Stability and Bifurcation Analysis of Complex-Valued Fractional-Order Gene Regulatory Networks with Mixed Delays and Impulsive Noise

This paper investigates the stability and bifurcation of Complex-Valued Fractional-Order Gene Regulatory Networks with Mixed Time Delays (mDFCGRNs). Initially, by integrating the properties of distributed time delays and the decomposition method of the real and imaginary parts of complex numbers, the original system is transformed to a fractional-order gene network with different orders in the field of real numbers. The time-delay is used as the bifurcation parameter to study the stability of the system, and the critical time-delay conditions for the occurrence of Hopf bifurcation are obtained. Second, in order to simulate the internal activities of GRNs more realistically, we introduce periodic impulsive noise with time-delay to simulate the noise stimulation inside the genetic system. For mDFCGRNs affected by noninstantaneous impulsive noise, we prove that it has equivalent local stability characteristics with its linearized average system, and then analyze its local stability and Hopf bifurcation. It is found that the positive or negative effect of impulsive noise on mDFCGRNs depends on the gain parameters of impulsive noise. Finally, we carry out some numerical simulations to verify the theorem conclusions obtained by the theoretical analysis in this paper. In addition, more dynamic behaviors of mDFCGRNs are discussed based on simulations, such as chaos, the characteristics of Hopf bifurcation and so on.