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Time-delay chaotic systems refer to the hyperchaotic systems with multiplepositive Lyapunov exponents. It is characterized by more complex dynamics and awider range of applications as compared to those non-time-delay chaoticsystems. In a three-dimensional general Lorenz chaotic system, time delays canbe applied at different positions to build multiple heterogeneous Lorenzsystems with a single time delay. Despite the same equilibrium point formultiple heterogeneous Lorenz systems with single time delay, their stabilityand Hopf bifurcation conditions are different due to the difference in timedelay position. In this paper, the theory of nonlinear dynamics is applied toinvestigate the stability of the heterogeneous single-time-delay Lorenz systemat the zero equilibrium point and the conditions required for the occurrence ofHopf bifurcation. First of all, the equilibrium point of each heterogeneousLorenz system is calculated, so as to determine the condition that only zeroequilibrium point exists. Then, an analysis is conducted on the distribution ofthe corresponding characteristic equation roots at the zero equilibrium pointof the system to obtain the critical point of time delay at which the system isasymptotically stable at the zero equilibrium point and the Hopf bifurcation.Finally, mathematical software is applied to carry out simulation verification.Heterogeneous Lorenz systems with time delay have potential applications insecure communication and other fields.

Research on Hopf Bifurcation and Stability of Heterogeneous Lorenz System with Single Time Delay