Constraining control method for time-delay coupling network system

Abstract

The invention provides a time-delay coupling network system containment control method, and relates to the technical field of control and information. For a network system described by a time-delay differential equation, a time-delay containment control law is designed to realize a synchronization target or a de-synchronization target, and a cooperative influence mechanism of node dynamics, global coupling strength, a network topology structure and a time-delay and containment control strategy on system synchronization stability is disclosed. According to the method, the influence of time delay in an actual network system is fully considered, the time delay characteristic and the network dynamics rule are combined, the network synchronization domain range is determined, the action mechanism of time delay on the complexity and stability of the system is revealed, key nodes in the network are efficiently recognized, and the global dynamic behavior is effectively restrained by using limited control resources. The method not only can overcome the negative influence of time delay on the system stability, but also can remarkably improve the control efficiency and robustness of the complex network.

Description

Technical Field

[0001] The present invention relates to the field of control and information technology, and particularly to a pinning control method for a time-delay coupled network system. Background Art

[0002] Time delays widely exist in actual complex network systems, such as communication networks, transportation networks, biological networks, and power systems, etc. The time delays in these systems are usually caused by signal transmission, processing delays, or reaction lags, and have a significant impact on the dynamic behavior and system stability of the network. The existence of time delays may lead to system oscillations, chaos, or even instability, making control and optimization more complex. Therefore, studying how to achieve effective control of the network under time-delay conditions not only has important theoretical significance but also has guiding value for practical engineering applications.

[0003] Pinning control is a method that affects the dynamic behavior of the entire network through a small number of key nodes or control signals, and has received attention in many fields. However, in a network with time-delay coupling, the design and implementation of pinning control face new challenges, such as how to overcome the weakening of the control effect by time delays, and how to determine the selection and optimization strategy of pinning nodes. The solution of these problems helps to achieve stable control and performance improvement of complex network systems under limited resources, and provides technical support for network security, intelligent scheduling, and dynamic optimization. Summary of the Invention

[0004] Aiming at the deficiencies of the prior art, the present invention provides a pinning control method for a time-delay coupled network system; for a complex network system described by time-delay differential equations, the present invention designs a time-delay pinning control law to achieve a synchronization target or a desynchronization target, and reveals the cooperative influence mechanism of node self-dynamics, global coupling strength, network topology structure, time delay, and pinning control strategy on system synchronization stability. Compared with the prior art, the present invention fully considers the influence of time delays in actual network systems, and the provided pinning control method can effectively regulate a time-delay coupled network system, improve the robustness of the system, reduce the control cost, and has a wider application range in practical applications.

[0005] A pinning control method for a time-delay coupled network system includes the following steps:

[0006] Step 1: Construct a time-delay coupled network system with N nodes:

[0007] Construct a time-delay coupled network system with N nodes, and the dynamic equation of the i-th node is expressed as follows:

[0008]

[0009] Wherein, The state variable representing the \(i\)-th node, \(x\) in (t) represents the \(n\)-th component of the state of the \(i\)-th node, where \(n\) represents the dimension of a single node, represents the first-order derivative of the state of the \(i\)-th node with respect to time \(t\), represents the set of \(n\)-dimensional real vectors; \(f(·)\) represents the self-dynamics function of a single node; the constant \(\sigma>0\) is the global coupling strength of the network; \(h(·)\) is the internal coupling function, representing the coupling relationship between the state variables of each node; is the Laplacian matrix characterizing the network topology, defined as: when there is an edge connecting node \(i\) and node \(j\), \(l\) ij \(=l\) ji \(=-1\), otherwise \(l\) ij \(=l\) ji \(=0\), and satisfies the dissipative coupling condition \(\sum\) j \(l\) ij \(=0\), \(i,j = 1,2,\cdots,N\), where \(N\) represents the total number of nodes in the network system; \(\tau\) is the time delay, and the term \(h(x\) j (t - \tau))\) indicates that the time delay occurs in the internal coupling, so it is called a time-delay coupled network system; the topology of the time-delay coupled network system is an undirected connected graph;

