–101. MR 312923, DOI 10.1016/0025-5564(73)90046-1
J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367. MR 44116, DOI 10.2140/pjm.1951.1.353
F. Brock Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133–148. MR 209600, DOI 10.2307/2373103
R. B. Grafton, A periodicity theorem for autonomous functional differential equations, J. Differential Equations 6 (1969), 87–109. MR 243176, DOI 10.1016/0022-0396(69)90119-3
R. B. Grafton, Periodic solutions of certain Liénard equations with delay, J. Differential Equations 11 (1972), 519–527. MR 293207, DOI 10.1016/0022-0396(72)90064-2
A. Granas, The theory of compact vector fields and some of its applications to topology of functional spaces. I, Rozprawy Mat. 30 (1962), 93. MR 149253
Jack K. Hale, Functional differential equations, Applied Mathematical Sciences, Vol. 3, Springer-Verlag New York, New York-Heidelberg, 1971. MR 0466837, DOI 10.1007/978-1-4615-9968-5
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Jorge Ize, Bifurcation theory for Fredholm operators, Mem. Amer. Math. Soc. 7 (1976), no. 174, viii+128. MR 425696, DOI 10.1090/memo/0174

Global bifurcation of periodic orbits

G. Stephen Jones, The existence of periodic solutions of $f^{\prime } (x)=-\alpha f(x-1)\{1+f(x)\}$, J. Math. Anal. Appl. 5 (1962), 435–450. MR 141837, DOI 10.1016/0022-247X(62)90017-3
G. Stephen Jones, Periodic motions in Banach space and applications to functional-differential equations, Contributions to Differential Equations 3 (1964), 75–106. MR 163039
James L. Kaplan and James A. Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, J. Math. Anal. Appl. 48 (1974), 317–324. MR 364815, DOI 10.1016/0022-247X(74)90162-0
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
John Mallet-Paret, Generic periodic solutions of functional differential equations, J. Differential Equations 25 (1977), no. 2, 163–183. MR 442995, DOI 10.1016/0022-0396(77)90198-x

Stability and complexity in model ecosystems

L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973–1974. MR 0488102
Roger D. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Functional Analysis 19 (1975), no. 4, 319–338. MR 0385656, DOI 10.1016/0022-1236(75)90061-0
Roger D. Nussbaum, Periodic solutions of some nonlinear, autonomous functional differential equations. II, J. Differential Equations 14 (1973), 360–394. MR 372370, DOI 10.1016/0022-0396(73)90053-3
Roger D. Nussbaum, Global bifurcation of periodic solutions of some autonomous functional differential equations, J. Math. Anal. Appl. 55 (1976), no. 3, 699–725. MR 430473, DOI 10.1016/0022-247X(76)90076-7
Roger D. Nussbaum, The range of periods of periodic solutions of $x’(t)=-\alpha f(x(t-1))$, J. Math. Anal. Appl. 58 (1977), no. 2, 280–292. MR 445086, DOI 10.1016/0022-247X(77)90206-2

Periodic solutions of differential-delay equations with two time lags

W. V. Petryshyn, On the approximation-solvability of equations involving $A$-proper and psuedo-$A$-proper mappings, Bull. Amer. Math. Soc. 81 (1975), 223–312. MR 388173, DOI 10.1090/S0002-9904-1975-13728-1
Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9
Gordon Thomas Whyburn, Topological analysis, Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N. J., 1958. MR 0099642