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This paper considers the finite-time and fixed-time synchronization of delayed complex-valued recurrent neural networks (CVRNNs) with discontinuous activation functions and nonidentical parameters via sliding mode control. Firstly, we design a sliding surface involving integral structure and a discontinuous control. Secondly, by constructing Lyapunov functional and using the differential inequality technique, some sufficient conditions are derived to guarantee the finite-time and fixed-time synchronization of delayed complex-valued recurrent neural networks with discontinuous activation functions and nonidentical parameters. Finally, two simulation examples are shown to illustrate the proposed methods.

Although the number of investigation fruits on neural networks is growing explosively, the majority of such research effort is devoted to integer-order neural networks, while only a few are on fractional-order neural networks (FONNs). By arguing the associated characteristic equation of the proposed network, we establish delay-dependent stability conditions and the bifurcation point. Then selecting the communication delay as the bifurcation parameter and the other delay as the constant in its stability interval, the conditions for the occurrence of Hopf bifurcation are established. Then, we confirm the conditions by numerical simulation. It is indicated that the stability of the FONN remains unchanged with the lesser control delay, and will not exist once the delay outnumbers its critical value. And we discover that compared with integer-order neural networks the convergence time to the equilibrium point of FONN is shorter for the same system parameters. It detects that fractional orders are able to advance(postpone) the generation of the bifurcations of the developed FONN. The paper demonstrates that the fractional orders have significant effects on the stability of the FONN. Finally, the theoretical results are authenticated by numerical simulations.

The synchronization of complex networks, as an important and captivating dynamic phenomenon, has been investigated across diverse domains ranging from social activities to ecosystems and power systems. Furthermore, the synchronization of networks proves instrumental in solving engineering quandaries, such as cryptography and image encryption. And finite-time synchronization (FTS) controls exhibit substantial resistance to interference, accelerating network convergence speed and heightening control efficiency. In this paper, finite-time synchronization (FTS) is investigated for a class of fractional-order nonidentical complex networks under uncertain parameters (FONCNUPs). Firstly, some new FTS criteria for FONCNUPs are proposed based on Lyapunov theory and fractional calculus theory. Then, the new controller is designed based on inequality theory. Compared to the general controller, it controls all nodes and adds additional control to some of them. When compared to other controllers, it has lower control costs and higher efficiency. Finally, a numerical example is presented to validate the effectiveness and rationality of the obtained results.

This paper is concerned with the asymptotic and pinning synchronization of fractional-order nonidentical complex dynamical networks with uncertain parameters (FONCDNUP). First of all, some synchronization criteria of FONCDNUP are proposed by using the stability of fractional-order dynamical systems and inequality theory. Moreover, a novel controller is derived by using the Lyapunov direct method and the differential inclusion theory. Next, based on the Lyapunov stability theory and pinning control techniques, a new group of sufficient conditions to assure the synchronization for FONCDNUP are obtained by adding controllers to the sub-nodes of networks. At last, two numerical simulations are utilized to illustrate the validity and rationality of the acquired results.