1 Introduction

Complex networks are characterized by many interacting components that exhibit emergent behavior [1]. Synchronization in complex networks refers to the phenomenon where the system’s components coordinate their internal dynamics to achieve a common state [2, 3]. The study of synchronization phenomena has important implications in a wide range of fields, including neuroscience [4,5,6], economics [7], biology [8, 9], and physics [10, 11]. Various studies have explored synchronization in networks of coupled chaotic systems [12,13,14,15]. For instance, Balootaki et al. [15] explored the synchronization and control of fractional-order chaotic systems. Meanwhile, Chen et al. [14] examined exponential synchronization and anti-synchronization of nonautonomous chaotic systems with uncertain parameters. The exploration of fixed-time synchronization and its control strategies has been addressed in [12].

Synchronization stability in complex networks is the ability of individual components to achieve and maintain a synchronized state [16]. This concept is crucial for the proper function of many natural and artificial networks [16, 17]. Pecora and Carroll [18] introduced the master stability function (MSF) approach for analyzing the stability of synchronized states in complex networks. This method provides a mathematical determination of the necessary conditions for the local stability of the synchronous manifold within coupled identical oscillators. Afterward, others broadened the MSF concept to include non-identical oscillators [19], multilayer networks [20], and higher-order networks [21].

Various factors, such as the oscillators’ dynamics, network topology, and the coupling between oscillators, influence synchronization and its stability in complex networks [18]. One of the significant issues is adjusting these factors and proposing techniques toward reaching optimum synchronization [22]. This is achieving synchronization with a lower cost (usually smaller coupling strength), resulting in a maximized stability region. Several studies have attempted to present methodologies to ensure optimum and enhanced synchronization. Some studies have proved that synchronization enhancement can be assessed through delay coupling, time-delayed feedback control, dynamical weights, intermittent noise, etc. [23,24,25,26]. Fan et al. [27] found that a proper selection of phase lag modulations can facilitate network synchronization. Estrada et al. [28] revealed the improvement of synchronization in the presence of long-range interactions. Dayani et al. [29] proposed a time-varying coupling function composed of diffusive single-variable couplings to enhance synchronization.

Oscillators within a network can be connected through single-variable or multi-variable coupling [30]. In single-variable coupling, the oscillators are linked to each other through a single state variable, while in multi-variable coupling, the oscillators are simultaneously interconnected through multiple state variables. Some researchers [30,31,32] have used the MSF approach to examine the impact of both couplings on network synchronization. Sevilla-Escoboza et al. [32] searched for the optimum multi-variable coupling for maximizing synchronization stability in coupled Rössler systems. Nazarimehr et al. [31] demonstrated that implementing multi-variable coupling among oscillators in circulant chaotic systems can result in more optimal synchronization than single-variable coupling. Recently, they extended their study to non-circulant chaotic systems [30], revealing that multi-variable coupling may not always lead to optimum synchronization than single-variable coupling. Only the diagonal couplings with equal coefficients have been established in these studies for the multi-variable coupling.

This study aims to propose the optimized multi-variable coupling based on an optimization algorithm in networks of chaotic oscillators. To this aim, an optimization algorithm is employed to find the strengths of multi-variable couplings based on the master stability function. The algorithm is applied to networks of chaotic Rössler, Hindmarsh-Rose, Lorenz, and Chen’s systems. The results show that multi-variable coupling leads to a more efficient synchronization than single-variable and diagonal couplings.

2 Methodology

The governing equations of a network with N interconnected identical oscillators can be written as follows,

$$\begin{aligned} \dot{X}_{i}=F(X_{i})-\sigma \sum \limits _{j=1}^N L_{ij} H(X_{j}),\hspace{0.05in}i=1,2,3,\ldots ,N. \end{aligned}$$
(1)

In Eq. (1), X represents the m-dimensional vector of state variables, and F characterizes the dynamics of the individual oscillators. The Laplacian matrix and the network coupling function are denoted by L and H, respectively. L represents the connection between the network oscillators and obeys the condition of \(\sum \nolimits _{j=1}^N L_{ij}=0\). The parameter \(\sigma \) denotes the global coupling strength between the oscillators. In the following, firstly, the MSF method for analyzing synchronization stability is described. Then, the optimization algorithm for obtaining the strengths of multi-variable coupling for optimum synchronization is introduced.

