Synchronization in Finite-Time of Delayed Fractional-Order Fully Complex-Valued Dynamical Networks via Non-Separation Method

In recent years, as a characteristic collective behavior of complex networks, the synchronization problem of complex networks has received more and more attention from different fields because of its outstanding potential applications and outstanding achievements in nature, social, and technological fields [1,2,3,4]. The complexity of the system can be expressed by entropy. The more complex the system, the higher the entropy is. In the process of controlling the system to achieve synchronization, the entropy value will also decrease. In practice, it is usually hoped to realize faster synchronization or even finite-time synchronization for complex networks [5]. Many academics and researchers have been interested in the FNTS of complex networks, and there have been numerous good achievements in this area [6,7,8,9,10,11,12,13,14].

Compared with integer-order differential equations, fractional calculus is more suitable for describing the memory and genetic characteristics of various materials and dynamic processes [15,16,17,18]. The advantage of the Caputo fractional derivative is that the initial conditions of fractional differential equations with Caputo derivatives are similar to those of integral differential equations. Therefore, introducing fractional order into complex networks has theoretical and practical significance. The research on FNTS problems of fractional-order complex networks (FCNs) has become the focus and research hotspot of engineers, technicians, and scientists. For example, in [19], the FNTS for a class of FCNs was investigated by using the hybrid feedback control technique. In [20], the FNTS of FCNs with a strongly connected topology was studied. The FNTS problem between different dimensional fractional-order complex dynamical networks was investigated in [21]. In [22], the FNTS of FCNs was studied via intermittent control. In the above literature about the FNTS of FCNs, the models were all assumed to be real-valued (RV) models.

However, compared with RV dynamic networks, because dynamic systems in complex spaces can develop in different directions, CV dynamic networks allow the transmitted signals to obtain more comprehensive versatility and anti-attack performance [23]. Therefore, it is fascinating to introduce complex values into complex dynamical networks because of their practical applications in engineering technology fields [24]. By decomposing the CV systems into two RV subsystems, the synchronization or stability problems for fractional-order CV complex networks were extensively investigated in [25,26,27,28,29,30,31,32], respectively. Although the separation technique is practical, the dimension of two RV systems is double that of the originals, significantly increasing the complexity and triviality of theoretical analysis and mathematical derivation. As a result, the accuracy and simplicity of the theoretical results are low by the separation technique. Furthermore, due to the high difficulty of the model, it is not easy to transform the CV system into two RV systems in a realistic operational procedure. Hence, in [33], unlike the traditional separation method, the FNTS problem for fractional-order CV dynamical networks was investigated by introducing signum functions for complex numbers and complex-valued vectors. Similarly, in [34], Xu et al. discussed FNTS for fractional-order complex-valued coupled systems based on the complex variable function instead of the separation approach. However, the coupling strengths and inner and outer coupling matrices of the mathematical model considered in [33,34] were RV, although the state variables and system function were CV. Integrating fully CV coupling strengths and couplings into synchronization studies for complex networks is more realistic and requires more extensive analysis. To overcome this bottleneck, Zheng et al. [35] designed the power law control strategies for CV networks by introducing the signum function in the complex domain and studied the FNTS for a class of fully CV neural networks In [36], based on the introduced CV vector signum function, the complex value control strategy was directly designed for the fully CV integer-order complex networks. The FNTS and fixed-time synchronization problems of the integer-order fully CV network were studied.

On the other hand, it is well known that delay is a common phenomenon in the real world [37,38,39]. Time delay widely exists in complex network systems such as medicine circulation systems, population dynamics models, disease infection models, neural network models, communication networks, power networks, economic systems, etc. In complex networks, there exist internal delay and coupling delay, which will show finite speed and propagation, as well as the impact of traffic congestion on node behavior, respectively [40]. The existence of time delay will increase the difficulty of analysis. When the controlled object has internal delay in the control system, the control difficulty of the system will increase. Besides, time delay systems have a richer dynamic behavior, which is more widely used for secure communication. Therefore, it is meaningful to study dynamic networks with internal delay and coupling delay. However, the FNTS results of the fully CV dynamical networks mentioned above do not demonstrate internal delay and coupling delay. Introducing internal delay and coupling delay into complex dynamics networks needs further analysis. Regretfully, as far as we know, there are few or no results on FNTS of FFCDNs with internal delay, as well as linearly non-delayed and delayed couplings. Our current research is motivated by this condition. To better illustrate the contribution of our study, we compared our paper with other similar papers published in the last three years. The differences are shown in Table 1, where fixed-time synchronization and adaptive synchronization are abbreviated as FXTS and ADS, respectively. The signum ✔ means the object is included in the paper. The signum ✕ means the object is not included in the paper.

Through comparison, it can be found that it is difficult and challenging to comprehensively study a class of fully fractional-order CV dynamic networks with internal delay, no delay, and time delay coupling. The main contribution of this research may be described as follows:

(1) The complex dynamical networks studied in this paper are novel. The state variables, system function, coupling strengths, inner coupling matrices, and outer coupling matrices in the considered dynamical networks are all CV. In addition, the mathematical model is the fractional-order case, which is more consistent with practical applications.

(2) Lyapunov functions are constructed based on the quadratic norm and the new norm composed of the absolute value norm by introducing the signum function. The fractional-order complex-valued networks do not need to be separated into real and imaginary parts, which reduces conservatism, complexity, and trivialness.

(3) In order to overcome the limitation of a single controller, two kinds of different controllers (based on the quadratic norm and 1-norm for a complex vector, respectively) are deployed in realizing FNTS, and a series of straightforward and flexible synchronization criteria is acquired.

