  Time-delay chaotic systems refer to the hyperchaotic systems with multiple
positive Lyapunov exponents. It is characterized by more complex dynamics and a
wider range of applications as compared to those non-time-delay chaotic
systems. In a three-dimensional general Lorenz chaotic system, time delays can
be applied at different positions to build multiple heterogeneous Lorenz
systems with a single time delay. Despite the same equilibrium point for
multiple heterogeneous Lorenz systems with single time delay, their stability
and Hopf bifurcation conditions are different due to the difference in time
delay position. In this paper, the theory of nonlinear dynamics is applied to
investigate the stability of the heterogeneous single-time-delay Lorenz system
at the zero equilibrium point and the conditions required for the occurrence of
Hopf bifurcation. First of all, the equilibrium point of each heterogeneous
Lorenz system is calculated, so as to determine the condition that only zero
equilibrium point exists. Then, an analysis is conducted on the distribution of
the corresponding characteristic equation roots at the zero equilibrium point
of the system to obtain the critical point of time delay at which the system is
asymptotically 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 in
secure communication and other fields.