2.1 Master stability function (MSF)

MSF is a powerful mathematical approach to obtain the necessary conditions for the local stability of the synchronization manifold in a network [18]. Let us assume that the synchronous manifold of the network \((X_{s})\) is stable; then, all the network oscillators are attracted to this manifold over time \((X_{1}=X_{2}=...=X_{N}=X_{s})\) and thus,

$$\begin{aligned} \dot{X}_{s}=F(X_{s}). \end{aligned}$$
(2)

Entering the disturbance d to the oscillators causes them to deviate from the synchronous manifold \((d_{i}=X_{s}-X_{i})\). Then, the variational equations can be obtained by substituting the disturbances in Eq. (1) and linearizing around the synchronous manifold as follows,

$$\begin{aligned}{} & {} \dot{d}_{i}=(DF(X_{s})-\sigma \sum \limits _{j=1}^N L_{ij} DH(X_{s}))d_{i},\nonumber \\{} & {} \qquad i=1,2,3,\ldots ,N. \end{aligned}$$
(3)

In Eq. (3), D represents the Jacobian of the matrices. By utilizing the eigenvalues and eigenvectors properties of the Laplacian matrix L, Eq. (3) can be simplified and changed to the decoupled form as Eq. (4).

$$\begin{aligned} \dot{\delta }=(DF(X_{s})-KDH(X_{s}))\delta \end{aligned}$$
(4)

In Eq. (4), \(K=\sigma \lambda _{i},\hspace{0.01in}i=1,2,3,\ldots ,N\), where \(\lambda _{i}\) represents the ith eigenvalue of the Laplacian matrix L. The largest Lyapunov exponent of Eq. (4) is referred to as the MSF of the system concerning the normalized coupling strength K. A negative value of MSF shows the attraction of the oscillators to the stable synchronous manifold, i.e. indicates stability.

2.2 Optimization algorithm

This study is focused on the linear diffusive coupling function, i.e., \(H(X_{j})=h.X_{j}\), where h is a constant matrix. For simplicity, the following descriptions are given for 3-dimensional systems \((m=3)\); however, the same procedure can be used for higher-dimensional systems. The systems are assumed to be coupled through all-variable coupling with different coupling strengths for each coupling. Hence,

$$\begin{aligned} h_{ij}= & {} 1(i,j=1,\ldots ,m),\,{\text {and}}\, \varvec{\sigma }\\= & {} \begin{bmatrix} \sigma _{1} &{} \sigma _{2} &{} \sigma _{3} \\ \sigma _{4} &{} \sigma _{5} &{} \sigma _{6} \\ \sigma _{7} &{} \sigma _{8} &{} \sigma _{9} \\ \end{bmatrix}, \sum \limits _{l=1}^9\sigma _{l}=\sigma . \end{aligned}$$

Note that instead of one coupling strength \((\sigma )\) for all coupling components as in Eq. (1), it is divided into \( m\times {m}\) parts \((\sigma _{1},\ldots ,\sigma _{(m\times {m})})\). Therefore, the equivalent coupling parameter matrix (refer to Eq. 4) for computing the MSF can be written as

$$\begin{aligned} \textbf{K}=\begin{bmatrix} k_{1} &{} k_{2} &{} k_{3} \\ k_{4} &{} k_{5} &{} k_{6} \\ k_{7} &{} k_{8} &{} k_{9} \\ \end{bmatrix}, \sum \limits _{l=1}^9k_{l}=K. \end{aligned}$$

As a result, the coupling parameters \(k_{1},\ldots ,k_{9}\) refer respectively to the couplings \(1\rightarrow 1\), \(2\rightarrow 1\), \(3\rightarrow 1\), \(1\rightarrow 2\), \(2\rightarrow 2\), \(3\rightarrow 2\), \(1\rightarrow 3\), \(2\rightarrow 3\), and \(3\rightarrow 3\). The notation of \(i\rightarrow j\) coupling shows that the difference between the ith variables is added to the jth variable.

This study aims to find the elements of the coupling parameter matrix at a given value of K for achieving optimum synchronization. The proposed algorithm is executed through five steps, as detailed below:

Step 1: First, the initial values are assigned to the normalized coupling parameters \((k_{1},\ldots ,k_{9})\) for each K value. Ten different initial conditions are considered for the coupling parameters to avoid getting stuck in local optima. In nine of these initial conditions, K is assigned to only one of the coupling parameters, and the remaining ones are set to 0. In the 10th initial condition, an initial value of K/9 is assigned to each coupling parameter.

Step 2: The MSF value of the system is calculated for the initial normalized coupling parameters. Then, two of the nine coupling parameters are selected, e.g. \(k_{i}\) and \(k_{j}\). Subsequently, two scenarios are considered for \(k_{i}\) and \(k_{j}\), but the other coupling parameters remain unchanged; then, the MSF is calculated for these two scenarios. In the first scenario, K/100 is subtracted from \(k_{i}\) and added to \(k_{j}\); and in the second one, K/100 is subtracted from the coupling parameter \(k_{j}\) and added to \(k_{i}\).

