×

zbMATH — the first resource for mathematics

Local and global Hopf bifurcation in a scalar delay differential equation. (English) Zbl 1126.34371

MSC:
34K18 Bifurcation theory of functional-differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] an der Heiden, U., Delay in physiological systems, J. math. biol., 8, 345-364, (1979) · Zbl 0429.92009
[2] an der Heiden, U.; Mackey, M.C., The dynamics of production and destruction: analytic insight into complex behaviour, J. math. biol., 16, 75-101, (1982) · Zbl 0523.93038
[3] Bélair, J., Stability in a model of a delayed neural network, J. dynam. differential equations, 5, 607-623, (1993) · Zbl 0796.34063
[4] Bose, F.G., The stability chart for the linearized cushing equation with a discrete delay and with gamma-distributed delays, J. math. anal. appl., 140, 510-536, (1989) · Zbl 0677.92015
[5] Cao, Y., Uniqueness of the periodic solution for differential delay equations, J. differential equations, 128, 46-57, (1996) · Zbl 0853.34063
[6] Chow, S.-N.; Hale, J.K., Methods of bifurcation theory, (1982), Springer-Verlag New York
[7] Chow, S.-N.; Mallet-Paret, J., Integral averaging and bifurcation, J. differential equations, 26, 112-159, (1977) · Zbl 0367.34033
[8] Chow, S.-N.; Mallet-Paret, J., The fuller index and global Hopf bifurcation, J. differential equations, 29, 66-85, (1978) · Zbl 0369.34020
[9] Diekmann, O.; van Gils, S.A.; Verduyn Lunel, S.M.; Walther, H.O., Delay equations, functional-, complex- and nonlinear analysis, (1995), Springer-Verlag New York · Zbl 0826.34002
[10] Doedel, E.J.; Leung, P.C., A numerical technique for bifurcation problems in delay differential equations, Congr. numer., 34, 225-237, (1982)
[11] Erbe, L.H.; Krawcewicz, W.; Geba, K.; Wu, J., S1-degree and global Hopf bifurcation theory of functional differential equations, J. differential equations, 98, 227-298, (1992) · Zbl 0765.34023
[12] Faria, T.; Magalhães, L.T., Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. differential equations, 122, 181-200, (1995) · Zbl 0836.34068
[13] Faria, T.; Magalhães, L.T., Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities, J. dynam. differential equations, 8, 35-70, (1995) · Zbl 0853.34064
[14] Doyne Farmer, J., Chaotic attractors of an infinite-dimensional dynamical system, Phys. D, 4, 366-393, (1982) · Zbl 1194.37052
[15] Gibbs, H.M.; Hopf, F.A.; Kaplan, D.L.; Shoemaker, R.L., Observation of chaos in optical bistability, Phys. rev. lett., 46, 474-477, (1981)
[16] Glass, L.; Mackey, M.C., From clocks to chaos, (1988), Princeton Univ. Press Princeton · Zbl 0705.92004
[17] Hadeler, K.P.; Tomiuk, J., Periodic solutions of difference – differential equations, Arch. rational mech. anal., 65, 87-95, (1977) · Zbl 0426.34058
[18] Hale, J.K.; Sternberg, N., Onset of chaos in differential delay equations, J. comput. phys., 77, 271-287, (1988)
[19] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002
[20] Hayes, N.D., Roots of the transcendential equation associated with a certain difference – differential equation, J. London math. soc., 25, 226-232, (1950) · Zbl 0038.24102
[21] Ikeda, K.; Kondo, K.; Akimoto, O., Successive higher-harmonic bifurcations in systems with delayed feedback, Phys. rev. lett., 49, 1467-1470, (1982)
[22] Mackey, M.C.; Glass, L., Oscillation and chaos in physiological control systems, Science, 197, 287-289, (1977) · Zbl 1383.92036
[23] Mallet-Paret, J.; Nussbaum, R.D., Global continuation and asymptotic behaviour for periodic solutions of a differential – delay equation, Ann. mat. pura. appl. (4), 145, 33-128, (1986) · Zbl 0617.34071
[24] Mallet-Paret, J.; Nussbaum, R.D., A differential – delay equation arising in optics and physiology, SIAM J. math. anal., 20, 249-292, (1989) · Zbl 0676.34043
[25] Mallet-Paret, J.; Sell, G.R., Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. differential equations, 125, 385-440, (1996) · Zbl 0849.34055
[26] Marcus, C.M.; Westervelt, R.M., Stability of analog neural networks with delay, Phys. rev. A, 39, 347-359, (1989)
[27] Martelli, M.; Schmitt, K.; Smith, H.L., Periodic solutions of some nonlinear delay-differential equations, J. math. anal. appl., 74, 494-503, (1980)
[28] Milton, J., Dynamics of small neural populations, CRM monograph series, 7, (1996), Amer. Math. Soc Providence
[29] Nussbaum, R.D., A Hopf global bifurcation theorem for retarded functional differential equations, Trans. amer. math. soc., 238, 139-164, (1978) · Zbl 0389.34050
[30] Stech, H.W., Hopf bifurcation calculations for functional differential equations, J. math. anal. appl., 109, 472-491, (1985) · Zbl 0592.34048
[31] Walther, H.-O., The 2-dimensional attractor of x′(t)=−μx(t)+f(x(t−1)), Mem. amer. math. soc., 544, (1995) · Zbl 0829.34063
[32] J. Wu, Delay-induced discrete waves of large amplitudes in neural networks with circulant connection matrices, preprint, 1997.
[33] Wu, J.; Zou, X., Patterns of sustained oscillations in neural networks with delayed interactions, Appl. math. math. comput., 73, 55-75, (1995) · Zbl 0857.92003
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.