This research studied bipartite leader and leaderless synchronization of fractional-order communication delay in coupled memristor neural networks by utilizing the decoupling approach and the Laplace transform. Further, the synchronization was analyzed under delay-independent criteria. Finally, numerical examples were provided to show the effectiveness of theoretical parts.

Higher-order interactions play a critical role in driving the rapid propagation of rumors. We propose a susceptible–exposed–infected–removed (SEIR) model that incorporates group interactions as higher-order terms. Detailed analysis reveals the emergence of bistability in both homogeneous and heterogeneous networks and shows that the bistable region expands as group interactions intensify. By utilizing group interactions as the bifurcation parameter, we derive explicit bifurcation conditions for both network types. For heterogeneous networks, we establish global stability criteria and propose an effective control strategy. Numerical simulations demonstrate the significant influence of group interactions on the bistable region. This study advances the theoretical understanding of rumor propagation dynamics and provides a practical strategy for controlling rumor propagation.

This paper primarily addresses the design of distributed optimal cooperative controllers and the utilization of a reinforcement learning (RL)-based event-triggered mechanism for multi-agent systems (MASs) with unknown dynamics. By setting an extra compensator, the augmented system is constructed to overcome the dependence for system dynamics. Then, to address the issue of computational burden, we utilize an event-triggered mechanism based on reinforcement learning (RL) and neural networks (NNs) to implement the adaptive dynamic programming (ADP) algorithm. Additionally, we take into consideration the trade-off between computational burden and achieving consensus control by introducing a weighting factor in the reward design for MASs. With this reward design, we present an algorithm based on the deep deterministic policy gradient (DDPG) algorithm to learn the event-triggered condition for MASs and achieve a balance between these two factors. The event-triggered mechanism of our algorithm can also identify constraints such as time limitations or computational resource restrictions, aiming to achieve consensus control without violating these constraints. We demonstrate the absence of Zeno behavior and the uniform ultimate boundedness (UUB) of both local consensus error and weight estimation error. Finally, simulation results illustrate the effectiveness of the control algorithm and the weighting factor.

This study presented a novel approach to investigating the existence, uniqueness, and stability of solutions for an initial value problem involving fractional differential equations of variable order. In contrast to conventional methods in the literature, which often utilized generalized intervals and piecewise constant functions, we introduced a new fractional operator that is more appropriate for this problem. The existence and uniqueness of the solutions ware demonstrated through Leray-Schauder fixed point theorem and Banach's theorem, with an analysis of the uniform stability of the problem. The strength of our approach lies in its straightforwardness and reliance on fewer restrictive assumptions. The study concluded with an application that features a practical example, accompanied by visual illustrations.

<p>This paper is mainly concerned with the existence of mild solutions and approximate controllability for a class of fractional semilinear systems with instantaneous and non-instantaneous impulses. By applying the Kuratowski measure of noncompactness and $ \rho $-set contractive fixed-point theorem, the results for the considered system were obtained. In the end, an example was studied to support the main results.</p>

<p>In this paper, the existence of nonoscillatory solutions for a class of higher-order nonlinear differential equations is investigated. Notably, the equations are of mixed neutral type with a forcing term, which distinguished the equations in this paper from the existing ones and made the qualitative analysis of the solution more complicated. By means of the Schauder-Tychonoff fixed point theorem and inequality techniques, some new sufficient conditions for the existence of nonoscillatory solutions were established. The results in this paper improved and generalized some known results in the existing works. Finally, an example was given to illustrate the effectiveness of the proposed method.</p>

<abstract><p>This paper mainly studied the problem of solving interval type-2 fuzzy relation equations $ \widetilde A \circ \widetilde X = \widetilde B $. First, to solve the interval type-2 fuzzy relation equations, we extend the semi-tensor product of matrices to interval matrices and give its specific definition. Second, the interval type-2 fuzzy relation equation was divided into two parts: primary fuzzy matrix equation $ {\widetilde A_\mu } \circ {\widetilde X_\mu }{\rm{ = }}{\widetilde B_\mu} $ and secondary fuzzy matrix equation $ {\widetilde A_f} \circ {\widetilde X_f} = {\widetilde B_f} $. Since all elements of $ {\widetilde X_f} $ equal to one, only the principal fuzzy matrix equation needs to be considered. Furthermore, it was proved that all solutions can be obtained from the parameter set solutions if the primary fuzzy matrix equation is solvable. Finally, with semi-tensor product of interval matrices, the primary fuzzy matrix equation was transformed into an algebraic equation and the specific algorithm for solving an interval type-2 fuzzy relation equation was proposed.</p></abstract>

<abstract><p>In this paper, the problem of existence-uniqueness and global exponential stability of periodic solution (i.e., stationary oscillation) for a class of nonlinear delay systems with impulses was studied. Some new sufficient conditions ensuring the existence of stationary oscillation for the addressed equations were derived by using the inequality technique that has been reported in recent publications. Our proposed method, which is different with the existing results in the literature, shows that nonlinear delay systems may admit a stationary oscillation using proper impulsive control strategies even if it was originally unstable or divergent. As an application, we considered the single species logarithmic population model and established a new criterion to guarantee the existence of positive stationary oscillation. Some numerical examples and their computer simulations were also given at the end of this paper to show the effectiveness of our development control method.</p></abstract>

<abstract><p>In this paper, the initial-value problem for a class of first order delay differential equations, which emerges as a model for population dynamics, is considered. To solve this problem numerically, using the finite difference method including interpolating quadrature rules with the basis functions, we construct a fitted difference scheme on a uniform mesh. Although this scheme has the same rate of convergence, it has more efficiency and accuracy compared to the classical Euler scheme. The different models, Nicolson's blowfly and Mackey–Glass models, in population dynamics are solved by using the proposed method and the classical Euler method. The numerical results obtained from here show that the proposed method is reliable, efficient, and accurate.</p></abstract>

<abstract><p>This paper studies the continuous Cucker–Smale model with time-varying topological structures and reaction-type delay. The goal of this paper is to establish a sufficient framework for flocking behaviors. Our method combines strict Lyapunov design with the derivation of an appropriate persistence condition for multi-agent systems. First, to prove that position fluctuations are uniformly bounded, a strict and trajectory-dependent Lyapunov functional is constructed via reparametrization of the time variable. Then, by constructing a global Lyapunov functional and using a novel backward-forward estimate, it is deduced that velocity fluctuations converge to zero. Finally, flocking behaviors are analyzed separately in terms of time delays and communication failures.</p></abstract>