This article proposes a novel adaptive predefined-time control scheme that guarantees global prescribed performance and exact tracking for a class of uncertain nonlinear systems characterized by nonstrict feedback dynamics, actuator faults, external disturbances, and sensor faults. Anovel switching feedback gain method is introduced to simultaneously handle unknown nonlinearities, disturbances, and actuator faults, while a tan-type error transformation and a new Lyapunov-like energy function are designed to ensure that all signals in the closed-loop systems remain globally bounded without requiring knowledge of control coefficients. In contrast to existing finite-time or asymptotic tracking control strategies, the proposed method ensures exact convergence to zero tracking error within a predefined time—independent of initial conditions—while strictly maintaining prescribed transient performance bounds. Simulation results validate the effectiveness of the proposed approach and its superiority over conventional methods, demonstrating faster convergence and smaller overshoot, thereby establishing a new benchmark for high-precision control of nonstrict feedback systems.

We propose IPSI, a general iterative framework for structural inference in interacting dynamical systems. It integrates a pretrained structural estimator and a joint inference module based on the Variational Autoencoder (VAE); these components are alternately updated to progressively refine the inferred structures. Initially, the structural estimator is trained on labels from either a meta-dataset or a baseline model to extract features and generate structural priors, which provide multi-level guidance for training the joint inference module. In subsequent iterations, pseudolabels from the joint module replace the initial labels. IPSI is compatible with various VAE-based models. Experiments on synthetic datasets of physical systems demonstrate that IPSI significantly enhances the performance of structural inference models such as Neural Relational Inference (NRI). Ablation studies reveal that feature and structural prior inputs to the joint module offer complementary improvements from representational and generative perspectives.

This article proposes a dynamic self-triggered distributed model predictive control algorithm for coupled nonlinear systems facing external disturbances and constraints on state and input variables. A dynamic self-triggered mechanism that combines the advantages of event-triggered and self-triggered strategies is designed to simultaneously reduce the frequencies of both sampling and solving optimization problems. Particularly, the triggering threshold is adaptively adjusted using a dynamic variable, which can effectively balance control performance and computational resources. Furthermore, through the construction of a two-model optimal control problem and the analysis of input-to-state practical stability for the overall system, a single-mode distributed model predictive control framework is established for each subsystem within the proposed algorithm, which enables a fully distributed implementation. Sufficient conditions for recursive feasibility and robust stability are investigated, and conservatism is reduced by eliminating the requirement for the system state to reach the terminal region in finite time. Finally, the effectiveness of the developed algorithm is validated through two numerical examples with comparisons.

The principal innovative contribution of this study resides in the introduction of a category of fractional delayed large-scale neural networks characterized by intricate topological structures. Additionally, this article provides a comprehensive exploration of novel outcomes linked to fractional order-induced bifurcations in large-scale networks. In the initial step, the correlation of the artificial neural network and the graphical neural network is established through the Mason's diagram method. Subsequently, the system's characteristic equations are derived by employing the Coates' flow graph decomposition method. Moving on, through the concept of the global element, an exhaustive investigation delves into the distribution of eigenroots. The sum of synaptic transmission delays among neurons is considered as a bifurcation parameter, with an analysis focused on the stability of the trivial equilibrium and the existence of the Hopf bifurcation. Following this, the optimal fractional order-dependent stability interval is determined using the implicit function array curve method, presenting a novel approach for critical value determination. Finally, the drawn conclusions are substantiated through multiple sets of computer simulations. It is indicated that an increase in delay precipitates the onset of Hopf bifurcation. Moreover, a reduction in the fractional order significantly improves the steady-state performance of the system. However, once the fractional order value descends below the left stability boundary, the system's stability is compromised, leading to the emergence of periodic oscillations. The prediction algorithm proposed in this article offers valuable insights into selecting the appropriate fractional order for large-scale complex networks.

Over the past decade, numerous synchronization-based identification methods have been proposed to address the challenge of identifying unknown network topologies. The linear independence condition (LIC) is an essential requirement in these methods, however, there are issues with this condition. In this article, we propose an improved LIC-free synchronization-based identification method to address above issues. Specifically, a drive network consisting of isolated nodes that satisfy specific conditions is constructed, and the network containing an unknown topology is defined as the response network. Through the design of appropriate controllers and update laws, the drive network and the response network achieve synchronization, while the estimation matrix accurately identifies the unknown topology matrix. Our method is proven to be a generalized form of the existing LIC-free identification methods. Furthermore, we introduce a novel proof framework to theoretically demonstrate the effectiveness of our method. Finally, two simulation examples demonstrate the effectiveness of the proposed method.

