Stability and Hopf bifurcation research of fractional-order B2B Supply Chain Finance model with time delays

Financing difficulties for small and medium-sized enterprises (SMEs) have become a bottleneck restricting their development. Supply chain finance (SCF) serves as a core means to alleviate this dilemma. Exploring the dynamic evolution mechanism of B2B SCF is crucial for ensuring the stable operation of banks and B2B platforms. Under the dynamic reward and punishment mechanism, this paper constructs integer-order and fractional-order evolutionary game models, analyzes the dynamic evolutionary dynamics between B2B platforms and banks, proves the existence and uniqueness of the system’s solutions, and derives the critical conditions for the local system stability and the occurrence conditions of Hopf bifurcation under six time-delay scenarios. In addition, this paper further introduces the PDξ controller to expand the stable range of time delays. Finally, based on the results of theoretical and simulation analyses, policy recommendations for maintaining the stability of the SCF system are put forward.