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The increasing penetration of renewable energy leads to a continuous reduction in system inertia, for which conventional synchronization criteria based solely on frequency consistency can no longer accurately capture the coupled dynamics of frequency and voltage during transients. To address this issue, this paper employs the concept of complex frequency and develops an analysis framework that integrates theory, indices, and simulation for assessing synchronization stability in low-inertia power systems. Firstly, the basic concepts and mathematical formulation of complex frequency and complex frequency synchronization are introduced. Then, dynamic criteria for local and global complex synchronization are established, upon which a complex inertia index is proposed. This index unifies the supporting role of traditional frequency inertia and the voltage support capability associated with voltage inertia, enabling the quantitative evaluation of the strength of coordinated frequency–voltage support and disturbance rejection within a region. Finally, transient simulations on a modified WSCC nine-bus system are carried out to validate the proposed method. The results show that the method can clearly reveal the synchronization relationships between subnetworks and the overall system, providing a useful theoretical reference for stability analysis and control strategy design in low-inertia power systems.

A complex network is composed of nodes and connecting edges, which can be applied to describe the structure of many real systems. Synchronization is an important behavior of dynamic complex networks. The traditional methods, such as changing network coupling mode and external control strategies, etc., cannot achieve complete network synchronization. In order to solve the network synchronization problem, in the paper, we propose a method of constructing an improved small-world network that combines degree distribution and eigenvector criteria from the viewpoint of the topology of complex networks and analyze the effects of network topology on synchronization capability. Aiming at the problem that synchronization capability is suppressed due to the uneven degree distribution during the construction of network model, we present a method of degree distribution connection to select the nodes with a smaller degree preferentially when reconnecting edges. Combining eigenvector Criteria, we also present a method of building Enhanced Synchronization Small-World, which deletes connecting edges by selecting the nodes with a larger degree and reconnects edges according to the eigenvector criterion. On this basis, we further analyze the impacts of differences in different network topologies on synchronization capability. The experimental results show that our solution is effective to construct network through the combination of degree distribution and eigenvector criterion, which can solve the network synchronization problem well by improving network topology.

In this paper, we consider finite-time synchronization between two complex dynamical networks using periodically intermittent control. Based on finite-time stability theory, some novel and effective finite-time synchronization criteria are derived by applying stability analysis technique. The derivative of the Lyapunov function V ( t ) is smaller than β V ( t ) ( β is an arbitrary positive constant) when no controllers are added into networks. This means that networks can be self-synchronized without control inputs. As a result, the application scope of synchronization is greatly enlarged. Finally, a numerical example is given to verify the effectiveness and correctness of the synchronization criteria.

In this article, the problem of composite synchronization of three induction motors with the circular distributions is studied. Composite synchronization consists of self-synchronization and controlled synchronization. Firstly, the mechanical and electrical coupling dynamics model of the vibration system is established, the response of the system is obtained, the differential equation is analyzed by means of small parameter average method, synchronicity conditions and stability criteria for vibrating systems to achieve vibrationally synchronized motion are obtained. Then, the master–slave control strategy and fuzzy PID (proportional–integral–derivative) control method are introduced to realize the part of controlled synchronization in the vibration system. The stability of the control method is verified by Lyapunov criterion. The applicability of this control method and its feasibility for practical applications are proved by numerical simulation and experiment. The study in this paper illustrates the feasibility of designing some new vibratory transporters for industrial transportation.

This paper proposes a Lyapunov-based fixed-time stability (FXTS) theorem for nonlinear dynamical systems. An upper bound estimate of the settling time and the sufficient condition for FXTS of nonlinear systems are provided. An optimality criterion is derived to obtain the minimum value of the settling time’s upper bound estimate. Meanwhile, an insight into accurate estimation of the upper bound of settling time is provided. Based on the FXTS theorem, non-singular terminal sliding surfaces (NSTSM) and corresponding controllers are constructed to make the trajectories of the perturbed nonlinear dynamical system reach the equilibrium point robustly in some fixed-time. A secure communication scheme based on fixed-time synchronization of chaotic systems is also proposed as an application of the theoretical results obtained in this study. Numerical simulations are carried out for the illustration and validation of theoretical claims.

