Dynamic analysis of renewable energy duopoly games with three-stage delay structures

This paper proposes a dynamic Bertrand duopoly model in order to study the strategic interactions among renewable energy firms under delayed decision-making processes influence. The existing two-stage delay framework is extended by introducing a three-stage delay structure, where firms adjust prices according to a weighted combination of three historical price signals. The model, which consists of a system of nonlinear delay ordinary differential equations, allows to capture the memory effects inherent in real-world price-setting behavior. The mathematical analysis addressed to local stability and Hopf bifurcation analysis, allows to derive the conditions under which delays induce oscillatory and chaotic dynamics, highlighting the destabilizing role of extended memory in competitive markets. According to the results, the introduction of a third delay term significantly enhances the system sensitivity to parameter and new complex dynamical regimes are captured, such as transitions from stable equilibria to limit cycles and chaotic attractors, as delay distributions vary. The results are of interest in the delay-aware pricing strategies, regulatory interventions, and the stability of renewable energy markets, where firms are increasingly exposed to time-lagged information environments.