The dynamic analysis of discrete fractional-order two-gene map

The evolutionary processes are based on information transmission by nervous systems and inheritance by genes in DNA. Various continuous and discrete mathematical models have been presented for genes. Discrete gene models are particularly interesting due to their simple analysis and low computational costs. It is imperative to create genetic factors based on gene models that depend on the past. This paper proposes a discrete fractional-order two-gene map model. At first, the gene map is evaluated using the phase plane, bifurcation diagram, and Lyapunov exponent, and the periodic and chaotic behaviors of the system are shown. Then, the fractional-order gene map model is introduced. The system's dynamic behaviors are investigated using bifurcation diagrams according to system parameters and derivative order. It is shown that increasing the value of the fractional order increases complexity, leading to chaotic behavior in the model. While decreasing the fractional derivative order mostly changes the dynamics to periodic. Finally, the synchronization of two two-gene maps with discrete fractional order is investigated using the electrical connection. The results show that in contrast to the integer-order model, the fractional-order model can reach synchronization.