(4) Compared with the separation method, which needs to apply controllers to the multiple separate systems, the control strategy in this paper is simpler and more efficient, can effectively reduce the cost, and has high practical application value.

The following is the structure of the paper. In Section 2, the preliminaries and the model description are given; in Section 3, two different controllers are proposed to ensure FNTS for the addressed delayed FFCDNs; numerical simulations are presented in Section 4 to illustrate the validity and practicality of the proposed theoretical solutions; the conclusion is given in Section 5.

Notations: Throughout this study, $\mathbb{R}$ and $\mathbb{C}$ represent the real field and complex field, respectively. ${\mathbb{R}}^{+}$ denotes the positive real field. ${\mathbb{C}}^n$ symbolizes the n-dimensional complex space. For any $v=p+\mathbf{i}q\in \mathbb{C}$, $\overline{v}=p-\mathbf{i}q$ denotes the conjugate of v, ${\left|v\right|}_1=\left|p\right|+\left|q\right|$, ${\left|v\right|}_2=\sqrt{\overline{v}v}$, where $\mathbf{i}$ meets $\mathbf{i}=\sqrt{-1}$, and $p,q\in \mathbb{R}$ are the real and imaginary parts of v, respectively, that is $\mathrm{Re}\left(v\right)=p$ and $\mathrm{Im}\left(v\right)=q$. $\left[v\right]=\mathrm{sign}\left(\mathrm{Re}\right(v\left)\right)+\mathbf{i}\mathrm{sign}\left(\mathrm{Im}\right(v\left)\right)$ is said to be the signum function of v. For any $v=(v_1,v_2,\cdots ,v_n)\in {\mathbb{C}}^n$, $v=\mathrm{Re}\left(v\right)+\mathbf{i}\mathrm{Im}\left(v\right)$, $v^{\mathrm{H}}$ denotes its conjugate transposition, ${\parallel v\parallel }_1={\parallel \mathrm{Re}\left(v\right)\parallel }_1+\mathbf{i}{\parallel \mathrm{Im}\left(v\right)\parallel }_1$, ${\parallel v\parallel }_2=\sqrt{v^{\mathrm{H}}v}$, and $\left[v\right]={\left(\mathrm{sign}\left(\mathrm{Re}\left(v_1\right)\right)+\mathbf{i}\mathrm{sign}\left(\mathrm{Im}\left(v_1\right)\right),\cdots ,\mathrm{sign}\left(\mathrm{Re}\left(v_n\right)\right)+\mathbf{i}\mathrm{sign}(\mathrm{Im}\left(v_n\right)\right)}^{\mathrm{T}}$. ${\mathbb{C}}^{m\times n}$ denotes the set of all $m\times n$-dimensional complex matrices. $I_N$ denotes the n-dimensional column vector with each element equal to 1, and $E_N$ represents the n-dimensional diagonal identity matrix. The notation ${\mathcal{C}}^n([t_0,+\infty ),\mathbb{C})$ denotes the family of all continuous n-differential functions from $[t_0,+\infty )$ into $\mathbb{C}$.

In this work, we selected the $\alpha$-order Caputo derivative to depict the dynamic behavior of delayed FFCDNs.

The delayed FFCDNs can be described as follows: where $0<\alpha <1$, $\alpha \in (0,1)$, $x_i\left(t\right)={(x_{i1}\left(t\right),x_{i2}\left(t\right),\cdots ,x_{in}\left(t\right))}^{\mathrm{T}}\in {\mathbb{C}}^n$ represents the state vector of the ith node at time t, $f:{\mathbb{C}}^n\times {\mathbb{C}}^n\to {\mathbb{C}}^n$ is a nonlinear vector function, $c_l\in \mathbb{C}$$(l=1,2)$ is the coupling strength, $G_l=\mathrm{diag}({\delta }_1^{\left(l\right)},{\delta }_2^{\left(l\right)},\cdots ,{\delta }_n^{\left(l\right)})\in {\mathbb{C}}^{n\times n}$$(l=1,2)$ is the inner matrix linking the coupled variables, ${\tau }_1$ is the internal delay occurring inside the dynamical node, ${\tau }_2$ represents the coupling delay, and $A={\left(a_{ij}\right)}_{N\times N}$ and $B={\left(b_{ij}\right)}_{N\times N}$ represent the topological structure of FFCDNs without and with time delays, respectively. The initial conditions associated with Equation (1) are given as $x_i\left(s\right)={\phi }_i\left(s\right)\in C([-\tau ,0],{\mathbb{C}}^n),i=1,2,3\dots N,$ where $C([-\tau ,0],{\mathbb{C}}^n)$ represents the set of all n-dimensional continuous differentiable functions defined on the interval $[-\tau ,0]$ and $\tau$ = max{${\tau }_1,{\tau }_2$}.

The topological structure of delayed FFCDNs should match the following criteria: if node i and node j$(i\ne j)$ have a link, then $a_{ij},b_{ij}\ne 0\in \mathbb{C}$; else $a_{ij}=b_{ij}=0$$(i\ne j)$, and the diagonal elements are

It follows from (1) and the definitions of A and B that

Define the error states as $e_i\left(t\right)=x_i\left(t\right)-s\left(t\right)$ with $i=1,2,\cdots ,N$. It is clear that where $i=1,2,\cdots ,N$, $g(e_i\left(t\right),e_i(t-{\tau }_1))=f(x_i\left(t\right),x_i(t-{\tau }_1))-f(s\left(t\right),s(t-{\tau }_1))$, and $u_i\left(t\right)$ is a controller that will be created later.

The following assumptions, definitions, and lemmas are required to reach our major conclusions.

The FNTS problems for a class of FFCDNs with internal delay and linearly non-delayed and delayed couplings were explored by designing two different controllers. The following are the key results.