Hopf Bifurcation and Stability of the Double-Delay Lorenz System

Abstract

Based on the nonlinear dynamics theory, the stability of the double-delay Lorenz system is investigated at the equilibrium points, and the conditions of the occurrence of Hopf bifurcation are analyzed. The double-delay Lorenz system has more complex dynamic behaviors, and it is applicable to many fields. Firstly, the equilibrium points of the system are calculated. Subsequently, the local stability of the system at the equilibrium points is determined by analyzing the distribution of the roots of the characteristic equation of the system, and the critical values of the time delays for generating Hopf bifurcation are yielded. With the time delays as the bifurcation parameter, the conditions of the existence of Hopf bifurcation in the system under the same and different time delays are analyzed. Lastly, it is confirmed numerically that the conclusions are drawn complying with the theoretical analysis and applied in the field of secure communication to make the encrypted information more secure and difficult to decipher during transmission.