This study explores the synchronization issue for uncertain multi-link complex networks incorporating stochastic characteristics and hybrid delays. Unlike previous works, internal delays, coupling delays, and stochastic delays considered in our model change over time; meanwhile, the impulse strength and position change with time evolution. To actualize network synchronization, a strategy called hybrid impulsive pinning control is applied, which combines the virtue of impulsive control and pinning control as well as two categories of impulses (i.e., synchronization and desynchronization). By decomposing the complicated topological structures into diagonal items and off-diagonal items, multiple nonlinear coupling terms are linearly decomposed in the process of theoretical analysis. Combining inequality technology and matrix decomposition theory, several novel synchronization criteria have been gained to ensure synchronization for the concerning multi-link model. The criteria get in touch with the uncertain strengths, coupling strengths, hybrid impulse strengths, delay sizes, impulsive intervals, and network topologies.

Generally, the synchronization studies on two or multiple exciters are preconditioned by being a single frequency, while the multiple-frequency synchronization problems in a vibrating system, including double-frequency and triple-frequency, are less considered, which are also very significant in engineering. This paper attempts to solve this issue by considering a dynamical model with an isolation frame, driven by the four exciters. The synchronization for the four exciters and its stability under the double-frequency and triple-frequency conditions are studied in detail. Firstly, the mathematical modeling of the system is established, and the corresponding motion differential equations are derived. Using the asymptotic method and the average method, yields the theoretical condition of implementing multiple-frequency synchronization, and the theoretical condition for stability of the system complies with the Routh–Hurwitz criterion. The dynamic characteristics of the system, including stable phase differences, stability abilities, responses of the system, and relative motion relationship, are qualitatively discussed by numeric. Finally, simulations are performed by applying a Runge–Kutta program to validate the theoretical and numerical qualitative results. It is shown that, by reasonably matching the key parameters of the system, the stronger, stable, and valuable motion states of vibrating machines, including vibration amplitudes, frequencies, and motion trajectory, can be realized, which are exactly the desires in engineering.

This article is concerned with fixed-time synchronization and preassigned-time synchronization of Cohen–Grossberg quaternion-valued neural networks with discontinuous activation functions and generalized time-varying delays. Firstly, a dynamic model of Cohen–Grossberg neural networks is introduced in the quaternion field, where the time delay successfully integrates discrete-time delay and proportional delay. Secondly, two types of discontinuous controllers employing the quaternion-valued signum function are designed. Without utilizing the conventional separation technique, by developing a direct analytical approach and using the theory of non-smooth analysis, several adequate criteria are derived to achieve fixed-time synchronization of Cohen–Grossberg neural networks and some more precise convergence times are estimated. To cater to practical requirements, preassigned-time synchronization is also addressed, which shows that the drive-slave networks reach synchronization within a specified time. Finally, two numerical simulations are presented to validate the effectiveness of the designed controllers and criteria.

In present work, the double and triple-frequency synchronization behaviors of a vibrating mechanical system with two different driving frequencies, driven by three reversed rotating exciters, are investigated by theory, numeric, and experiment. Based on Lagrange’s equations, the dynamic model corresponding to vibrating machine is proposed and motion differential equations are constructed. The Bogoliubov standard formal equations for three exciters are established, by introducing the asymptotic method, in which the synchronization problem is converted into that of the existence and stability of zero-valued solution of the average differential equations. The synchronization criterion of satisfying the synchronous operation is deduced. According to the Routh–Hurwitz criterion, the stability criterion of the synchronous states is achieved analytically. Based on the obtained theory results, the stability characteristics of the system, are numerically discussed in detail, including the stability ability coefficients and stable phase differences. Finally, simulations and experiments under the condition of two different driving frequencies, are performed to further examine the validity of the theoretical and numerical qualitative results. The present work can provide a theoretical reference for designing some new types of the vibrating machines with adjustable frequencies and motion trajectories.

Chaotic behavior in power systems that are integrated with permanent-magnet synchronous generators (PMSGs) poses a significant threat to stability and security. Existing control methods often suffer from slow convergence, reliance on precise system models, or the inability to guarantee convergence within a predefined time. To address these issues, this paper develops a predefined-time synchronization control scheme for chaotic PMSG systems under unknown nonlinearities and external disturbances. First, an adaptive neural network with variable exponent coefficients is constructed to approximate unknown system dynamics online. Second, a predefined-time stability criterion is established, ensuring global convergence of synchronization errors within a user-specified time, independently of initial conditions. Third, the proposed controller achieves superior disturbance rejection without requiring prior knowledge of disturbance bounds. Numerical simulations demonstrate that the proposed method outperforms conventional finite-time control in convergence speed, control smoothness, and robustness to parameter variations—offering a practical and theoretically guaranteed solution for enhancing the stability of PMSG-based power systems.