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

The authors consider the nonlinear differential-difference equation

x ˙(t)=f(x(t),x(t-τ 1 ),x(t-τ 2 )),(1)

where τ 1 ,τ 2 are positive constants, f(0,0,0)=0, and f:×× 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)].

34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations