Fractional-order Complex Networks Research Articles

Novel Controller Design for Finite-Time Synchronization of Fractional-Order Nonidentical Complex Dynamical Networks under Uncertain Parameters

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.

Cluster Synchronization of Fractional-Order Complex Community Networks in Quaternion-Valued Field

Without separating the fractional-order quaternionvalued complex networks (FOQVCNs) into four real-valued systems, cluster synchronization is investigated in this paper. First, the FOQVCNs are divided into several clusters, and the dynamic behaviors of the clusters are different. For the case of different dynamics behaviors of the network nodes, hybrid adaptive control is proposed, which means that coupling strength and feedback control gain are simultaneously regulated with the evolution of states. Particularly, some new analytical approaches for fractional-order adaptive schemes, including the method of contradiction, the Mean Value Theorem are developed to establish synchronization criteria of the network system. Furthermore, asymptotical synchronization of FOQVCNs with identical nodes is also studied by adaptive feedback controller and adjusting coupling strength as a special case. At last, two fractional-order quaternion-valued complex community networks are divided to synchronize to the desired trajectory, respectively and numerical simulations are provided to illustrate the validity of theoretical results.

Synchronization of fractional-order fuzzy complex networks with time-varying couplings and proportional delay

The topic of synchronization of fractional-order fuzzy complex networks with time-varying couplings and proportional delay is addressed in this paper. By combining graph theory, the Lyapunov method, and the Razumikhin method, the synchronization criteria are obtained by designing a new time-varying graph-theoretical Lyapunov function. Different from the typical quadratic Lyapunov function, the time-varying graph-theoretical Lyapunov function contains the product of three-term functions and poses the difficulty of fractional derivative due to the unavailability of the general Leibniz formula and the chain rule. A new inequality about the fractional derivative rule of the time-varying graph-theoretical Lyapunov function is established based on the convex function creatively. In addition, compared with relevant literatures, the synchronization criteria are related to the proportionality coefficient and the order of fractional derivative. Then, the synchronization of fractional-order power system established on fuzzy complex networks with time-varying couplings and proportional delay is studied for the first time by applying the feedback control. Lastly, some simulations are exhibited to demonstrate the effectiveness of the theoretical results.

Quasi-synchronization of fractional-order complex networks with random coupling via quantized control

We investigate the quasi-synchronization of fractional-order complex networks (FCNs) with random coupling via quantized control. Firstly, based on the logarithmic quantizer theory and the Lyapunov stability theory, a new quantized feedback controller, which can make all nodes of complex networks quasi-synchronization and eliminate the disturbance of random coupling in the system state, is designed under non-delay conditions. Secondly, we extend the theoretical results under non-delay conditions to time-varying delay conditions and design another form of quantization feedback controller to ensure that the network achieves quasi-synchronization. Furthermore, the error bound of quasi-synchronization is obtained. Finally, we verify the accuracy of our results using two numerical simulation examples.

Fixed-Time Synchronization of Fractional-Order Multilayer Complex Networks Via a New Fixed-Time Stability Theorem

AbstractFixed-time synchronization of fractional-order multilayer complex networks is studied in this paper. At first, a novel fixed-time stability theorem for the fractional-order nonlinear system is presented. The stability theorem is a generalization of the integer order stability theorem and plays an important role on the synchronization schemes. Based on the proposed stability theorem, the fixed-time synchronization of fractional-order multilayer complex networks is investigated, and a fixed-time synchronization criterion is presented. Simulation results are given to demonstrate the effectiveness of our results.

Passivity and Finite-Time Passivity for Multi-Weighted Fractional-Order Complex Networks With Fixed and Adaptive Couplings.

This article presents several new α -passivity and α -finite-time passivity ( α -FTP) concepts for the fractional-order systems with different input and output dimensions, which are distinct from the concepts for integer-order systems and extend the existing passivity and FTP definitions to some extent. On one hand, we not only develop some sufficient conditions for ensuring the α -passivity of the multi-weighted fractional-order complex dynamical networks (MWFOCDNs) with fixed and adaptive couplings, but also discuss the synchronization for the MWFOCDNs based on the α -output-strict passivity ( α -OSP). On the other hand, the α -FTP for the MWFOCDNs with fixed and adaptive couplings are also studied on the basis of the designed state feedback controller, and the relationship between finite-time synchronization (FTS) and α -FTP for the MWFOCDNs is also illustrated. Finally, two numerical examples with simulation results are used to demonstrate the validity of the obtained criteria.

Complex modified projective synchronization of fractional-order complex-valued dynamic network with time-varying coupling and parameters estimation

The problem of Complex Modified Projective Synchronization (CMPS) for fractional-order complex-valued dynamic networks (FOCDNs) is investigated in this paper. Compared with real-valued dynamic networks, complex-valued dynamic networks have expansive application space. First, coupling weight has been generally considered as static constant in most situations, we consider time-varying coupling weight and the unknown parameters in the FOCDNs. Meanwhile, several parameter update laws are designed to estimate unknown parameters and an adaptive coupling weight is defined to adjust coupling nodes in order to solve CMPS problem of the network. Second, some effective criteria for CMPS of FOCDNs are derived by fractional calculus and Lyapunov stability theory. According to the theoretical analysis, FOCDNs with time-varying delay can achieve CMPS via adaptive controller. Finally, the effectiveness of the modified projective strategy for fractional-order complex networks is illustrated via two numerical examples in the complex-valued space.