[0010] Step 2: Design a pinning control law with time delay to achieve the synchronization goal or the desynchronization goal:

[0011] The pinning control law with time delay is designed as follows:

[0012] \(u\) i (t)=-\sigma d i [h(x i (t - \tau)) - h(s(t - \tau))], \(i = 1,2,\cdots,N\) (2)

[0013] where the constant \(\sigma>0\) is the global coupling strength of the network, \(\tau\) is the time delay; \(d\) i represents the control gain. If control is applied to node \(i\), then the control gain \(d\) i \(>0\), otherwise, \(d\) i \(=0\); \(s(t)\) is the synchronization state;

[0014] Controlling all nodes of the entire network to the synchronization state \(s(t)\), the time-delay coupled network system under pinning control is expressed as follows:

[0015]

[0016] where, represents the state variable of the \(i\)-th node, \(n\) represents the dimension of a single node, represents the first-order derivative of the state of the \(i\)-th node with respect to time \(t\), denotes the set of n-dimensional real vectors; f(·) denotes the self-dynamics function of a single node, which is continuously differentiable; the constant σ > 0 is the global coupling strength of the network; h(·) is the inner-coupling function, representing the coupling relationship between the state variables of each node; is the Laplacian matrix characterizing the network topology, defined as: when there is an edge connecting node i and node j, l ij = l ji = -1, otherwise l ij = l ji = 0, and satisfies the dissipative coupling condition ∑ j l ij = 0, i, j = 1, 2, …, N; τ is the time delay, h(x j (t - τ)) represents that the time delay occurs in the inner-coupling; d i represents the control gain. If control is applied to node i, then ξ i = 1 and the control gain d i > 0, otherwise, ξ i = 0 and d i = 0; the symbol ξ i represents whether control is applied to the i-th node. Thus, the number of controlled nodes is ∑ i ξ i , and ∑ i ξ i < N, s(t) is the synchronous state;

[0017] When the network system (1) is stable in the synchronous state by itself, through designing the pinning control law described by Equation (2), the synchronization suppression goal, i.e., desynchronization, is achieved, which is applicable to the actual scenarios where synchronization is harmful;

[0018] Step 3: According to the master stability function method, analyze the stability discrimination conditions of the time-delay coupled network system under pinning control;

[0019] Step 3.1: Define the node error vector e i (t) = x i (t) - s(t), i = 1, 2, …, N, then Equation (3) becomes:

[0020]

[0021] where, represents the error vector of the i-th node, N represents the total number of nodes in the network system, the superscript T represents the transpose, denotes the set of nN-dimensional real vectors; J f (s(t)) is the Jacobian matrix of the node self-dynamics function f(·) at the synchronous state s(t), J h(s(t - τ)) is the Jacobian matrix of the internal coupling function h(·) at s(t - τ); the coupling control matrix C = L + D, where L is the Laplacian matrix and D is a diagonal matrix defined as D = diag{ξ 1 d 1 , ξ 2 d 2 , …, ξ N d N}, where d i represents the control gain and ξ i indicates whether control is applied to the i-th node; the constant σ is the global coupling strength and τ is the time delay; I N represents the N-dimensional identity matrix;

[0022] Step 3.2: Since the topological structure of the time-delay coupled network system is an undirected connected graph, the coupling control matrix C is diagonalized to obtain the matrix G = P -1 CP, where P is the transformation matrix composed of the eigenvectors corresponding to the eigenvalues of the matrix C. Introduce the variables and Then equation (4) becomes:

[0023]

[0024] where I n is the n-dimensional identity matrix;

[0025] Step 3.3: Let the parameter θ = σμ i , where σ is the global coupling strength and μ i is the eigenvalue of the matrix C; define the auxiliary variable y(t) to obtain:

[0026]

[0027] Equation (6) is the main stability function of the time-delay coupled network system under pinning control;

[0028] The condition for the local stability of the time-delay coupled network system (3) under pinning control in the synchronous state is that the maximum Lyapunov exponent of the time-delay-containing main stability function (6) is less than 0;

[0029] Step 4: Determine the synchronization domain S of the time-delay coupled network system according to the time-delay-containing main stability function R, on this basis, further determine the control gain, the number of controlled nodes and their position distribution of the pinning control law, so that the product of the global coupling strength σ and all the eigenvalues of the coupling control matrix C all fall into the synchronization domain, thereby realizing the complete synchronization of the network; when synchronization is harmful and needs to be suppressed, on the basis of determining the synchronization domain, design a time-delay pinning control law, so that the product of the global coupling strength σ and the eigenvalues of the coupling control matrix C partially fall outside the synchronization domain, thereby realizing the desynchronization of the network.

[0030] The beneficial effects produced by adopting the above technical solutions are as follows:

[0031] The present invention provides a pinning control method for a time-delay coupled network system, which can effectively solve the problem of controlling the dynamic behavior of a time-delay coupled network system, and has important theoretical significance and practical application value. By combining the time-delay characteristics with the laws of network dynamics, this method can efficiently identify the key nodes in the network and use limited control resources to effectively pin the global dynamic behavior. This method can not only overcome the negative impact of time delay on the system stability, but also significantly improve the control efficiency and robustness of complex networks. In addition, when it is necessary to suppress synchronization in the actual scenario, this method can also make the originally synchronously stable system achieve the desynchronization goal. The present invention has strong applicability and is applicable to various actual scenarios such as communication networks, transportation networks, and biological networks, providing a theoretical basis and technical support for solving the problem of stable control of time-delay coupled networks, and having important application value for network security, intelligent scheduling, and system optimization. Brief Description of the Drawings

[0032] Figure 1 It is a flowchart of the pinning control method for a time-delay coupled network system provided for the implementation of the present invention;

[0033] Figure 2 It is a synchronization domain diagram in the implementation of the present invention;

[0034] Figure 3 It is a network structure diagram in the implementation of the present invention;

[0035] Figure 4 It is an evolution diagram of the node state over time when the system parameter b > 0 in the implementation of the present invention;

[0036] Among them, (a) - the node state diagram without applying pinning control, (b) - the node state diagram with applying pinning control;

[0037] Figure 5 It is an evolution diagram of the node state over time when the system parameter b < 0 in the implementation of the present invention;

[0038] Among them, (a) - the node state diagram without applying pinning control when the global coupling strength σ = 2, (b) - the node state diagram without applying pinning control when the global coupling strength σ = 5, (c) - the node state diagram with applying pinning control when the global coupling strength σ = 2. Detailed implementation manners

[0039] The following combines the accompanying drawings and embodiments to further describe in detail the detailed implementation manners of the present invention. The following embodiments are used to illustrate the present invention, but are not used to limit the scope of the present invention.

[0040] A method for pinning control of a time-delay coupled network system, as Figure 1 shown, includes the following steps:

[0041] Step 1: Construct a time-delay coupled network system with N nodes:

[0042] Construct a time-delay coupled network system with N nodes. The dynamic equation of the i-th node is expressed as follows:

[0043]

[0044] Among them, represents the state variable of the i-th node, and x in (t) represents the n-th component of the state of the i-th node, where n represents the dimension of a single node, represents the first-order derivative of the state of the i-th node with respect to time t, represents the set of n-dimensional real vectors; f(·) represents the self-dynamic function of a single node; the constant σ>0 is the global coupling strength of the network; h(·) is the inner-coupling function, representing the coupling relationship between the state variables of each node; is the Laplacian matrix characterizing the network topology, defined as: when there is an edge connection between node i and node j, l ij =l ji =-1, otherwise l ij =l ji =0, and satisfies the dissipative coupling condition ∑ j l ij =0, i, j = 1, 2,..., N, where N represents the total number of nodes in the network system; τ is the time delay, and the term h(x j (t - τ)) represents that the time delay occurs in the inner coupling, so it is called a time-delay coupled network system; the topology of the time-delay coupled network system is an undirected connected graph;

[0045] In this embodiment, it is considered that the node dynamics is a one-dimensional linear system b is a system parameter, i.e., n = 1 in Equation (1); when b > 0, the equilibrium point x = 0 of the system is unstable; when b < 0, the equilibrium point x = 0 of the system is stable; the system parameter b here can vary with time, which can better reflect the characteristics of the actual system;

[0046] Step 2: Design a pinning control law with time delay to achieve the synchronization goal or the desynchronization goal:

[0047] The pinning control law with time delay is designed as follows:

[0048] u i (t) = -σd i [h(x i (t - τ)) - h(s(t - τ))], i = 1, 2, …, N (2)

[0049] where the constant σ > 0 is the global coupling strength of the network, τ is the time delay; d i represents the control gain. If control is applied to node i, then the control gain d i > 0, otherwise, d i = 0; s(t) is the synchronization state;

[0050] All nodes of the entire network are controlled to the synchronization state s(t), and the time-delay coupled network system under pinning control is expressed as follows:

[0051]

[0052] where, represents the state variable of the i-th node, n represents the dimension of a single node, represents the first derivative of the state of the i-th node with respect to time t, represents the set of n-dimensional real vectors; f(·) represents the self-dynamics function of a single node, which is continuously differentiable; the constant σ > 0 is the global coupling strength of the network; h(·) is the inner coupling function, representing the coupling relationship between the state variables of each node; is the Laplacian matrix characterizing the network topology, defined as: when there is an edge connecting node i and node j, l ij = l ji = -1, otherwise l ij = l ji = 0, and it satisfies the dissipative coupling condition ∑ j l ij = 0, i, j = 1, 2, …, N; τ is the time delay, h(x j (t - τ)) indicates that the time delay occurs in the inner coupling; d i represents the control gain. If control is applied to node i, then ξ i = 1 and the control gain di > 0, otherwise, ξ i = 0 and d i = 0; The symbol ξ i indicates whether there is control applied to the i-th node. Thus, the number of controlled nodes is ∑ i ξ i , and ∑ i ξ i < N, that is, only a small number of nodes in the network are controlled; s(t) is the synchronous state;

[0053] When the network system (1) is stable in the synchronous state by itself, through designing the pinning control law described by Equation (2), the synchronization suppression goal, that is, desynchronization, is achieved, which is applicable to the actual scenarios where synchronization is harmful;

[0054] Step 3: According to the master stability function method, analyze the stability discrimination conditions of the time-delay coupled network system under pinning control;

[0055] Step 3.1: Define the node error vector e i (t) = x i (t) - s(t), i = 1, 2,..., N, then Equation (3) becomes:

[0056]

[0057] where, represents the error vector of the i-th node, N represents the total number of nodes in the network system, the superscript T represents the transpose, represents the set of nN-dimensional real vectors; J f (s(t)) is the Jacobian matrix of the node's own dynamics function f(·) at the synchronous state s(t), and J h (s(t - τ)) is the Jacobian matrix of the inner coupling function h(·) at s(t - τ); The coupling control matrix C = L + D, L is the Laplacian matrix, and D is a diagonal matrix defined as D = diag{ξ 1 d 1 , ξ 2 d 2 , …, ξ N d N},d i represents the control gain, and ξ i indicates whether there is control applied to the i-th node; The constant σ is the global coupling strength, and τ is the time delay; I N represents the N-dimensional identity matrix;

[0058] Step 3.2: Since the topological structure of the time-delay coupled network system is an undirected connected graph, the coupling control matrix C is diagonalized to obtain the matrix G = P -1CP, where P is the transformation matrix composed of the eigenvectors corresponding to the eigenvalues of matrix C. Introduce variables and Then equation (4) becomes:

[0059]

[0060] where I n is the n-dimensional identity matrix;

[0061] Step 3.3: Let the parameter θ = σμ i , where σ is the global coupling strength and μ i is the eigenvalue of matrix C; Define the auxiliary variable y(t), and we get:

[0062]

[0063] Equation (6) is the master stability function of the time-delay coupled network system under pinning control; Different from the general master stability function, equation (6) contains time delay, indicating that time delay not only increases the complexity of the network system but also affects the synchronization stability of the system;

[0064] Through theoretical analysis, it can be known that the condition for the local stability of the time-delay coupled network system (3) under pinning control in the synchronous state is that the maximum Lyapunov exponent of the master stability function with time delay (6) is less than 0;

[0065] Step 4: Determine the synchronization domain S R of the time-delay coupled network system according to the master stability function with time delay. On this basis, further determine the control gain, the number of controlled nodes and their position distribution of the pinning control law, so that the product of the global coupling strength σ and all the eigenvalues of the coupling control matrix C all fall into the synchronization domain, thereby realizing the complete synchronization of the network; When synchronization is harmful and needs to be suppressed, on the basis of determining the synchronization domain, design the time-delay pinning control law so that the product of the global coupling strength σ and the eigenvalues of the coupling control matrix C partially fall outside the synchronization domain, thereby realizing the desynchronization of the network.

[0066] In this embodiment, the one-dimensional linear system The characteristic equation of the master stability function (6) is:

[0067] F(λ, b, θ) = λ - b + θe -λτ = 0 (7)

[0068] In equation (7), the real part of the eigenvalue λ determines the stability of the master stability function. To separate the real part and the imaginary part in the characteristic equation, let λ = iω, and we can get

[0069]

[0070] Thus, we obtain

[0071]

[0072] Then, the synchronization domain diagrams of the constrained control time-delay coupled network system in this embodiment under different time delays can be obtained and are summarized as follows;

[0073] When b ≤ 0, When At this time, When At this time, where S R represents the synchronization domain, ω(·) represents the characteristic frequency, τ is the time delay, represents the empty set;

[0074] From the synchronization domain diagram, that is, Figure 2 it can be seen that when τ = 0.1, the synchronization domain bifurcation point is b = 10; when τ = 0.2, the synchronization domain bifurcation point is b = 5; when τ = 1, the synchronization domain bifurcation point is b = 1. Figure 2 The inner part surrounded by the curve in is the synchronization domain, S R1 is the synchronization domain for τ = 0.1, S R2 is the synchronization domain for τ = 0.2, S R3 is the synchronization domain for τ = 1. It can be seen from Figure 2 that as the time delay τ increases, the synchronization domain gradually becomes smaller, which reduces the probability of non-zero characteristic modules falling into the synchronization domain and decreases the synchronization of the network system. This shows that the time delay has a negative impact on the system stability in this case.

[0075] Next, the effectiveness of the proposed pinning control method of the present invention is verified. In the embodiment, the network is a ring graph composed of 5 nodes, as Figure 3 shown, satisfying the condition of a connected graph. The Laplacian matrix of this network is

[0076]

[0077] First, consider the case where the system parameter b > 0. Figure 4 (a) is the evolution diagram of the node states over time when the system parameters are b = 2, the coupling strength σ = 2, the time delay τ = 0.1, and no pinning control is applied; it can be seen from the figure that the node states diverge and cannot synchronize. On this basis, the pinning control method is introduced, and pinning control is applied to nodes 1, 2, 3, and 4. The control gain is designed as d 1 = d 2 = d 3 = d 4 = 2, and the coupling control matrix can be obtained as

[0078]

[0079] Through calculation, the eigenvalues of the coupling control matrix C are obtained as 1.1392, 2.7459, 3.382, 5.1149, and 5.618. Further, the product range of the global coupling strength σ and all eigenvalues of the coupling control matrix C is [2.2784, 11.236]. According to Figure 2 , when the time delay τ = 0.1 is determined, the range of the synchronization domain is S R1 region. Obviously, the product of the global coupling strength σ and all eigenvalues of the coupling control matrix C after introducing the pinning control all fall into the synchronization domain S R1 , so the complete synchronization of the network can be achieved. Figure 4 (b) is the evolution diagram of the states of all nodes over time when the system parameters are b = 2, the coupling strength σ = 2, the time delay τ = 0.1, and pinning control is applied at nodes 1, 2, 3, and 4, where the control gain d 1 = d 2 = d 3 = d 4 = 2; it can be seen from the figure that the node states can be synchronized. This shows that the pinning control method for the time-delay coupled network system proposed in the present invention is reasonable and effective, can overcome the negative impact of time delay on the synchronization stability of the system, and achieve the synchronization goal.

[0080] To further illustrate the effectiveness of the pinning control method proposed in the present invention, as a comparative experiment, the case of b < 0 is considered next. When the system parameter b = -1, it is easy to know that at this time, a single system is stable at the equilibrium point x = 0. When forming the Figure 3 network system as shown, and when the time delay τ = 0.1 and the global coupling strength σ = 2, the network nodes can self-organize synchronization, as shown in Figure 5 (a); however, when the global coupling strength increases to σ = 5, the network nodes oscillate divergently and cannot be synchronized, as shown in Figure 5 (b). Figure 5 (a) and Figure 5 (b) illustrate the complexity of the time-delay coupled network system. The existence of time delay makes the over-strong coupling between nodes instead inhibit synchronization. Figure 5 (c) is the evolution diagram of the node state over time when pinning control is applied at node 1 on the basis of Figure 5 (a), and the control gain d 1 = 2. It can be seen from the figure that the originally synchronized nodes oscillate divergently under the control action, achieving the synchronization inhibition goal. Figure 5 (c)'s conclusion provides guidance for suppressing synchronization when synchronization is harmful in practical applications. For example, when router synchronization in a computer communication network causes network congestion, it is necessary to suppress synchronization. Figure 4 and Figure 5The simulation experiment results shown above further reveal the collaborative influence mechanism of node's own dynamics, global coupling strength, network topology, time delay, and pinning control strategy on the synchronization stability of the system.

[0081] The above description is only the preferred embodiments of the present disclosure and the explanation of the applied technical principles. Those skilled in the art should understand that the scope of the invention involved in the embodiments of the present disclosure is not limited to the technical solutions formed by the specific combination of the above technical features, but should also cover other technical solutions formed by any combination of the above technical features or their equivalent features without departing from the above inventive concept. For example, the technical solutions formed by mutually replacing the above features with the technical features (but not limited to) disclosed in the embodiments of the present disclosure that have similar functions.

Claims

1. A time-delay coupled network system control method, characterized in that: The following steps are involved: Step 1: Construct a time-delay coupled network system with N nodes: Construct a time-delay coupled network system with N nodes, and the dynamic equation of the i-th node is expressed as follows: in, represents the state variable of the i-th node, x in (t) represents the nth component of the state of the ith node, n represents the dimension of a single node, represents the first-order derivative of the state of the i-th node with respect to time t, represents a set of n-dimensional real vectors; f(·) represents the self-dynamic function of a single node; the constant σ>0 is the global coupling strength of the network; h(·) is the internal coupling function, which represents the coupling relationship between the state variables of each node; The Laplacian matrix that describes the network topology is defined as: when there is an edge connecting node i and node j, l ij = l ji = -1, otherwise l ij = l ji =0, and the dissipative coupling condition ∑ j l ij =0,i,j=1,2,…,N, where N represents the total number of nodes in the network system; τ is the time delay, h(x j The term (t-τ)) indicates that the time delay occurs on the internal coupling, so it is called a time-delay coupled network system; the topological structure of the time-delay coupled network system is an undirected connected graph; Step 2: Design a time-delayed control law to achieve synchronization or desynchronization goals: Step 3: According to the master stability function method, analyze the stability judgment conditions of the time-delay coupled network system under pinning control; Step 4: Determine the synchronization domain S of the time-delay coupled network system based on the master stability function with time delay R On this basis, the control gain, number of controlled nodes and position distribution of the pinning control law are determined so that the product of the global coupling intensity σ and all eigenvalues ​​of the coupling control matrix C falls within the synchronization domain, thereby achieving complete synchronization of the network.