Note: To prevent negative coupling parameters, if a coupling parameter is zero, K/100 is added to it and subtracted from the other coupling parameter. Suppose one coupling parameter becomes negative after subtracting K/100. In that case, the corresponding parameter is set to zero, and the distance of its previous value from zero is added to the other coupling parameter.

Step 3: The calculated MSF values from the previous step are compared, and the matrix of normalized coupling parameters associated with the lowest MSF value is retained for the algorithm’s subsequent steps. Since there are 36 unique configurations to select two couplings among nine coupling parameters, each iteration necessitates the algorithm to be repeated 36 times.

Step 4: The algorithm termination criterion is evaluated by comparing the normalized coupling parameter matrices obtained after two successive iterations. If these matrices are identical, the algorithm is converged to the optimized coupling parameters. The matrix of the optimized coupling parameters and the corresponding MSF is then preserved. If the matrices differ, steps 2 to 4 are repeated, considering the last coupling parameter matrix and its associated MSF for the subsequent iteration.

Step 5: Steps 1 to 4 are repeated for every ten different initial conditions, and the matrix of normalized coupling parameters corresponding to the smallest MSF is considered the algorithm’s global optimum.

3 Results

The proposed algorithm is applied to networks of chaotic Rössler, Hindmarsh-Rose, Lorenz, and Chen’s systems [33]. The systems’ dynamics are described below in Eqs. (5)–(8).

Rössler system:

$$\begin{aligned} \begin{array}{l} \dot{x}=-y-z,\\ \dot{y}=x+\alpha y,\\ \dot{z}=\beta +(x-\gamma )z,\\ \end{array} \end{aligned}$$
(5)

with the parameters \(\alpha =0.2,\beta =0.2, \gamma =9,\) and the initial conditions \((x_{0},y_{0},z_{0})=(0,0,0)\).

Hindmarsh-Rose system:

$$\begin{aligned} \begin{array}{l} \dot{x}=y+3x^2-x^3-z+I,\\ \dot{y}=1-5x^2-y,\\ \dot{z}=-rz+rs(x+1.6),\\ \end{array} \end{aligned}$$
(6)

with the initial conditions \((x_{0},y_{0},z_{0})=(0.1,0.1,0.1)\) and the parameters \(I=3.2,r=0.006,\) and \(s=4\).

Lorenz system:

$$\begin{aligned} \begin{array}{l} \dot{x}=a(y-x),\\ \dot{y}=x(\rho -z)-y,\\ \dot{z}=xy-\beta z,\\ \end{array} \end{aligned}$$
(7)

with the initial conditions \((x_{0},y_{0},z_{0})=(0.5,0.5,0.5)\) and the parameters \(a=10,\rho =28,\) and \(\beta =2\).

Chen’s system:

$$\begin{aligned} \begin{array}{l} \dot{x}=a(y-x),\\ \dot{y}=(c-a-z)x+cy,\\ \dot{z}=xy-\beta z,\\ \end{array} \end{aligned}$$
(8)

with the initial conditions \((x_{0},y_{0},z_{0})=(0.5,0.5,0.5)\) and parameters \(a=35,c=28,\) and \(\beta =8/3\).

These systems demonstrate chaotic behavior under the specified initial conditions. Figures 1a–d show the 2D projection of the chaotic attractors for the Rössler, HR, Lorenz, and Chen’s systems, respectively.

The optimum normalized coupling parameters are obtained for each system, and the corresponding MSFs are discussed. All simulations are performed with runtime \(T=10000\), time step 0.01, and initial conditions mentioned previously. Figures 2, 3, 4 and 5 depict the relative normalized coupling strengths \(( k_{i}/K,\hspace{0.05in}i=1,2,\ldots ,9)\) for varying K across different iterations until attaining optimized coupling parameters for the Rössler, HR, Lorenz, and Chen’s systems, respectively. These figures are obtained for the initial value of K/9 for all normalized coupling strengths. The analyses of these figures reveal that, in all four systems, the diagonal couplings involving \(k_{1}\), \(k_{5}\), and \(k_{9}\) exhibit higher strengths than other couplings across various K values. Specifically, in the Rössler system, the most prominent coupling strengths are associated with the coupling \(1\rightarrow {1}\) \((k_{1})\) for large values of K and the coupling \(2\rightarrow {2}\) \((k_{5})\) for low K values. In the HR system, the coupling \(3\rightarrow {3}\) with the coupling parameter \(k_{9}\) has the largest value. In the Lorenz and Chen systems, the \(2\rightarrow {2}\) coupling \((k_{5})\) for large values of K and the \(3\rightarrow {3}\) coupling \((k_{9})\) for small values of K represents the highest coupling strengths. Furthermore, the analyses indicate that Chen’s system requires more iterations than the others to converge toward the optimized multi-variable coupling.

