Hopf bifurcation in a love-triangle model with time delays

In this paper, a love-triangle model with nonlinear romantic sentimental interactions and four time delays is proposed. Regarding time delay as bifurcating parameter, the dynamical behaviors which include local asymptotical stability and Hopf bifurcation are studied in detail by analyzing the characteristic equation corresponding to the linearized system of a love-triangle model. When the delay passes through a sequence of critical values, we find that Hopf bifurcation occurs. Furthermore, stability and direction of bifurcating periodic solution are derived by applying the normal form theory and the center manifold theorem in our love-triangle model. Finally, an illustrative example is also used to support our theoretical results.