This study investigates the dynamic game behaviors of dual-channel supply chains involving an oligopoly manufacturer selling low-carbon products to online and offline retailers. The price game models under government subsidy are discussed under three scenarios: (1) simultaneous decision, (2) manufacturer dominates the market, and (3) retailer dominates the market. The equilibrium strategies are compared under the government subsidy policy. Using numerical simulation,complex characteristics of the dual-channel supply chain under the carbon subsidy policy are investigated. The complexity of wholesale price and sales commission of each channel are analyzed by bifurcation, largest Lyapunov exponent and basin of attraction diagrams. Furthermore,parameter adjustment and delayed feedback control methods are proven to be effective approaches to chaos control.

翻译：本研究探讨了由寡头制造商向线上和线下零售商销售低碳产品的双渠道供应链中的动态博弈行为。在三种情景下讨论了政府补贴下的价格博弈模型：（1）同时决策，（2）制造商主导市场，以及（3）零售商主导市场。在政府补贴政策下比较了均衡策略。通过数值模拟，研究了碳补贴政策下双渠道供应链的复杂特性。利用分岔图、最大Lyapunov指数和吸引域图分析了各渠道批发价格和销售佣金的复杂性。此外，参数调整和延迟反馈控制方法被证明是控制混沌的有效手段。