Bifurcations and chaos of a discrete-time model in genetic regulatory networks

In this paper, the dynamics of a discrete-time genetic model is investigated. The existence and stability conditions of the fixed points are obtained. It is shown that the discrete-time genetic network undergoes fold bifurcation, flip bifurcation and Neimark–Sacker bifurcation. The biological parameter and discretization step size are taken as bifurcation parameters, respectively, and the explicit bifurcation criteria are derived based on the center manifold theorem and bifurcation theory. Numerical simulations validate the theoretical analysis and also show that the system can exhibit diverse dynamic behaviors such as period-7, -14, -5, -10 orbits and chaos. The overall results reveal much richer dynamics of the discrete-time genetic model than that of the original continuous-time model.