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Stability and bifurcations of equilibria in a multiple-delayed differential equation. (English) Zbl 0809.34077

Many biological and physical events are described by the delay- differential equation \[ x'(t)= f_ 1(x(t- T_ 1))+f_ 2(x(t- T_ 2)),\tag{1} \] where \(f_ i(u)= -A_ i\tanh(u)\), \(A_ i= \text{const}>0\) for \(i= 1,2\). In the paper under review a lot of problems concerning the equation (1) or some of its generalizations are investigated. For example, the linearized stability of the unique equilibrium solution \(x=0\) is completely analyzed with help of a characteristic equation involving two parameters. Moreover, bifurcations are determined and described and details by the construction of a centre manifold.

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
37G05 Normal forms for dynamical systems
92C20 Neural biology
34K20 Stability theory of functional-differential equations
34C23 Bifurcation theory for ordinary differential equations
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