Fig. 1
figure 1

The 2D projection of the chaotic attractors on the \(X-Y\) plane for a Rössler, b HR, c Lorenz, and d Chen’s systems

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Fig. 2
figure 2

The evolution of the relative normalized coupling parameters to reach the optimized multi-variable coupling as a function of K for the Rössler system. a \(k_{1}/K\), b \(k_{2}/K\), c \(k_{3}/K\), d \(k_{4}/K\), e \(k_{5}/K\), f \(k_{6}/K\), g \(k_{7}/K\), h \(k_{8}/K\), and i \(k_{9}/K\). The initial values of each coupling strength in the optimization algorithm are selected as K/9 for each K. The colors represent the value of the relative normalized coupling strength. The figures represent that the coupling parameters \(k_{1}\) and \(k_{5}\), corresponding to couplings \(1\rightarrow {1}\) and \(2\rightarrow {2}\), show greater strengths than other couplings for different K values. (Color figure online)

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Fig. 3
figure 3

The evolution of the relative normalized coupling parameters to reach the optimized multi-variable coupling as a function of K for the HR system. a \(k_{1}/K\), b \(k_{2}/K\), c \(k_{3}/K\), d \(k_{4}/K\), e \(k_{5}/K\), f \(k_{6}/K\), g \(k_{7}/K\), h \(k_{8}/K\), and i \(k_{9}/K\). The initial values of each coupling strength in the optimization algorithm are selected as K/9 for each K. The colors represent the value of the relative normalized coupling strength. The figures represent that the coupling parameters \(k_{1}\), \(k_{2}\), \(k_{5}\), and \(k_{9}\), show greater strengths than other couplings. (Color figure online)

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Fig. 4
figure 4

The evolution of the relative normalized coupling parameters to reach the optimized multi-variable coupling as a function of K for the Lorenz system. a \(k_{1}/K\), b \(k_{2}/K\), c \(k_{3}/K\), d \(k_{4}/K\), e \(k_{5}/K\), f \(k_{6}/K\), g \(k_{7}/K\), h \(k_{8}/K\), and i \(k_{9}/K\). The initial values of each coupling strength in the optimization algorithm are selected as K/9 for each K. The colors represent the value of the relative normalized coupling strength. The figures represent that the coupling parameters \(k_{5}\), and \(k_{9}\), show greater strengths than other couplings. (Color figure online)

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Fig. 5
figure 5

The evolution of the relative normalized coupling parameters to reach the optimized multi-variable coupling as a function of K for Chen’s system. a \(k_{1}/K\), b \(k_{2}/K\), c \(k_{3}/K\), d \(k_{4}/K\), e \(k_{5}/K\), f \(k_{6}/K\), g \(k_{7}/K\), h \(k_{8}/K\), and i \(k_{9}/K\). The initial values of each coupling strength in the optimization algorithm are selected as K/9 for each K. The colors represent the value of the relative normalized coupling strength. The figures represent that the coupling parameters \(k_{1}\) and \(k_{9}\) in small K values and \(k_{5}\) in large values of K show greater strengths than other couplings. (Color figure online)

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Figure 6 illustrates the obtained optimized coupling parameters for Rössler, HR, Lorenz, and Chen’s systems across varying K. It exhibits that, for the Rössler system, the \(1\rightarrow {1}\) and \(2\rightarrow {2}\) couplings exhibit superior strengths (\(k_{1}\) and \(k_{5}\)); while in the HR system, the couplings \(1\rightarrow {1}\), \(2\rightarrow {2}\) and \(3\rightarrow {3}\) demonstrate higher strengths (\(k_{2}\), \(k_{5}\), and \(k_{9}\)) than other couplings. Similarly, in the Lorenz system, the couplings \(2\rightarrow {2}\) and \(3\rightarrow {3}\) display large strengths (\(k_{5}\) and \(k_{9}\)) across different K values. Notably, within the Chen’s system, the coupling parameter \(k_{5}\) (\(2\rightarrow {2}\) coupling) is prominent for small K values, whereas for larger K values, the coupling parameters \(k_{5}\) and \(k_{9}\) (\(2\rightarrow {2}\) and \(3\rightarrow {3}\) couplings) surpass other couplings.

