Stability and Hopf bifurcation of a love model with two delays

Abstract

We analyze the stability and Hopf bifurcation of a proposed nonlinear love model with two delays. The nonlinearity stems from a logistic term injected to meticulously depict a couple’s cautious love affairs. The analysis is based on the linearization theory and starts with a single-delay case assuming that one of the two characters is always impulsive. The critical delay values and existence conditions for the Hopf bifurcation are obtained. Subsequently, we extend to the more practical but also more involved dual-delay case where the delays affect the emotions of both individuals. Accordingly, a non-trivial resultant-based analysis framework is proposed, and the corresponding Hopf bifurcation is detected intuitively and exhaustively by an obtained stability map in the delay domain. The results explain an interesting and well-known feature in our daily life, i.e., proper time delays stabilize the love dynamics. Furthermore, the spectral analysis based on a numerical case reveals another common emotional experience, that proper delays not only lead to a smoother transition process but also yield a shorter settling time for the love convergence. The numerical bifurcation diagram and computational cost are also considered to demonstrate the effectiveness and efficiency of the presented analysis framework. Finally, some additional suggestions are provided for the studied couple to earn a sweet love relationship.