Stability and bifurcation in delay-differential equations with two delays. (English) Zbl 0946.34066

The authors consider the nonlinear differential-difference equation \[ \dot x(t)=f(x(t),x(t-\tau_1),x(t-\tau_2)),\tag{1} \] where \(\tau_1,\;\tau_2\) are positive constants, \(f(0,0,0)=0\), and \(f:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is continuously differentiable. First, the local stability of the zero solution to (1) is investigated. Second, it is shown that the two delay equation exhibits Hopf bifurcation and that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are orbitally stable under certain conditions. Results of the paper improve some of the results obtained by J. Bélair and S. A. Campbell [SIAM J. Appl. Math. 54, No. 5, 1402-1424 (1994; Zbl 0809.34077)].


34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations


Zbl 0809.34077
Full Text: DOI


[1] Bélair, J.; Campbell, S.A., Stability and bifurcations of equilibrium in a multiple-delayed differential equation, SIAM J. appl. math., 54, 1402-1424, (1994) · Zbl 0809.34077
[2] Bélair, J.; Mackey, M.C.; Mahaffy, J.M., Age-structured and two delay models for erythropoiesis, Math. biosci., 128, 317-346, (1995) · Zbl 0832.92005
[3] Bellman, R.; Cooke, K.L., Differential – difference equations, (1963), Academic Press New York · Zbl 0118.08201
[4] Beuter, A.; Bélair, J.; Labrie, C., Feedback and delay in neurological diseases: A modeling study using dynamical systems, Bull. math. biol., 55, 525-541, (1993) · Zbl 0825.92072
[5] Beuter, A.; Larocque, D.; Glass, L., Complex oscillations in a human motor system, J. motor behavior, 21, 277-289, (1989)
[6] Braddock, R.D.; van den Driessche, P., On a two lag differential delay equation, J. austral. math. soc. ser. B, 24, 292-317, (1983) · Zbl 0513.92016
[7] Campbell, S.A.; Bélair, J., Analytically and symbolically-assisted investigation of Hopf bifurcations in delay – differential equations, Canad. appl. math. quart., 3, 137-154, (1995) · Zbl 0840.34074
[8] Chow, S.-N.; Mallet-Paret, J., Integral averaging and Hopf bifurcation, J. differential equations, 26, 112-159, (1977) · Zbl 0367.34033
[9] Cooke, K.L.; Grossman, Z., Discrete delay, distributed delay and stability switches, J. math. anal. appl., 86, 592-627, (1982) · Zbl 0492.34064
[10] Cooke, K.L.; van den Driessche, P., On zeros of some transcendental equations, Funkcial. ekvac., 29, 77-90, (1986) · Zbl 0603.34069
[11] Cooke, K.L.; Yorke, J.A., Some equations modelling growth processes and gonorrhea epidemics, Math. biosci., 16, 75-101, (1973) · Zbl 0251.92011
[12] Dieudonné, J., Foundations of modern analysis, (1960), Academic Press New York · Zbl 0100.04201
[13] Gopalsamy, K., Global stability in the delay – logistic equation with discrete delays, Houston J. math., 16, 347-356, (1990) · Zbl 0714.34113
[14] Hale, J.K., Nonlinear oscillations in equations with delays, (), 157-185
[15] Hale, J.K.; Huang, W., Global geometry of the stable regions for two delay differential equations, J. math. anal. appl., 178, 344-362, (1993) · Zbl 0787.34062
[16] Hale, J.K.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer-Verlag New York
[17] J. K. Hale, and, S. M. Tanaka, Square and Pulse Waves with Two Delays, CDSNS97-283. · Zbl 0949.34062
[18] Hassard, B.D.; Kazarinoff, N.D.; Wan, Y.H., Theory and applications of Hopf bifurcation, (1981), Cambridge Univ. Press Cambridge · Zbl 0474.34002
[19] Huang, W., On asymptotic stability for linear delay equations, Differential integral equations, 4, 1303-1310, (1991) · Zbl 0737.34054
[20] Kazarinoff, N.D.; Wan, Y.H.; van den Driessche, P., Hopf bifurcation and stability of periodic solutions of differential – difference with integrodifferential equations, J. inst. math. appl., 21, 461-477, (1978) · Zbl 0379.45021
[21] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002
[22] Mahaffy, J.M.; Zak, P.J.; Joiner, K.M., A geometric analysis of the stability regions for a linear differential equation with two delays, Internat. J. bifur. chaos appl. sci. engrg., 5, 779-796, (1995) · Zbl 0887.34070
[23] Marriot, C.; Vallée, R.; Delisle, C., Analysis of a first order delay differential – delay equation containing two delays, Phys. rev. A, 40, 3420-3428, (1989)
[24] Mizuno, M.; Ikeda, K., An unstable mode selection rule: frustrated optical instability due to two competing boundary conditions, Physcia D, 36, 327-342, (1989)
[25] Nussbaum, R.D., Differential delay equations with two delays, Mem. amer. math. soc., 16, 1-62, (1978)
[26] Claeyssen, J.Ruiz, Effect of delays on functional differential equations, J. differential equations, 20, 404-440, (1976) · Zbl 0345.34052
[27] Stech, H., The Hopf bifurcation: A stability result and application, J. math. anal. appl., 71, 525-546, (1979) · Zbl 0418.34073
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