Superdiffusion refers to the faster diffusion process in a multiplex network compared to that in an individual network. In this work, we study how interlayer connectivity affects the diffusion performance of a multiplex network. Based on spectral graph theory, we explore the principles of superdiffusion in multiplex networks. We prove that in a duplex network with identical structures, superdiffusion cannot occur under one-to-one interlayer connections. In addition, we prove that the dissimilarity of the Fiedler vector significantly enhances the network superdiffusion performance, which can lead to superdiffusion when selecting nodes with differential eigenvector components in the Fiedler vector for interlayer connections. We also prove that the upper bound of network diffusion with interlayer crossing-connections is limited by the maximum difference of the eigenvector components in the Fiedler vector. Finally, we verify the effectiveness of the theoretical results by numerical analysis.

Pinning control has been attracting wide attention for the study of various complex networks for decades. This article explores grounded theory on the pinning synchronization of the emerging multiplex dynamical networks. The multiplex dynamical networks under study can describe many real-world scenarios, in which different layers have distinct individual dynamics of node. In this work, we build the bridge between multiplex structures and network dynamics by using the Lyapunov stability theory and the spectral graph theory. Furthermore, by analyzing spectral properties of the grounded super-Laplacian matrices, we set up several graph-based synchronization criteria for multiplex networks via pinning control. In addition, we overcome the difficulties induced by distinct node dynamics in different layers, and find that interlayer coupling strengths promote intralayer synchronization of multiplex networks. Finally, a collection of numerical simulations verifies the effectiveness of theoretical results.

For strict-feedback systems with mismatched uncertainties, adaptive fuzzy control techniques are developed to provide global prescribed performance with prescribed-time convergence. First, a class of prescribed-time prescribed performance functions are designed to quantify the performance constraints of the tracking error. Additionally, a novel error transformation function is provided to eliminate the initial value limitations and resolve the singularity issue in previous research. To ensure the convergence of the tracking error into a prescribed bounded region within a prescribed time and satisfactory transient performance, controllers with or without approximating structures are established. Notably, the settling time and initial condition of the prescribed performance function are completely independent of the initial tracking error and system parameters, thereby improving upon existing results. Furthermore, the disadvantage of the semi-global boundedness of tracking error induced by dynamic surface control can be eliminated through the use of a novel Lyapunov-like energy function. Finally, the effectiveness of the proposed strategies is validated through numerical simulations performed on practical examples.

This article pays attention to finite-time (FnT)$p$th moment asymptotically bounded (MAB) for stochastic nonlinear systems (SNSs) and its application to$p$th moment quasi-synchronization (MQS) of stochastic neural networks (SNNs) based on parameter mismatches. First, this article develops FnT asymptotically bounded theorems SNSs. In detail, several new FnT$p$th MAB theorems of the SNSs are proposed, the mathematical expression of finite settlement time is obtained, and the bounds of MAB are estimated. Second, new sufficient conditions are designed to ensure FnT$p$th MQS of SNNs. In particular, novel FnT$p$th MQS conditions of the SNNs enlarge the value of$p$. Moreover, when the initial value of the system meets certain conditions, the smaller the$p$is, the smaller the system synchronization control energy consumption is, which can be more meaningful. Finally, a numeric example illustrates the validity of the methods.

This article investigates uniformly predefined-time bounded consensus of leader-following multiagent systems (MASs) with unknown system nonlinearity and external disturbance via distributed adaptive fuzzy control. First, uniformly predefined-time-bounded stability is analyzed and a sufficient condition is derived for the system to achieve semiglobally (globally) uniformly predefined-time-bounded consensus. Therein, the settling time is independent of initial conditions and can be defined in advance. Then, for first-order MASs, distributed adaptive fuzzy controllers are designed by combining neighboring consensus errors to drive all following agents to globally track the leader's state within predefined time. For second-order MASs, by formulating filtered errors, the consensus errors between following agents and the leader are shown to be bounded if the filtered errors are bounded. Furthermore, with the distributed controllers designed based on filtered errors, second-order MASs achieve semiglobally uniformly predefined-time-bounded leader-following consensus. Finally, two numerical examples are simulated, including: 1) a first-order leader-following MAS and 2) a second-order Lagrangian system consisting of single-link manipulators, to demonstrate the performance of the proposed controllers.

