This work addresses quasi-synchronization (QS) and finite-time synchronization (FTS) in fractional-order neural networks (FONNs) by utilizing aperiodically intermittent event-triggered control. First, a new stability lemma is proposed under the intermittent control framework for the inequality tkCDt alpha V(t)<=-phi 1V(t)-phi 2V-delta(t)+phi 0. Unlike existing results limited to practical FTS (Cao and Zhang 2024 Int. J. Fuzzy Syst. 26 1507-18; Wei et al 2025 IEEE Trans. Circuits Syst. I Regul. Pap. 72 4103-14; Du et al 2024 Inf. Sci. 667 120457), this lemma extends the synchronization scope of FONNs to cover both QS and FTS. Second, by integrating an event-triggered mechanism into aperiodically intermittent control, the proposed hybrid control strategy effectively reduces control updates, thereby conserving energy consumption. Sufficient synchronization conditions are established, along with explicit settling-time estimates for FTS, and Zeno behavior is strictly excluded. Numerical simulations verify the theoretical results.

In this paper, an efficient numerical method is presented for solving the generalized time-fractional Boussinesq equation. The Caputo fractional derivative is approximated by the L1 formula and the space is discretized by using the Fourier spectral method. The a priori estimates, solvability and convergence of the numerical scheme are rigorously established. The presented numerical results demonstrate the effectiveness of the Fourier spectral method.

Fractional laser chaotic systems can exhibit more complex dynamical behavior, so its projective synchronization and applications to image encryption have been widely investigated in recent years. This work devotes to realize its function matrix projective synchronization, whose projective scale factor is a function matrix varying with time t, and applies the function matrix projective synchronization to the color image encryption. First, two different fractional laser chaotic systems with model uncertainty and external disturbance are constructed and the function matrix projective synchronization is defined. Second, the fractional sliding mode surface and the sliding mode controller are designed to achieve the function matrix projective synchronization. And, the synchronization analysis is confirmed by numerical simulation. Finally, by Zigzag transformation and DNA coding, the function matrix projective synchronization is applied to the color image encryption. And the comparison results shows that an excellent encryption scheme can be designed by the complex projection factor. This work not only extends the concept of the projective synchronization, but also provides an effective method for the image encryption.

Three kinds of Darboux transformations are constructed by means of the loop group method for the complex reverse space-time (RST) nonlocal modified Korteweg–de Vries equation, which are different from that for the PT symmetric (reverse space) and reverse time nonlocal models. The N-periodic, the N-soliton, and the N-breather-like solutions, which are, respectively, associated with real, pure imaginary, and general complex eigenvalues on a finite background are presented in compact determinant forms. Some typical localized wave patterns such as the doubly periodic lattice-like wave, the asymmetric double-peak breather-like wave, and the solitons on singly or doubly periodic waves are graphically shown. The essential differences and links between the complex RST nonlocal equations and their local or PT symmetric nonlocal counterparts are revealed through these explicit solutions and the solving process.

In this paper,by using the Sobolev inequality,the Green formula coupling and the elaborate energy method,we study the quasi-neutral limit of compressible Planck-NernestPoisson-Navier-Stokes（PNPNS） system with the general mobilities of two kinds of charges,which arises in the electro-hydrodynamics.

In this paper, we study the initial layer problem of the Euler–Poisson in collisionless plasma that is rigorously proved by using the weighted energy method coupled with multiscaling asymptotic expansions. We apply a formal expansion for Debye length and derive the inner and the initial layer equations, which are used for deriving the error equations, and give out uniform estimate.

It is well known that the variability and complexity of projection proportionality factors of dual projective synchronization (DPS) can effectively enhance signal confidentiality. However, in most literatures, the proportionality factors are some simple fixed constants, which can't ensure high security of information. For two pairs of fractional-order hyperchaotic systems (FOHS), how to expand the projection proportionality factors to increase its complexity? Then, our work will propose a new synchronization type, i.e., Dual Function Matrix Projective Synchronization (DFMPS) and realize the DFMPS for FOHS for the first time. Firstly, based on the traditional DPS, we generalize the proportionality factors to a function matrix depending on time t, present the error functions and define the DFMPS. Then, for FOHS, the active controller and synchronization condition are designed and proved. At the same time, when the system is affected by parameter disturbances, the active controller can eliminate the influence of parameter disturbances to the system's DFMPS, which indicates that the proposed control strategy has strong robustness. Finally, the DFMPS of two pairs of fractional-order hyperchaotic Chen and Rabinovich systems are realized, and synchronizing analysis and system robustness analysis are verified by numerical simulation. Particularly, the DFMPS can be degenerated to dual antisynchronization, dual complete synchronization, DPS, modified DPS and dual matrix projective synchronization. This work extends the synchronization types for FOHS and offers a useful method to explore DFMPS for other fractional-order systems.

