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Delay-induced bifurcations in a nonautonomous system with delayed velocity feedbacks. (English) Zbl 1074.34068
The authors study the following oscillatory system with external excitation and delayed feedback control \[ \ddot x + \omega^2_0 x - \alpha_1 \dot x + \alpha_3 \dot x^3 = k \cos (\Omega t) + A (\dot x_\tau - \dot x) + B (\dot x_\tau - \dot x)^3, \] where \(\tau\) is the time delay and \(x_\tau = x(t-\tau)\), \(\alpha_1\alpha_3>0\), \(k>0\). \(A\) and \(B\) are feedback parameters. The main purpose of the paper is to study bifurcations in that system. First, for the case without excitation, the stability of the zero equilibrium is studied in detail. For the case with excitation, the center manifold theory and perturbative approach is used to establish periodic solutions and their stability analytically.

MSC:
34K18 Bifurcation theory of functional-differential equations
93D15 Stabilization of systems by feedback
34K19 Invariant manifolds of functional-differential equations
34K35 Control problems for functional-differential equations
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