Local Bifurcations Analysis of a State-Dependent Delay Diﬀerential Equation - V. O. Golubenets (Моделирование и анализ информационных систем, 2015)
In this paper, a ﬁrst-order equation with state-dependent delay and with a nonlinear right-hand side is considered. Conditions of existence and uniqueness of the solution of initial value problem aresupposed to be executed. The task is to study the behavior of solutions of the considered equation in a small neighborhood of its zero equilibrium. Local dynamics depends on real parameters which are coeﬃcients of equation right-hand side decomposition in a Taylor series. The parameter which is a coeﬃcient at the linear part of this decomposition has two critical values which determine a stability domain of zero equilibrium. We introduce a small positive parameter and use the asymtotic method of normal forms in order to investigate local dynamics modiﬁcations of the equation near each two critical values. We show that the stability exchange bifurcation occurs in the considered equation near the ﬁrst of these critical values, and the supercritical Andronov – Hopf bifurcation occurs near the second of them (if the suﬃcient condition is executed). Asymptotic decompositions according to correspondent small parameters are obtained for each stable solution. Next, a logistic equation with state-dependent delay is considered as an example. The bifurcation parameter of this equation has one critical value. A simple suﬃcient condition of Andronov – Hopf bifurcation occurence in the considered equation near a critical value is obtained as a result of applying the method of normal forms.