Analysis of Parameter Space, Bifurcation and Symbolic Dynamics of Evolutionary Strategies on a One-Dimensional Regular Lattice

The payoff matrix parameters, bifurcation, topological conjugacy and symbolic dynamics of the evolutionary game dynamical systems on a one-dimensional regular lattice are thoroughly investigated in this paper. Based on the properties of addition and scalar multiplication invariance, an effective dimensionality reduction method is proposed to classify the payoff matrix parameters, resulting in 27 evolutionary game dynamical systems. Furthermore, their qualitative dynamics is presented and 14 topological conjugacy classes are obtained through two homeomorphisms, revealing the parametric bifurcation in the payoff matrix. Subsequently, their symbolic dynamics is examined in the context of the snowdrift game, and chaotic characteristics such as nonzero topological entropy and mixing property are derived. These analytical results indicate that even simple interactive elements have diverse dynamical expressiveness to capture the complexity or simplicity of spatial evolutionary games. Therefore, it is recommended that numerical simulations with different parameters should focus on parameter selection, dynamical properties and their combined effects to understand the factors affecting the evolutionary behaviors.