Dynamical Properties of a Two-gene Network with Hysteresis

A mathematical model for a two-gene regulatory network is derived and several oftheir properties analyzed. Due to the presence of mixed continuous/discrete dynam-ics and hysteresis, we employ a hybrid systems model to capture the dynamics of thesystem. The proposed model incorporates binary hysteresis with different thresholdscapturing the interaction between the genes. We analyze properties of the solutionsand asymptotic stability of equilibria in the system as a function of its parameters. Ouranalysis reveals the presence of limit cycles for a certain range of parameters, behaviorthat is associated with hysteresis. The set of points defining the limit cycle is charac-terized and its asymptotic stability properties are studied. Furthermore, the stabilityproperty of the limit cycle is robust to small perturbations. Numerical simulations arepresented to illustrate the results. Q. Shu and R. G. Sanfelice are with the Department of Aerospace and Mechanical Engineer-ing, University of Arizona 1130 N. Mountain Ave, AZ 85721. Email: [email protected],[email protected]. This research has been partially supported by the National Science Founda-tion under CAREER Grant no. ECS-1150306 and by the Air Force Office of Scientific Research under Grantno. FA9550-12-1-0366.