The time-triggered impulsive controls were widely used to study the collective behavior of homogeneous dynamical networks due to their low control cost, which was a bit conservative in the occupation of communication channels. This article addresses designing the event-triggered impulsive controls for the quasisynchronization, namely, a weak cooperative behavior with the synchronization error no more than a positive constant in the leader-following heterogeneous dynamical network, which thus can reduce the occupation of resources significantly. The centralized and distributed impulsive controls are designed to lead the followers to synchronize approximately to the leader within a nonzero bound, where the impulsive instants are triggered, respectively, by the global or local state-dependent conditions. Numerical results are put forward to verify the effectiveness of the proposed methods.

Interval bipartite consensus of multiagents described by signed graphs has received extensive concern recently, and the rooted cycles play a critical role in stabilization, while the structurally balanced graphs are essential to achieve bipartite consensus. However, the gauge transformation used in the linear system is no longer feasible in the nonlinear case. This article addresses interval bipartite synchronization of multiple neural networks (NNs) in a signed graph via a Lyapunov-based approach, extending the existing work to a more practical but complicated case. A general matrix M in signed graphs is introduced to construct the novel Lyapunov functions, and sufficient conditions are obtained. We find that the rooted cycles and the structurally balanced graphs are essential to stabilize and achieve bipartite synchronization. More importantly, we discover that the nonrooted cycles are crucial in reaching interval bipartite synchronization, not previously mentioned. Several examples are presented to illustrate interval bipartite synchronization of multiple NNs with signed graphs.

The cooperation of linear multiagent systems (MASs) in the signed graphs has received wide attention. However, time delays and nonlinearity are ignored. This article deals with the cooperation behavior, especially interval bipartite synchronization (IBS) of delayed nonlinear neural networks (NNs) with signed graphs. A generalized matrix is proposed for the construction of the Lyapunov functionals, establishing sufficient conditions in a linear matrix inequality related to the coupling strength, the delays, and the network structure. It suggests that the negative rooted cycles and the negative nonrooted cycles take an important part in stabilizing delayed nonlinear NNs and leading to their diversity, and time delays, especially communication delays, significantly impact cooperation performance. Numerical examples are employed to validate our derived results.

Topology identification of multilayer networks is of great significance in the fields of information, engineering, and society. Current research on topology identification of multilayer networks requires the knowledge of the specific number of layers and the corresponding nodes in each layer in advance. However, the information is often unknown in reality. Herein, for a multilayer network with unknown layers, where node dynamic functions are in the quadratic form, we can obtain the specific layers and the topology structure based on compressive sensing. Notably, the number of layers can be obtained in two ways: 1) by the number of diverse node dynamic functions or 2) inner coupling matrices. The effectiveness and robustness of our method are verified through numerical simulations.

As the bounds of convergence time (CT) for fixed-time (FxT) stability have connections with some parameters of the system, the existing methods of FxT stability are still not really FxT. Therefore, this article studies FxT synchronization of complex networks (CNs) by proposing a new FxT stability theorem. First, the new FxT stability theorem is proposed, the CT of the system is any given time in advance, which is not concerning the starting value and parameters of the system. Second, a set of new controllers, including the continuous controller (CCr) and discontinuous controller (DCCr) are, respectively, designed to obtain the new FxT synchronization criteria for the CNs. Moreover, the control methods effectively solve the CT boundary problem associated with the system parameters in the previous FxT stability methods. Finally, numerical simulations and analog circuits are used to verify the effectiveness of the discussed method.

Fixed-time synchronization of complex networks is investigated in this article. First, a completely novel lemma is introduced to prove the fixed-time stability of the equilibrium of a general ordinary differential system, which is less conservative and has a simpler form than those in the existing literature. Then, sufficient conditions are presented to realize synchronization of a complex network (with a target system) within a settling time via three different kinds of simple controllers. In general, controllers designed to achieve fixed-time stability consist of three terms and are discontinuous. However, in our mechanisms, the controllers only contain two terms or even one term and are continuous. Thus, our controllers are simpler and of more practical applicability. Finally, three examples are provided to illustrate the correctness and effectiveness of our results.

Traditional classification learning algorithms have several limitations: 1) they are time consuming for the large-scale training multivariate time-series (MTS) data, and unsuitable for the dynamically added training data; 2) as the number of the training MTS data becomes larger, they could not achieve the desired classification accuracy; 3) most of them do not consider how to make use of the unlabeled samples to enhance the classifier performance; and 4) due to the high dimension of MTS and complex relationship among variables, existing online learning algorithms are not effective to update shapelet-based association rules. Up to now, few work touched online classification learning for dynamically added unlabeled examples. To efficiently address these issues, we propose an online rule-based classifier learning framework on dynamically added unlabeled MTS data (ORCL-U). This framework integrates a confidence-based labeling strategy (CLS) and an online rule-based classifier learning approach (ORBCL). Extensive experiments on ten datasets show the effectiveness and efficiency of our proposed approach.

Since the speed that the$p$th moment of an output signal norm tends to 0 can be used to measure the quality of the output signal, the stability of the$p$th moment of complex networks has been widely concerned. This article studies fixed-time synchronization in the$p$th moment for time-varying delay stochastic multilayer networks, where fixed-time synchronization of multilayer networks is realized by using nonlinear feedback or delay feedback, and the concrete expression of fixed settling time is evaluated. Compared with some existing fixed-time control methods, the optimal combination relationship between the exponential power of fixed-time controllers and the$p$th moment is given, and the designed controllers are continuous function. Finally, the effectiveness of the method is verified by numerical simulations.

The article pays attention to a two-layer multi-weighted network, and studies finite-time (FT) intra-/inter-layer quasi-synchronization of two-layer multi-weighted networks. Firstly, FT stability and quasi-stability theorems of dynamical system are discussed, and the results show that the maximum convergence time of FT quasi-stability is smaller than that of FT stability for the dynamic system. Secondly, novel sufficient criteria are gained for FT intra-/inter-layer quasi-synchronization of two-layer multi-weighted networks. Thirdly, the relationship among multiple weights number, the topological structure, inner coupling modes, coupling strengthes and across layers are established. In particular, the results show that the smaller multiple weights number do not necessarily lead to faster quasi-synchronization, and the multiple weights number may have a positive feedback effect on FT intra-/inter-layer quasi-synchronization. When the multiple weights number is greater than a certain value, FT intra-layer quasi-synchronization and FT inter-layer quasi-synchronization of networks can be realized simultaneously, and the conditions of taking the minimum convergence time are also given. Finally, numerical examples verify the effectiveness of the proposed method.

In reality, individuals will spread their opinions by word of mouth, meanwhile sharing their opinions on social platforms. To gain a clear insight into this kind of behavior, we propose a diffusion model of various opinions in a two-layer interconnected network using some statistical characteristics of network structures. Theoretical analysis reveals that the final fraction of any opinion in one layer will get identical to that of the same opinion in the other layer. In particular, when the seed fraction of an opinion in one layer is different from that in the other layer, the diffusion behavior and final prevalence of opinions not only rely on the seed fractions of opinions, but also depend on the attributes of individuals that are active on different layers, including their inter-layer linking patterns and linking number. Further analysis shows that mass media will promote the spread of the opinion that is in accordance with its own. Finally, we illustrate the effectiveness of analytical results by simulating the spread of two opinions under four inter-layer linking patterns on three types of two-layer interconnected networks. The findings throw new light on some interesting phenomena in society, and facilitate decision-makers to orientate the prevalence of a special product.

This paper discusses finite/fixed-time (FNT/FXT) synchronization of multi-layer networks based on energy consumption estimation. Firstly, a novel FXT stability theorem of dynamical system is presented. Secondly, sufficient criteria for FNT/FXT synchronization in multi-layer networks are given, and the estimation of the energy consumption are also gained. Especially, the range of the number of layers of network is gained based on the least convergence time and the estimation of the least energy consumption. The results show that shorter synchronization time and less energy consumption does not mean less network layers. Finally, the validity of the theoretical method is verified by numerical simulation.

