A four-dimensional hyperchaotic Lu system with multiple time-delay controllers is con-sidered in this paper. Based on the theory of Hopf bifurcation in delay system, we obtain a simple relationship between the parameters when the system has a periodic solution. Numerical simulations show that the assumption is a rational condition, choosing parame-ter in the determined region can control hyperchaotic Lu system well, the chaotic state is transformed to the periodic orbit. Finally, we consider the differences between the analysis of the hyperchaotic Lorenz system, hyperchaotic Chen system and hyperchaotic Lu system.

A new class of numerical methods for Volterra integro-differential equations with memory is developed. The methods are based on the combination of general linear methods with compound quadrature rules. Sufficient conditions that guarantee global and asymptotic stability of the solution of the differential equation and its numerical approximation are established. Numerical examples illustrate the convergence and effectiveness of the numerical methods.

This paper deals with the adapted Milstein method for solving linear stochastic delay differential equations. It is proved that the numerical method is mean-square (MS) stable under suitable conditions. The obtained result shows that the method preserves the stability property of a class of linear-coefficient problems. This is also verified by several numerical examples.

This paper deals with the stability of Runge–Kutta methods for a class of stiff systems of nonlinear Volterra delay-integro-differential equations. Two classes of methods are considered: Runge-Kutta methods extended with a compound quadrature rule, and Runge-Kutta methods extended with a Pouzet type quadrature technique. Global and asymptotic stability criteria for both types of methods are derived.

In functional differential equations (FDEs), there is a class of infinite delay-differential equations (IDDEs) with porportional delay, which aries in many scientific fields such as electric me-chanics, quantum mechanics, and optics. Ones have found that there exist very different mathematical challenges between FDEs with proportional delay and those with constant delays. Some research on the numerical solutions and the corresponding analysis for the linear FDEs with proportional delays have been presented by several authors. However, up to now, the research for nonlinear case still remains to be done. For a class of IDDEs with proportional delays. It is shown under the suitable conditions that a(k, l)-algebraically stable RK method for this kins of nonlinear IDDE IS globally and asymptotically stable.