# Exponential Outer Synchronization between Two Uncertain Time-Varying Complex Networks with Nonlinear Coupling

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem formulation and preliminaries

_{i}, y

_{i}∈ R

^{n}, i = 1, 2,…, N. F

_{i}(·) ∈ R

^{n}is a continuously differentiable nonlinear vector field, ${\theta}_{i}={({\theta}_{i}^{(1)},{\theta}_{i}^{(2)},\dots ,{\theta}_{i}^{(m)})}^{T}\in {R}^{m}$ are unknown parameters, ${\tilde{\theta}}_{i}={({\tilde{\theta}}_{i}^{(1)},{\tilde{\theta}}_{i}^{(2)},\dots ,{\tilde{\theta}}_{i}^{(m)})}^{T}\in {R}^{m}$ are the estimation of the unknown θ

_{i}, h(·) ∈ R

^{n}is the nonlinear inner-coupling function, τ

_{1}is node delay and τ

_{2}is coupling delay. A(t) = (a

_{ij}(t))

_{N}

_{×}

_{N}, B(t) = (b

_{ij}(t))

_{N}

_{×}

_{N}are respective time-varying coupling matrices representing the topological structures of networks X and Y. The entries a

_{ij}(t) (b

_{ij}(t)) are defined as follows: a

_{ij}(t) (b

_{ij}(t)) > 0 if there is a connection between node i and node j (i ≠ j); otherwise a

_{ij}(t) (b

_{ij}(t)) = 0 (i ≠ j), and the diagonal entries ${a}_{ii}(t)=-{\displaystyle {\sum}_{j=1,j\ne i}^{N}{a}_{ij}(t)}$, ${b}_{ii}(t)=-{\displaystyle {\sum}_{j=1,j\ne i}^{N}{b}_{ij}(t)}$, i = 1, 2,…, N. u

_{i}is a controller to be designed.

_{i}is linearly dependent of the nonlinear functions of the ith node, and we can rewrite networks Equations (1) and (2) in the following forms:

_{i}

_{1}(·) ∈ R

^{n}is a continuous vector function and f

_{i}

_{2}(·) ∈ R

^{n}

^{×}

^{m}is a continuous matrix function, i = 1, 2,…, N.

_{i}(t) = y

_{i}(t) − x

_{i}(t), ${\overline{\theta}}_{i}={\tilde{\theta}}_{i}-{\theta}_{i}$. According to the drive network Equation (3) and the response network Equation (4), the error dynamical network is then described by

_{i}(·, θ

_{i}) = f

_{i}

_{1}(·) + f

_{i}

_{2}(·) θ

_{i}, i = 1, 2,…, N.

**Assumption 1.**For any x

_{i}(t); y

_{i}(t) ∈ R

^{n}, there exists a positive constant L satisfying

**Assumption 2.**There exists a positive constant α such that

^{n}.

**Assumption 3.**Denote${f}_{i2}(t,{y}_{i}(t),{y}_{i}(t-{\tau}_{1}))=({f}_{i2}^{(1)}(t,{y}_{i}(t),{y}_{i}(t-{\tau}_{1})),{f}_{i2}^{(2)}(t,{y}_{i}(t),{y}_{i}(t-{\tau}_{1})),\dots ,{f}_{i2}^{(m)}(t,{y}_{i}(t),{y}_{i}(t-{\tau}_{1})))$. Assume that${f}_{i2}^{(k)}(t,{y}_{i}(t),{y}_{i}(t-{\tau}_{1}))$, h(y

_{j}(t − τ

_{2})) (k = 1, 2,…, m, j = 1, 2,…, N) are linearly independent on the synchronized orbit x

_{i}(t) = y

_{i}(t) of synchronization manifold for any given i ∈ {1, 2,…, N}.

**Lemma 1.**For any vectors x, y ∈ R

^{n}, the following inequality holds:

**Definition 1.**We say that networks X and Y achieve exponential outer synchronization if there exist positive constants M and μ such that

## 3. Theoretical Results

**Theorem 1.**Suppose that Assumptions 1–3 hold. Then exponential outer synchronization between the drive network Equation (3) and the response network Equation (4) can be achieved and the unknown parameters θ

_{i}can be identified by using the estimation${\tilde{\theta}}_{i}$ with the following adaptive controllers and the corresponding updating laws:

_{i}, l

_{i}and k

_{i}are positive constants, i = 1, 2,…, N.

**Proof.**Construct the following Lyapunov functional:

^{*}is fixed, then we can choose $d*\ge \frac{\mu +3L}{2}+\frac{L}{2}\mathrm{exp}(\mu {\tau}_{1})+\frac{\alpha NH*}{2}(1+\mathrm{exp}(\mu {\tau}_{2}))+1$. Thus, one has $\dot{V}(t)\le -{\displaystyle {\sum}_{i=1}^{N}{e}_{i}^{T}(t){e}_{i}(t)\mathrm{exp}(\mu t)\le 0}$. It follows that V (t) ≤ V (0) for any t ≤ 0. From the Lyapunov function Equation (7), one gets

_{i}(t) of network Equation (4) must synchronize exponentially toward the x

_{i}(t) with a convergence rate of $\frac{\mu}{2}$. It implies that exponential outer synchronization between the drive network Equation (3) and the response network Equation (4) has been achieved. Thus, $\underset{t\to \infty}{\mathrm{lim}}\Vert {e}_{i}(t)\Vert =0\phantom{\rule{0.2em}{0ex}}(i=1,2,\dots ,N)$.