Fractional-order neural networks (FONNs) can improve the computational ability and facilitate the information transmission of neurons, whose projective synchronization and applications have been widely used in the fields of security communication. Our main work will propose a new synchronization type, i.e., function matrix projective synchronization (FMPS), and realize the FMPS for delayed and unknown FONNs for the first time. Firstly, based on the traditional PS, we generalize the projective proportionality factors to any function matrix depending on time t , present the error function and define the FMPS, which is more extensive and practical than other synchronization types. Then, for delayed and unknown FONNs, the adaptive control strategy with the controlling strength updated rules and the unknown parameter adaptive rules are designed and the FMPS is realized by establishing a Lyapunov function. Finally, for a numerical example, trajectories of the synchronization errors approach to 0 and unknown parameters converge to the fixed constants, which illustrate the efficiency of the proposed theory analysis. This work may offer a useful method to explore FMPS for fractional-order systems.

Fractional Order Memristor-Based Neural Networks (FOMBNNs) has the strong sensitivity to initial values and shows more complex paths, so its Projective Synchronization (PS) and applications have been widely used in the fields of security communication. Our main work intends to extend the scaling factor of PS to a function matrix depending on time t and proposes a new synchronization type for the first time, i.e., Function Matrix Projective Synchronization (FMPS) for FOMBNNs, whose scaling factor is highly variable over time and difficult to predict. However, the FOMBNNs is a state dependent discontinuous system and it is easy to produce complex nonlinearity, which makes the study of the FMPS becomes a challenge. Therefore, our work will commit to solving this problem and realizing the FMPS for multi-time delayed FOMBNNs with parameter uncertainty. Firstly, the error functions and FMPS are defined, which can be degenerated to matrix PS, modified PS, PS, antisynchronization and complete synchronization. Then, for the multi-time delayed FOMBNNs with parameter uncertainty, the active controller is designed and the sufficient condition for realizing the FMPS is proved by using a Lyapunov functional and some Lemmas of fractional calculus. Finally, the FMPS of four numerical examples are given and trajectories of their synchronization errors approach to 0, which illustrate the efficiency of the proposed synchronization analysis. This research will provide a general method for studying the FMPS of other dynamical systems.

根据分数阶Lü 混沌系统,提出具有非线性时滞项的分数阶Lü 混沌系统.首先,用Adomian分解算法(ADM)对分数阶Lü 混沌系统进行数值求解;其次,用MATLAB软件绘制系统相轨迹图;最后,用仿真技术及分岔图、复杂度和相轨迹等动力学分析工具,分析系统参数对系统的影响.数值仿真结果表明,该系统具有丰富的动力学特性.

针对分数阶Lü超混沌系统,采用Adomian分解法对其非线性项进行分解,并采用MATLAB软件绘制了系统的相图,同时从系统的分岔图、谱熵(spectral entropy,SE)复杂度、C0复杂度等数值仿真分析研究了 0.90阶分数阶Lü超混沌系统丰富的动力学特性.同时采用QR分解算法,将Lyapunov指数计算展开,利用MATLAB软件仿真,得出Lyapunov指数谱与复杂度具有一致性的结论.

This work generalizes the projection scaling factor to a general constant matrix and proposes the matrix-projection synchronization (MPS) for fractional-order neural networks (FNNs) based on sliding mode control firstly. This kind of scaling factor is far more complex than the constant scaling factor, and it is highly variable and difficult to predict in the process of realizing the synchronization for the driving and response systems, which can ensure high security and strong confidentiality. Then, the fractional-order integral sliding surface and sliding mode controller for FNNs are designed. Furthermore, the criterion for realizing MPS is proved, and the reachability and stability of the synchronization error system are analyzed, so that the global MPS is realized for FNNs. Finally, a numerical application is given to demonstrate the feasibility of theory analysis. MPS is more general, so it is reduced to antisynchronization, complete synchronization, projective synchronization (PS), and modified PS when selecting different projective matrices. This work will enrich the synchronization theory of FNNs and provide a feasible method to study the MPS of other fractional-order dynamical models.

