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A three-neuron network model with mixed delays involving multiple discrete and distributed delays is considered in this paper. Taking the discrete and distribute delays as bifurcation parameters respectively, we investigate the stability of the system structure and the conditions for the generation of Hopf bifurcation from the perspective of the distribution of the root of the characteristic equation of the linearized system at the equilibrium state of the nonlinear system. The intervals of parameters that make the system stable and unstable are also given. In addition, when the conditions of Hopf bifurcation theorem are satisfied, the calculation formulas for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solution are presented by means of the central manifold theorem and normal form method. Finally, numerical experiments are carried out to support the correctness of the theoretical results in different cases. It can be concluded that the oscillation and instability caused by delays obviously affect the stability of the network.

Discrete Fourier transform (DFT), inverse DFT (IDFT), and circular convolution are important tools for analyzing and designing discrete signals and systems, and are widely used in various industries. In order to pursue faster operational efficiencies or more accurate operational results, engineering calculations are often required to be quick and easy. Therefore, it is necessary to reduce the local computational overhead required to perform DFT, IDFT, and circular convolution. In addition to improving the algorithms themselves, cloud computing outsourcing is also an optional method. In this paper, in response to the new challenges brought by cloud computing outsourcing, we design and propose an efficient and secure cloud computing outsourcing protocol for DFT, IDFT, and circular convolution. Through a theoretical explanation and simulation experiment, we show the efficiency, security, and verifiability of the proposed protocol.

The consensus problem is investigated in this paper for multi-agent systems subjected to sampled data and packet losses. The switching system, which simultaneously contains both stable and unstable subsystems, is applied to model the packet dropout in a deterministic way. The focus of this paper is to derive an improved condition stated elegantly in terms of linear matrix inequality (LMI) with less conservative and lower complexity such that the multi-agent system considered can reach consensus. In line with this, a novel nonlinear function is introduced in the construction of the piecewise differentiable Lyapunov functions, and the Wirtinger inequality is applied as well. Furthermore, several comparison results against other related ones are presented, which yield that our consensus protocols significantly improve upon the existing ones advocated thus far. We verify the validity, the effectiveness, and the practical applicability of our results in the simulation examples.