**4. Numerical example**

In this section, a numerical example is used to verify the effectiveness of the proposed methods.

**Example 1.** Consider a dynamical network consisting of 10 nodes that are identical Chua's circuits. A single Chua's circuit is described by

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathcal{G}(-\mathbf{x}\_1 + \mathbf{x}\_2 - \boldsymbol{\zeta}'(\mathbf{x}\_1)) \\ \dot{\mathbf{x}}\_2 = \mathbf{x}\_1 - \mathbf{x}\_2 + \mathbf{x}\_3 \\ \dot{\mathbf{x}}\_3 = -\alpha \mathbf{x}\_2 \end{cases} \tag{38}$$

where *ϑ* = 10, *ω* = 14.87, *ζ*(*x*1) <sup>=</sup>*bx*<sup>1</sup> <sup>+</sup> *<sup>a</sup>* <sup>−</sup> *<sup>b</sup>* <sup>2</sup> (|*x*<sup>1</sup> + 1| − | *x*<sup>1</sup> −1|), *a* = − 1.27, and *b* = − 0.68. It is known that the Chua's system has a chaotic attractor which is shown in **Figure 1**.

**Figure 1.** The chaotic attractor of Chua's circuit.

It is obvious that system (38) is also be rewritten as

$$
\dot{\mathbf{x}} = H\mathbf{x} + \mathbf{g}(\mathbf{x})\tag{39}
$$

where

New Stabilization of Complex Networks with Non-delayed Couplings over Random Exchanges 43 http://dx.doi.org/10.5772/62504

$$\begin{aligned} \mathbf{x} &= \begin{bmatrix} \mathbf{x}\_1 & \mathbf{x}\_2 & \mathbf{x}\_3 \end{bmatrix}^T \\\\ H &= \begin{bmatrix} -\mathcal{G} & \mathcal{G} & \mathbf{0} \\\\ 1 & -1 & 1 \\\\ \mathbf{0} & -\mathcal{O} & \mathbf{0} \end{bmatrix} \\\\ \mathbf{g}(\mathbf{x}) &= \begin{bmatrix} -\mathcal{G}\zeta'(\mathbf{x}\_1) & \mathbf{0} & \mathbf{0} \end{bmatrix}^T \end{aligned}$$

Without loss of generality, matrix P here is selected as P = diag{1, ω, 1}. Next, we will check whether there is a suitable η satisfying condition (11) in Assumption 1. It is obtained that

$$\begin{aligned} \mathbf{x}^{\top}P(H\mathbf{x} + \mathbf{g}(\mathbf{x})) &\leq \frac{1}{2}\mathbf{x}^{\top}(PH + H^{\top}P)\mathbf{x} - \mathcal{G}\mathbf{a}\mathbf{x}\_{\text{l}}^{2} \\ &= \frac{1}{2}\mathbf{x}^{\top}(\tilde{H} + \tilde{H}^{\top})\mathbf{x} \\ &\leq \frac{1}{2}\lambda\_{\text{max}}(\tilde{H} + \tilde{H}^{\top})\mathbf{x}^{\top}\mathbf{x} \\ &= \eta\mathbf{x}^{\top}\mathbf{x} \end{aligned} \tag{40}$$

where H = PH + diag( - &, 0, 0} and n = = 1 ma(H + H + )=9.0620. Thus, condition (11) is satisfied. Then, the resulting network closed by controller (4) is described as

