Stability and Bifurcation Analysis of Fractional Transcriptional Regulatory Networks

Fractional differential equations capture changes in genes over time better than integer-order differential equations. Due to the complexity of the system caused by fractional order integrals, the regulatory network for fractional order differential equations lacks stability analysis and HOPE bifurcation condition analysis. In this paper, the dynamical behavior of the time-delay system is studied based on fractional differential equations. By adding two delay factors and PD controllers, the stability and HOPF bifurcation conditions of fractional-order gene regulatory networks with time delay are deduced through analyzing Caputo fractional derivatives theory. The influence of order on system bifurcation is verified by the use of numerical simulations. At the same time, it is found that changes in controller parameters will make the HOPF bifurcation of fractional order systems occur in advance or be delayed.