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A new kind of generalized heterochronous synchronization phenomenon is reported. Different kinds of generalized synchronous states (including generalized anticipated, isochronous and lag projective synchronization) coexist among different state variables between two unidirectionally coupled time-delayed chaotic systems. The analytical conditions for generalized heterochronous synchronization are obtained. We also find that the synchronization conditions are independent of the delay times in the original time-delayed system. The theoretical results are well confirmed by the numerical simulations and electronic circuit experiments.

A general method for formulating first-order time-delayed chaotic systems with simple linear time-delayed term is proposed. The formulated systems are realized with electronic circuit experiments. In order to determine the unknown coefficients in a general delayed differential equations for having chaotic solutions, we follow the route of period-doubling bifurcation to chaos. Firstly, the conditions for a time-delayed system having a stable periodic solution, generating from a destablized steady state, is analyzed with Hopf bifurcation theory. Then the delay time parameter is changed according to the bifurcation direction to search the chaotic state, which is identified by the Lyapunov exponents spectra. The theoretical analysis is well confirmed by numerical simulations and circuit experiments.

Consider the nonlinear delay difference equation \(x_{n + 1} - x_n + \sum\limits_{j = 1}^m {p_j f_j (x_n - x_j ) = 0.} \)

An experimental setup has been built, based on the control method of the impulsive proportional feedback in the system variable (IPF-SV), and the results on stabilizing unstable periodic orbits (UPO) embedded in chaotic and hyperchaotic attractors are obtained. In order to enhance the effectiveness of controlling the UPO, a modified IPF-SV method has been suggested and applied to the circuit successfully.

Single-species nonautonomous delay diffusion models with nonlinear growth rates are investigated. Some sufficient conditions are determined that guarantee the permanence of the species and the existence of a positive periodic solution which is global attractively.