$$\begin{cases} \dot{\mathbf{x}}\_{i1} = 10(-\mathbf{x}\_{i1} + \mathbf{x}\_{i2} - \zeta(\mathbf{x}\_{i1})) + c \sum\_{j=1}^{N} a\_{ij} \mathbf{x}\_{j1} + c \sum\_{j=1}^{N} b\_{ij} \mathbf{x}\_{j1} (t - \tau) + \mathbf{u}\_{i1} \\\\ \dot{\mathbf{x}}\_{i2} = \mathbf{x}\_{i1} - \mathbf{x}\_{i2} + \mathbf{x}\_{i3} + c \sum\_{j=1}^{N} a\_{ij} \mathbf{x}\_{j2} + c \sum\_{j=1}^{N} b\_{ji} \mathbf{x}\_{j2} (t - \tau) + \mathbf{u}\_{i2} \\\\ \dot{\mathbf{x}}\_{i3} = -14.87 \mathbf{x}\_{i2} + c \sum\_{j=1}^{N} a\_{ij} \mathbf{x}\_{j3} + c \sum\_{j=1}^{N} b\_{ji} \mathbf{x}\_{j3} (t - \tau) + \mathbf{u}\_{i3} \end{cases} \tag{41}$$

Without loss of generality, the coupling matrices A and B are expressed by small-world and scale-free networks, which are depicted in Figures 2 and 3, respectively.

**Figure 2.** The simulation of coupling matrix *A*.

**Figure 3.** The simulation of coupling matrix *B*.

When such coupling matrices exchange randomly, under conditions such that *c* = 50, *α*\* = 0.85, *δ*\* = 3.6, and pinning fraction = 0.8, based on Theorem 1, we have the corresponding parameters computed as follows:

 $k\_i = 22.8791$ ,  $k\_{di} = 2.3840$ ,  $i \in \mathbb{S}\_{\ell^\vee}$  and  $i$ 

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Under the initial condition x;(t)=[0.1] 0.1 0.2]", where i = 1, 2, ..., 10 and τ = 0.005, we have the state response of the closed-loop network by the stochastic pinning controller (5) shown in Figure 4 and is stable.

Figure 4. The state response of the complex network by controller (5).

Based on the results in this chapter, it is known that probability a\* plays important roles in the stabilization of complex networks, where non-delay and delay control gains k, and k,, are very close to a\*. Let k = k .2 + k .; we have the relationship between parameters a\* and K, k .; and k, given in Table 1, where the more detailed correlation between o\* and k;, k;; and k, is simulated in Figure 5. From Table 1 and Figure 5, it is seen that both gains of k; and k;; have effects in the stabilization of the underlying complex network. It is also found that there is not a phenomenon that larger a\* results in larger k;, or smaller k;. This property further demonstrates the necessity of considering the probability distribution of non-delay states while the stabilization problem of delayed systems is considered. Particularly, it is seen that when a\* = 0, there are no solutions to k; and k; This is determined by condition (10), which is actually determined by the inherent property of pinning control of complex network with delayed coupling.


Table 1. The relations between a\* and k;; ka; ka·

Figure 5. The simulation of correlation between a\* and k; kai, Kai

When probability a\* is uncertain and described as (20) such that x=0.85 and µ=0.1, by Theorem 2, one has the corresponding parameters computed as follows:

k; = 150.8308, ka; = 63.5059, i ∈ Sg, and


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When probability a\* is inaccessible, a kind of adaptive pinning control method may be exploited. Let the corresponding parameters P, η, and δ\* same to the above values, by Theorem 3, one could get the related parameters computed as follows: k; = 102.2258, k;; = 22.3035, i ∈ Sq, and

Figure 6. The state response of the complex network by controller (29).


where ó is selected to be ó=5. Under the same initial condition and topologies having couplings exchanges, the simulations of the resulting complex network are given in Figures 6 and 7, where Figure 6 is state response of the closed-loop system through the desired adaptive pinning controller with form (29) and updating law with form (30), and Figure 7 is the curve of estimation a(t) with a =0.2.

Figure 7. The curve of estimation of a\*.

From these simulations, it is said that the desired partially delay-dependent controllers in terms of stochastic pinning controller (5) and adaptive controller (29) are both effective, where the resulting complex network is stable even if the coupling matrices experience random exchanges. On the other hand, when a is obtained exactly but ô\* is unavailable, using Theorem 4, we have the corresponding parameters obtained as follows: k; = 30.6104, k; = 16.7135, i ∈Sg) and


where & is selected to be § = 1.
