Stability switches and bifurcation analysis of a time delay model for the diffusion of a new technology

We deal with the time delay model for the diffusion of innovation technologies proposed by Fanelli and Maddalena [2012]. For this model, the stability switches and the occurrence of Hopf bifurcations are still largely undetermined, and in the present paper we perform some analysis on these topics. In particular, by applying the theory of delay differential equations and the analytical-geometrical approach developed by Beretta and Kuang [2002], we show that the equilibrium may lose stability and Hopf bifurcations may occur. Moreover, using the normal form theory and the center manifold theorem, we derive closed-form expressions that allow us to determine the direction of the Hopf bifurcations and the stability of the periodic solutions. Numerical results are presented which confirm and illustrate the theoretical predictions obtained.