This article is concerned with fixed-time synchronization of the nonlinearly coupled neural networks with discontinuous activation and mismatched parameters. First, a novel lemma is proposed to study fixed-time stability, which is less conservative than those in most existing results. Then, based on the new lemma, a discontinuous neural network with mismatched parameters will synchronize to the target state within a settling time via two kinds of unified and simple controllers. The settling time is theoretically estimated, which is independent of the initial values of the considered network. In particular, the estimated settling time is closer to the real synchronization time than those given in the existing literature. Finally, two numerical simulations are presented to illustrate the effectiveness and correctness of our results.

Different spreading processes in multiplex networks may interact with each other and display intertwined effects. In this paper, we propose a theoretical framework called infection-probability-dependent interlayer interaction propagation processes in multiplex networks with an arbitrary number of layers, to more precisely depict the intertwined effects which bring challenges to the existing state-dependent interlayer interaction models. Specifically, the spreading rate of each node is regulated by the proposed spreading rate function (SRF) which depends on both the intrinsic dynamics in its layer and the infection probabilities of its counterparts. We propose an algorithm to obtain the spreading threshold of each layer of the proposed theoretical framework. We analyze the three-layer tuberculosis-awareness-flu model with the SRF of each node being the expectation of infection-probability-dependent spreading rate. This paper gives a thorough and detailed numerical investigation of the impact and interaction of system settings and the spreading threshold of each layer. We find that for tuberculosis spreading which is in competing relation with awareness and cooperation relation with flu, the epidemic threshold is a constant when other layers' intrinsic spreading rates are small. The cooperation layer has dramatic influence on the constant while the competing layer has no effect on it.

Complex networks with time-varying nodal parameters are of considerable interest and significance in many areas of science and engineering. Reconstructing networks with unknown but continuously bounded time-varying nodal parameters from limited measured information is desirable and of significant interest for using and controlling these networks. Based on the Lasso method and the Taylor expansion approximation, we develop an efficient and feasible, completely data-driven approach to predicting the structures of networks with unknown but continuously bounded time-varying nodal parameters in the presence or absence of noise. In particular, the reconstruction framework is implemented on several different kinds of artificial, two-layer and real complex networks composed of various parameter-varying nodal dynamics. Through numerical simulations, we demonstrate that, networks structures can be fully reconstructed with limited available information and presence or absence of noise, though systemic parameters are continuously time-varying. In addition, our method is also applicable to structure identification of multilayer networks as well as networks with constant nodal parameters. We expect our method to be useful in addressing issues of significantly current concern in the information era, natural networks, and large-scale multilayer networks.

The coexistence of multiple types of interactions within social, technological, and biological networks has motivated the study of the multilayer nature of real-world networks. Meanwhile, identifying network structures from dynamical observations is an essential issue pervading over the current research on complex networks. This paper addresses the problem of structure identification for multilayer networks, which is an important topic but involves a challenging inverse problem. To clearly reveal the formalism, the simplest two-layer network model is considered and a new approach to identifying the structure of one layer is proposed. Specifically, if the interested layer is sparsely connected and the node behaviors of the other layer are observable at a few time points, then a theoretical framework is established based on compressive sensing and regularization. Some numerical examples illustrate the effectiveness of the identification scheme, its requirement of a relatively small number of observations, as well as its robustness against small noise. It is noteworthy that the framework can be straightforwardly extended to multilayer networks, thus applicable to a variety of real-world complex systems.

We present a variational method for subdivision surface reconstruction from a noisy dense mesh. A new set of subdivision rules with continuous sharpness control is introduced into Loop subdivision for better modeling subdivision surface features such as semi-sharp creases, creases, and corners. The key idea is to assign a sharpness value to each edge of the control mesh to continuously control the surface features. Based on the new subdivision rules, a variational model with L1 norm is formulated to find the control mesh and the corresponding sharpness values of the subdivision surface that best fits the input mesh. An iterative solver based on the augmented Lagrangian method and particle swarm optimization is used to solve the resulting non-linear, non-differentiable optimization problem. Our experimental results show that our method can handle meshes well with sharp/semi-sharp features and noise.

This paper presents a variational algorithm for feature-preserved mesh denoising. At the heart of the algorithm is a novel variational model composed of three components: fidelity, regularization and fairness, which are specifically designed to have their intuitive roles. In particular, the fidelity is formulated as an L1 data term, which makes the regularization process be less dependent on the exact value of outliers and noise. The regularization is formulated as the total absolute edge-lengthed supplementary angle of the dihedral angle, making the model capable of reconstructing meshes with sharp features. In addition, an augmented Lagrange method is provided to efficiently solve the proposed variational model. Compared to the prior art, the new algorithm has crucial advantages in handling large scale noise, noise along random directions, and different kinds of noise, including random impulsive noise, even in the presence of sharp features. Both visual and quantitative evaluation demonstrates the superiority of the new algorithm.

