Complex dynamics of discrete-time replicators in repeated Snowdrift Games with four strategies

Abstract

This paper investigates the dynamics of discrete-time replicators in repeated Snowdrift Games with four strategies. A three-dimensional discrete-time dynamical system is proposed to model these repeated Snowdrift Games. The existence of equilibrium points within the system is classified, and their local stabilities are thoroughly studied. Utilizing the center manifold theorem and bifurcation theory, it is demonstrated that the system undergoes flip bifurcations. The chaos control is studied by feedback control strategies for the understudied discrete system. Numerical simulations are conducted to validate the theoretical results, revealing that the system exhibits complex dynamical behaviors, including multiple periodic orbits and chaotic behavior. The maximum Lyapunov exponent, time series graphs, and bifurcation diagrams confirm the chaotic dynamical behaviors of the system.