Synchronization Rather Than Finite-Time Synchronization Results of Fractional-Order Multi-Weighted Complex Networks.

This article investigates the synchronization of fractional-order multi-weighted complex networks (FMWCNs) with order α ∈ (0,1) . A useful fractional-order inequality t0C Dtα V(x(t)) ≤ -μV(x(t)) is extended to a more general form t0C Dtα V(x(t)) ≤ -μVγ(x(t)),γ ∈ (0,1] , which plays a pivotal role in studies of synchronization for FMWCNs. However, the inequality t0C Dtα V(x(t)) ≤ -μVγ(x(t)),γ ∈ (0,1) has been applied to achieve the finite-time synchronization for fractional-order systems in the absence of rigorous mathematical proofs. Based on reduction to absurdity in this article, we prove that it cannot be used to obtain finite-time synchronization results under bounded nonzero initial value conditions. Moreover, by using feedback control strategy and Lyapunov direct approach, some sufficient conditions are presented in the forms of linear matrix inequalities (LMIs) to ensure the synchronization for FMWCNs in the sense of a widely accepted definition of synchronization. Meanwhile, these proposed sufficient results cannot guarantee the finite-time synchronization of FMWCNs. Finally, two chaotic systems are given to verify the feasibility of the theoretical results.

Dynamical effects of hypergraph links in a network of fractional-order complex systems.

In recent times, the fractional-order dynamical networks have gained lots of interest across various scientific communities because it admits some important properties like infinite memory, genetic characteristics, and more degrees of freedom than an integer-order system. Because of these potential applications, the study of the collective behaviors of fractional-order complex networks has been investigated in the literature. In this work, we investigate the influence of higher-order interactions in fractional-order complex systems. We consider both two-body and three-body diffusive interactions. To elucidate the role of higher-order interaction, we show how the network of oscillators is synchronized for different values of fractional-order. The stability of synchronization is studied with a master stability function analysis. Our results show that higher-order interactions among complex networks help the earlier synchronization of networks with a lesser value of first-order coupling strengths in fractional-order complex simplices. Besides that, the fractional-order also shows a notable impact on synchronization of complex simplices. For the lower value of fractional-order, the systems get synchronized earlier, with lesser coupling strengths in both two-body and three-body interactions. To show the generality in the outcome, two neuron models, namely, Hindmarsh-Rose and Morris-Leccar, and a nonlinear Rössler oscillator are considered for our analysis.

Cluster Synchronization for Multiweighted and Directed Fractional-Order Networks With Cooperative-Competitive Interactions

In this brief, we investigate the cluster synchronization of fractional-order multiweighted complex networks (FOMCNs) via pinning control. Recently, multiweighted complex network has been a hot model to replace classical single-weighted network. The outer coupling matrices (OCMs) are often assumed to be symmetric and strongly connected; moreover, the interactions between nodes are only cooperative. The main contribution of this brief is that we can remove all these limitations, i.e., OCMs can be asymmetric and not connected, while interactions inter-/intra-nodes can be cooperative or competitive. The approach is called re-arranging variables’ order technique (ROT), where we combine OCMs with elements in inner linking matrices (ILMs) as coefficients to form union matrices on the same dimension of each node. Some sufficient criteria are presented to guarantee cluster synchronization of FOMCNs, and numerical simulations are given to demonstrate its effectiveness.

Mean-square synchronization of fractional-order stochastic complex network via pinning control

In this paper, the mean-square synchronization problem is considered for the fractional-order stochastic complex network (FOSCN) via pinning control. Firstly, suppose that the dynamics of the node are modeled by continuous-time nonlinear fractional stochastic differential equations (FSDEs). Next, the pinning nodes are selected according to the low-dimensional pinning criteria. Furthermore, the controller is designed for each pinning node, and a set of sufficient conditions are given to guarantee the mean-square synchronization. Then, the convergence analysis of the closed-loop system is finished by directly utilizing the properties of the integral solution. It proved that the mean-square synchronization of FOSCN can be achieved with the designed controller under some conditions. In addition, the synchronization problem of general fractional-order complex networks (FOCN) without stochastic noise is solved by using a similar method. Finally, compared simulation examples are performed to demonstrate the effectiveness of the proposed methods.

Disturbance rejections and synchronization of fractional-order fuzzy complex networks

This paper deals with the problem of disturbance rejection and synchronization of fractional-order complex dynamical networks subject to nonlinear coupling strength and discontinuous nonlinear functions. Notably, the nonlinear coupling strength is linearised by using a well-known Takagi-Sugeno fuzzy approach. The considered system is transformed into a nominal form by employing the uncertainty and disturbance estimator-based control approach, which simplifies the control objective and improves the system performance. Second, the uncertainty and disturbance estimator is incorporated into the traditional feedback control scheme to reject the unknown disturbance and uncertainty. Then, the required synchronization conditions for both the discontinuous and continuous fractional-order systems are obtained by using Lyapunov stability and fractional calculus theories. Last, numerical examples are provided to illustrate the efficiency of the proposed control strategy, wherein it is shown that the system yields better satisfactory tracking performance and high robustness against possible disturbance and uncertainties and finite set of jump discontinuous nonlinear functions. Moreover, the selection of appropriate filter design is discussed for various kinds of disturbance signals.

