On the stability of hybrid limit cycles and isolated equilibria in a genetic network with binary hysteresis

A mathematical model for a two-gene regulatory network is derived and several of its properties are analyzed. Due to the presence of continuous dynamics and binary hysteresis, we propose a hybrid system model. Binary hysteresis with different thresholds captures the interaction between the genes. We analyze properties of the solutions and asymptotic stability of equilibria in the system as a function of its parameters. Our analysis reveals the presence of limit cycles for a certain range of parameters, a behavior that is associated with the presence of binary hysteresis. The set of points defining the limit cycle is characterized and its asymptotic stability properties are studied. Numerical simulations are presented to illustrate some of the results.