2. A time-delay coupled network system control method according to claim 1, characterized in that: The control law with time delay described in step 2 is designed as follows: you i (t)=-σd i [h(x i (t-τ))-h(s(t-τ))],i=1,2,…,N (2) Among them, the constant σ>0 is the global coupling strength of the network, τ is the time delay; d i Represents the control gain. If control is applied to node i, then the control gain d i >0, otherwise, d i =0; s(t) is the synchronous state; All nodes in the entire network are controlled to the synchronous state s(t), and the time-delay coupled network system under pinned control is expressed as follows: Among them, represents the state variable of the \(i\)-th node, \(n\) represents the dimension of a single node, represents the first-order derivative of the state of the \(i\)-th node with respect to time \(t\), represents the set of \(n\)-dimensional real vectors; \(f(\cdot)\) represents the self-dynamics function of a single node, which is continuously differentiable; the constant \(\sigma>0\) is the global coupling strength of the network; \(h(\cdot)\) is the internal coupling function, representing the coupling relationship between the state variables of each node; is the Laplacian matrix characterizing the network topology, defined as: when there is an edge connecting node \(i\) and node \(j\), \(l\) ij = \(l\) ji = -1, otherwise \(l\) ij = \(l\) ji = 0, and satisfies the dissipative coupling condition \(\sum\) j \(l\) ij = 0, \(i,j = 1,2,\cdots,N\); \(\tau\) is the time delay, \(h(x\) j (t - \tau)) represents that the time delay occurs in the internal coupling; \(d\) i represents the control gain. If control is applied to node \(i\), then \(\xi\) i = 1 and the control gain \(d\) i > 0, otherwise, \(\xi\) i = 0 and \(d\) i = 0; the symbol \(\xi\) i represents whether control is applied to the \(i\)-th node. Thus, the number of controlled nodes is \(\sum\) i \(\xi\) i , and \(\sum\) i \(\xi\) i < N, \(s(t)\) is the synchronous state; When the network system (1) itself is stable in the synchronous state, the synchronization suppression target, i.e., desynchronization, is achieved by designing the pinning control law described by equation (2), which is applicable to actual scenarios where synchronization is harmful.

3. The time-delay coupled network system control method according to claim 1, characterized in that: The step 3 specifically comprises the following steps: Step 3.1: Define the node error vector e i (t) = x i (t)-s(t), i=1,2,…,N, then equation (3) becomes: in, e i (t), i = 1, 2, ..., N represents the error vector of the i-th node, N represents the total number of nodes in the network system, and the superscript T represents the transposition. represents a set of nN-dimensional real vectors; J f (s(t)) is the Jacobian matrix of the node’s own dynamics function f(·) in the synchronous state s(t), J h (s(t-τ)) is the Jacobian matrix of the internal coupling function h(·) in s(t-τ); the coupling control matrix C = L + D, where L is the Laplacian matrix and D is a diagonal matrix, defined as D = diag{ξ1d1,ξ2d2,…,ξ N d N }, d i represents the control gain, ξ i Indicates whether there is control applied to the i-th node; the constant σ is the global coupling strength, τ is the time lag; I N represents the N-dimensional unit matrix; Step 3.2: Since the topological structure of the time-delay coupled network system is an undirected connected graph, the coupling control matrix C is diagonalized to obtain the matrix G = P -1 CP, P is a transformation matrix composed of eigenvectors corresponding to the eigenvalues ​​of matrix C, and the variable and Then formula (4) becomes: Among them, I n is the n-dimensional unit matrix; Step 3.3: Let parameter θ = σμ i , where σ is the global coupling strength, μ i is the eigenvalue of matrix C; define auxiliary variable y(t), and get: Formula (6) is the main stability function of the time-delay coupled network system under pinning control; The condition for the time-delay coupled network system (3) under pinned control to be locally stable in the synchronous state is that the maximum Lyapunov exponent of the main stability function (6) with time delay is less than 0.

4. The time-delay coupled network system control method according to claim 1, characterized in that: In step 4: when synchronization is harmful and needs to be suppressed, based on the determination of the synchronization domain, a time-delay control law is designed so that the product of the global coupling strength σ and the eigenvalue of the coupling control matrix C falls outside the synchronization domain, thereby achieving network desynchronization.

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