Comments on “Synchronization for a Class of Fractional-order Linear Complex Networks via Impulsive Control”

Comments on “Synchronization for a Class of Fractional-order Linear Complex Networks via Impulsive Control”

Finite-Time Projective Synchronization and Parameter Identification of Fractional-Order Complex Networks with Unknown External Disturbances

This paper is devoted to exploring the finite-time projective synchronization (FTPS) of fractional-order complex dynamical networks (FOCDNs) with unknown parameters and external disturbances. Based on the stability theory of fractional-order differential systems, synchronization criteria between drive-response networks were obtained and both the uncertain parameters and external disturbances were identified or conquered simultaneously. Moreover, the upper limit of the settling-time function was obtained. Finally, a numerical example was given to verify the effectiveness of the results.

Finite-time synchronization of uncertain fractional-order multi-weighted complex networks with external disturbances via adaptive quantized control

The finite-time synchronization of fractional-order multi-weighted complex networks (FMCNs) with uncertain parameters and external disturbances is studied. Firstly, based on fractional calculus characteristics and Lyapunov stability theory, quantized controllers are designed to guarantee that FMCNs can achieve synchronization in a limited time with and without coupling delay, respectively. Then, appropriate parameter update laws are obtained to identify the uncertain parameters in FMCNs. Finally, numerical simulation examples are given to validate the correctness of the theoretical results.

Identification and Synchronization of Switching Fractional-Order Complex Networks with Time-Varying Delays Based on a Fuzzy Method

Identification and Synchronization of Switching Fractional-Order Complex Networks with Time-Varying Delays Based on a Fuzzy Method

Synchronization of discrete fractional-order complex networks with and without unknown topology.

This paper studies the outer synchronization problem of discrete fractional complex networks (DFCNs) with and without the presence of unknown topology. A discrete complex network with a fractional difference is first established and analyzed. By constructing a suitable Lyapunov function and utilizing properties of the fractional difference, outer synchronization criteria for the DFCNs with and without unknown topology are established based on linear matrix inequalities. Meanwhile, the unknown parameters in the topology structure of the network can be identified by adaptive update laws. In the end, two numerical examples are given to exemplify the validity and applicability of the obtained results.

Cluster synchronization of fractional-order complex networks via uncertainty and disturbance estimator-based modified repetitive control

The combined problems of cluster synchronization and disturbance rejection for a family of fractional-order complex networks subject to coupling delay, unknown uncertainty and disturbances (UDs) are examined in this study. In particular, the existence of coupling delay is taken into the account with both known and unknown cases. First, a new uncertainty and disturbance estimator (UDE)-based control protocol is formulated for the concerned system to estimate and compensate for the effects of UD. Although the UDE strategy has proven to be a viable tool to deal with slowly changing UDs in control design, the presence of rapidly changing UDs or sinusoidal disturbances is not an effective tool. A well-known modified iterative control (MRC) block is built internally in a closed feedback control loop to solve this problem. After implementing UDE and MRC blocks into the feedback loop, the resulting system becomes almost UD-free. Moreover, a set of sufficient linear matrix inequality constraints are established to ensure the cluster synchronization of the resulting system. Lastly, the benefits, feasibility and robustness of the established UDE-based MRC scheme are confirmed by two illustrative examples.

Cluster synchronization in fractional-order network with nondelay and delay coupling

In this paper, cluster synchronization for fractional-order complex network with nondelay and delay coupling is investigated. Based on the stability theory of fractional-order systems and the properties of fractional derivative, both static and adaptive control schemes are adopted to design effective controllers. Sufficient condition for achieving cluster synchronization about static controllers is provided. From the condition, the needed feedback gains can be estimated by simple calculations. Further, adaptive control scheme is introduced to design unified controllers. Noticeably, in the adaptive controllers, the feedback gains need not be calculated in advance and can adjust themselves to the needed values according to updating laws. Finally, numerical simulations are given to demonstrate the correctness of the obtained results.

Intermittent control for finite-time synchronization of fractional-order complex networks

This paper is concerned with the finite-time synchronization problem for fractional-order complex dynamical networks (FCDNs) with intermittent control. Using the definition of Caputo’s fractional derivative and the properties of Beta function, the Caputo fractional-order derivative of the power function is evaluated. A general fractional-order intermittent differential inequality is obtained with fewer additional constraints. Then, the criteria are established for the finite-time convergence of FCDNs under intermittent feedback control, intermittent adaptive control and intermittent pinning control indicate that the setting time is related to order of FCDNs and initial conditions. Finally, these theoretical results are illustrated by numerical examples.