In this paper, by using the adaptive control method, the global matrix-projective synchronization of delayed fractional-order competitive neural network with different time scales is researched for the first time. Firstly, the fractional-order global matrix-projective synchronization is defined. Then, in order to achieve the matrix-projective synchronization, the sufficient condition is obtained under an adaptive controller and its effectiveness is proved by combining the fractional-order Barbalat theory with a suitable Lyapunov–Krasovskii functional as well as some fractional-order differential inequalities. And all unknown parameters are identified and estimated to the fixed constants successfully. Finally, as applications, a numerical example with simulations is employed to demonstrate the feasibility and efficiency of the new synchronization analysis.

针对分数阶超混沌系统,采用分数阶非线性系统的Lyapunov稳定性理论,提出了一种自适应同步控制方法,同时设计模拟电路实现了自适应同步控制;基于以往分数阶的局限性,给出了0.90~0.99阶系统电路,并采用Multisim仿真实现了该电路.电路仿真结果与理论结果一致,表明了控制器硬件的可实现性.

The research of finding hidden attractors in nonlinear dynamical systems has attracted much consideration because of its practical and theoretical importance. A new fractional order four-dimensional system, which can exhibit some hidden hyperchaotic attractors, is proposed in this paper. The predictor-corrector method of the Adams-Bashforth-Moulton algorithm and the parameter switching algorithm are used to numerically study this system. It is interesting that three different kinds of hidden hyperchaotic attractors with two positive Lyapunov exponents are found, and the fractional order system can have a line of equilibria, no equilibrium point, or only one stable equilibrium point. Moreover, a self-excited attractor is also recognized with the change of its parameters. Finally, the synchronization behavior is studied by using a linear feedback control method.

根据Lü混沌系统,结合时滞因素,提出了分数阶时滞Lü混沌系统,运用Adomian分解法算法,对非线性进行分解,得出分数阶时滞系统数值解;结合数值解的过程采用MATLAB仿真,通过系统的分岔图、复杂度以及吸引子相图等工具验证了参数对系统的影响;仿真结果表明了0.9阶次时滞系统丰富的动力学特性,为分数阶时滞系统应用于图像加密时的参数选择提供了理论基础.

Reaction–diffusion systems have received considerable attention during the last few decades due to their ability to model the dynamics of a wide range of natural physical phenomena in the fields of fluid dynamics [1], chemistry [2], ecology [3] and other disciplines. These systems take into account the nonhomogeneous spatial distribution of populations and the effect of that distribution on the diffusion process. Reaction diffusion systems have recently found applications in the fields of natural computing [4] and biological communication networks [5]. In addition, such models have given us a deep insight into the dynamics of diffusive neural networks and their control and applications [6].

This paper is concerned with the quasi-matrix and quasi-inverse-matrix projective synchronization between two nonidentical delayed fractional order neural networks subjected to external disturbances. First, the definitions of quasi-matrix and quasi-inverse-matrix projective synchronization are given, respectively. Then, in order to realize two types of synchronization for delayed and disturbed fractional order neural networks, two sufficient conditions are established and proved by constructing appropriate Lyapunov function in combination with some fractional order differential inequalities. And their estimated synchronization error bound is obtained, which can be reduced to the required standard as small as what we need by selecting appropriate control parameters. Because of the generality of the proposed synchronization, choosing different projective matrix and controllers, the two synchronization types can be reduced to some common synchronization types for delayed fractional order neural networks, like quasi-complete synchronization, quasi-antisynchronization, quasi-projective synchronization, quasi-inverse projective synchronization, quasi-modified projective synchronization, quasi-inverse-modified projective synchronization, and so on. Finally, as applications, two numerical examples with simulations are employed to illustrate the efficiency and feasibility of the new synchronization analysis.

This paper constructs a new physical system, i.e., the fractional-order Rabinovich system, and investigates its stability, chaotic behaviors, chaotic control and matrix projective synchronization. Firstly, two Lemmas of the new system's stability at three equilibrium points are given and proved. Next, the largest Lyapunov exponent, the corresponding bifurcation diagram and the chaotic behaviors are studied. Then, the linear and nonlinear feedback controllers are designed to realize the system's local asymptotical stability and the global asymptotical stability, respectively. It's particularly significant that, the fractional matrix projective synchronization between two Rabinovich systems is achieved and two kinds of proofs are provided for Theorem 4.1. Especially, under certain degenerative conditions, the fractional matrix projective synchronization can be reduced to the complete synchronization, anti-synchronization, projective synchronization and modified projective synchronization of the fractional-order Rabinovich systems. Finally, all the theoretical analysis is verified by numerical simulation.