Stability and Hopf Bifurcation with Optimal Control in a Delayed Tripartite Evolutionary Game

This paper studies new energy adoption using a tripartite evolutionary game involving government incentives, enterprise investment, and user adoption. Two time delays are introduced to describe delayed enterprise responses to policy incentives and delayed user perceptions of supply improvements, while social feedback captures interactions among users. The equilibria and their local stability are analyzed, and Hopf bifurcation conditions are derived for the single-delay and double-delay cases. Numerical results illustrate that, under representative parameter settings, both types of delays can induce a loss of local stability of the interior equilibrium and lead to oscillatory behavior near the corresponding critical delays. Social feedback also plays an important role in determining system stability. Finally, an optimal control model based on Pontryagin’s maximum principle is developed for the delay-free benchmark system to promote the transition from low to high adoption while accounting for intervention costs. The results highlight the effects of delayed responses, social feedback, and policy control on the evolutionary dynamics of new energy adoption.