Fig. 6
figure 6

Optimized multi-variable coupling parameters as a function of K for a Rössler, b HR, c Lorenz, and d Chen’s systems. In these diagrams, the horizontal axis shows the normalized coupling parameter K, and the vertical axis represents the relative normalized coupling parameter \((k_{i}/K),\hspace{0.05in}i=1,2,\ldots ,9\). According to these diagrams, in the Rössler system, the coupling parameters \(k_{1}\) and \(k_{5}\), in HR and Lorenz systems, the coupling parameters \(k_{5}\) and \(k_{9}\), and in Chen’s system, the coupling parameter \(k_{5}\) in all values and \(k_{9}\) in large K values have stronger values than other couplings. Also, in the HR system, the strength of the non-diagonal coupling \(k_{2}\) is also greater than other couplings

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Fig. 7
figure 7

The largest Lyapunov exponent \((\Lambda )\) of the variational equations (Eq. 4) across normalized coupling parameter K for different coupling schemes. a Rössler, b HR, c Lorenz, and d Chen’s systems. The solid lines are related to the single-variable couplings. The green dashed (OC) and red dotted (D) lines correspond to optimized multi-variable and diagonal coupling, respectively. For all systems, the largest Lyapunov corresponding to the optimized multi-variable coupling is lower than the other couplings. (Color figure online)

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Fig. 8
figure 8

Enlargement of the largest Lyapunov exponents presented in Fig. 7. a Rössler, b HR, c Lorenz, and d Chen’s systems. The circles show the zero-crossing points of the largest Lyapunov exponent curves. The values of zero-crossing points corresponding to the optimized multi-variable couplings are smaller than the other couplings

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Figure 7 represents the largest Lyapunov exponents \((\Lambda )\) of the variational equations (Eq. 4) for the optimized multi-variable coupling (green dashed line), each of single-variable couplings (solid lines) and diagonal coupling (red dotted lines) across various normalized coupling strength K. Note that in the single-variable coupling, only one variable is included in the coupling, i.e., only one element in the h matrix is equal to one. In the diagonal coupling, h equals the identity matrix of size m. According to this figure, for all four systems, the \(\Lambda \) values related to the optimized multi-variable coupling are smaller than other single-variable and diagonal couplings for all K values. Also, in the negative region of \(\Lambda \), the optimized multi-variable coupling results in a more negative value than other coupling schemes, suggesting a substantially enhanced synchronization stability.

To represent the zero-crossing points of the MSFs, the enlargement of the largest Lyapunov exponents \((\Lambda )\) across K are shown in Fig. 8 for all four systems. This figure indicates that for all systems, the MSF corresponding to the optimized multi-variable coupling intersects the horizontal axis at smaller values of K than other MSFs. Therefore, the optimized multi-variable coupling leads to achieving synchronization by expensing lower total coupling strength than any of the single-variable and diagonal couplings.

4 Conclusion

This research aimed to obtain the optimized multi-variable coupling to enhance the synchronization in networks of coupled chaotic oscillators. The coupling was considered to be linear diffusive. Ten different initial conditions were considered for the initial coupling. Then, an optimization algorithm was proposed to attain the optimum multi-variable coupling based on MSF. The optimization algorithm was applied to networks of chaotic Rössler, Hindmarsh-Rose, Lorenz, and Chen’s systems.

To evaluate the optimization algorithm’s efficiency, the MSF curves were calculated for the optimum multi-variable coupling, all single-variable couplings, and the diagonal coupling, keeping the total normalized coupling parameter the same. Our findings showed that in all investigated systems, the MSF values for the optimized multi-variable coupling were smaller than the MSF values for each single-variable and diagonal coupling. Also, the zero-crossing point of the MSF related to the optimized multi-variable coupling occurred at smaller K values than other MSFs. Therefore, the optimized multi-variable coupling resulted in an enhanced and more stable synchronization than the other couplings.

Furthermore, the optimized multi-variable coupling matrix had larger values in the diagonal elements than the non-diagonal elements in all K values. Consequently, the diagonal couplings have a more prominent role in providing optimum synchronization. The importance of diagonal couplings in enhanced synchronization has also been reported in networks of chaotic circulant systems [31].

In summary, the results of the present study demonstrate that the multi-variable coupling obtained by the optimization algorithm can lead to more efficient synchronization in networks consisting of chaotic oscillators. However, it’s important to note that the study’s assumptions-linear couplings through pairwise connections without time delay-may not capture the full complexity of real-world systems. Factors like nonlinear couplings, time delays, non-pairwise links, and time-varying connections are common in real systems and can significantly influence synchronization dynamics. Thus, exploring the proposed optimization method in more complex networks with these features could deepen our understanding of network dynamics.