The placement of knot vector and the determination of control points are two fundamental issues in B-spline surface reconstruction. This paper presents a variational approach to construct B-spline surfaces from a set of data points. The approach finds the optimal placement of knots and control points simultaneously while most previous methods determine the knots heuristically or in a separate step. Moreover, different from most previous methods using least squares metric, our approach adapts L_1-norm with total variation (TV) as regularization in the fitting procedure, which enables the approach to handle both Gaussian noise and outliers in the same manner and is able to automatically optimize the placement of knot vector to faithfully reconstruct the sharp features. A numerical solver based on the augmented Lagrangian method is also proposed in the paper to efficiently solve the TV-L_1 optimization. Experimental results demonstrate the effectiveness and efficiency of the proposed variational B-spline surface reconstruction.

It is well known that theoretical analysis based on the mathematical predictive models of middle-sized genetic regulatory networks are crucial to understand the life at system level for systems biology. Since there doesn't exist an efficient mathematical model for the middle-sized or large-sized biological systems, it is also a challenge to theoretically investigate these networks. To overcome the limitations of the traditional continuous differential equations and discrete Boolean models, this paper aims to introduce a hybrid modelling approach for the general middle-sized genetic regulatory networks. This hybrid modelling method is able to quantitatively investigate the middle-sized or large-sized biological networks by merging the advantages of both continuous differential equation models and Boolean models.

Complex networks have found widespread real-world applications. One of the key problems in research of complex networks is topology identification, which is concerned with deciding the interaction patterns from observed dynamical time series. This presents a very challenging problem, especially in the absence of the knowledge of nodal dynamics and in the presence of system noise. In this paper a simple and yet efficient approach is proposed for topology identification of complex networks in such challenging scenarios. The main idea behind the proposed approach is to use piecewise partial Granger causality, which measures the directed connections of nonlinear time series influenced by hidden variables. The effectiveness of the proposed approach in relation to network parameters is demonstrated by a commonly-used testing network.

This paper is concerned with applying the impulsive control scheme to generalized synchronization (GS) of complex networks of nonlinear dynamical systems. The auxiliary-system approach is utilized to show that complex dynamical networks consisting of nonidentical systems can reach generalized synchronization under impulsive control. Then the relations between the GS error and the topological parameter are examined for scale-free networks, which reveals that an increase in the topological parameter causes a decrease in the GS error. Also, the relations between the GS speed and the adding of random edges are investigated for small-world networks, which shows that increasing the probability of adding random edges can accelerate GS. Moreover, the effect of node dynamics on the GS speed is studied for both small-world and scale-free networks.

This paper is concerned with carrying out synchronization between two completely different complex dynamical networks, which is called generalized outer synchronization. Our objective is to develop a control scheme so as to achieve this generalized outer synchronization. The derived result is built upon Barbalat's Lemma. It is shown that when two complex dynamical networks to be synchronized have different topologies and diverse node dynamics, the designed controller is a nonlinear one. In particular, when two complex dynamical networks have the same topological structures or identical dynamics, the designed controller can be simplified, sometimes to a linear controller. Computer simulations are presented to show that generalized outer synchronization can be quickly realized under the designed controller.

Dynamical analysis and more detailed numerical analysis for understanding the compound structure of Chen's attractor are proposed in this paper. Routes to chaos and bifurcation of the Chen's system under a constant controller are demonstrated with numerical experiments. The systematic analysis method can also be applied to investigate chaotic attractor's compound structure shared by some typical attractors such as the Lorenz attractor, modified Lorenz attractor, Lu attractor, and PWL Chen's attractor, etc.

A new piecewise-linear chaotic system family, which bridges the gap between the PWL Lorenz and newly proposed PWL Chen's system, is introduced in this paper. The new system represents the continuous transition from the PWL Lorenz system to the PWL Chen system and is chaotic over the entire spectrum of one of the key system parameters. Some basic dynamical behaviors and its classification are also briefly described based on theoretical analysis and numerical experiments.