_{i}(t) converges to a constant as t→∞, one has $\underset{t\to \infty}{\mathrm{lim}}{\dot{e}}_{i}(t)=0$. According to the error system Equation (5), we have

_{ij}(t) − a

_{ij}(t) → 0 as t→∞ [42]. That is to say, the unknown parameters θ

_{i}can be identified by using the control scheme Equation (6). All this completes the proof.

## 4. Numerical Example

_{i}

_{1}is the membrane action potential, x

_{i}

_{2}is a recovery variable and x

_{i}

_{3}is a slow adaptation current, I is the external direct current, α

_{i}

_{1}, α

_{i}

_{2}, α

_{i}

_{3}, α

_{i}

_{4}, s, p and χ are constants, i = 1, 2,…, N. In the following simulations, let α

_{i}

_{1}= 1.0, α

_{i}

_{2}= 3.0, α

_{i}

_{3}= 1.0, α

_{i}

_{4}= 5.0, s = 4.0, p = 0.006, χ = −1.60 and I = 3.0, HR system shows the chaotic firing pattern [44]. For simplicity, we only assume that α

_{i}

_{4}are not known in advance, then system Equation (8) can be rewritten as

_{i}= α

_{i}

_{4}, i = 1, 2,…,N. For convenience, set the node delay τ

_{1}= 0.1 and the coupling delay τ

_{2}= 0.2. Let the nonlinear inner-coupling function h(x

_{i}) = (0, 1 + sin(x

_{i}

_{2}), 0)

^{T}, so it satisfies Assumptions 2 and 3.

_{i}= k

_{i}= l

_{i}= 20 and μ = 0.1. The network size is taken as N = 5, the initial values a

_{ij}(0) and b

_{ij}(0) (i, j = 1,…, 5) are randomly chosen in the interval (0, 0.5) and the interval (0.5, 1) respectively. The other initial values are randomly chosen in the interval (–1, 1). The time evolution of outer synchronization error E(t) = max{∥y

_{i}(t) − x

_{i}(t)∥ : i = 1, 2,…, 5} is shown in Figure 1. It is clear that outer synchronization error is rapidly converging to zero. Figure 2 shows the identification of unknown parameters θ

_{i}(i = 1,…, 5). Figure 3 displays the evolution of adaptive feedback gains d

_{i}(i = 1,…, 5). From Figures 4 and 5, we can find that a

_{ij}(t) and b

_{ij}(t) are converged to the same constants when outer synchronization appears.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Watts, D.J.; Strogatz, S.H. Collective dynamics of small-world networks. Nature
**1998**, 393, 440–442. [Google Scholar] - Barabási, A.-L.; Albert, R. Emergence of scaling in random networks. Science
**1999**, 286, 509–512. [Google Scholar] - Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.-U. Complex networks: Structure and dynamics. Phys. Rep
**2006**, 424, 175–308. [Google Scholar] - Newman, M.E.J. Networks: An Introduction; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- Sun, W. Random walks on generalized Koch networks. Phys. Scr
**2013**, 88, 045006. [Google Scholar] [CrossRef] - Lu, J.; Ho, D.W.C. Local and global synchronization in general complex dynamical networks with delay coupling. Chaos Solitons Fractals
**2008**, 37, 1497–1510. [Google Scholar] - Song, Q.; Cao, J.; Liu, F. Synchronization of complex dynamical networks with nonidentical nodes. Phys. Lett. A
**2010**, 374, 544–551. [Google Scholar] - Sun, W.; Yang, Y.; Li, C.; Liu, Z. Synchronization inside complex dynamical networks with double time-delays and nonlinear inner-coupling functions. Int. J. Mod. Phys. B
**2011**, 25, 1531–1541. [Google Scholar] - Yang, X.; Cao, J. Synchronization of Markovian coupled neural networks with nonidentical node-delays and random coupling strengths. IEEE Trans. Neural Netw. Learn. Syst
**2012**, 23, 60–71. [Google Scholar] - Yu, W.; Chen, G.; Lü, J.; Kurths, J. Synchronization via pinning control on general complex networks. SIAM J. Control Optim
**2013**, 51, 1395–1416. [Google Scholar] - He, W.; Cao, J. Exponential synchronization of hybrid coupled networks with delayed coupling. IEEE Trans. Neural Netw
**2010**, 21, 571–583. [Google Scholar] - Yang, Y.; Cao, J. Exponential synchronization of the complex dynamical networks with a coupling delay and impulsive effects. Nonlinear Anal. Real World Appl
**2010**, 11, 1650–1659. [Google Scholar] - Yang, X.; Cao, J.; Lu, J. Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control. IEEE Trans. Circuits Syst. I Regul. Pap
**2012**, 59, 371–384. [Google Scholar] - Wang, X.; Chen, G. Synchronization in small-world dynamical networks. Int. J. Bifurc. Chaos
**2002**, 12, 187–192. [Google Scholar] - Wang, X.; Chen, G. Synchronization in scale free dynamical networks: Robustness and fragility. IEEE Trans. Circuits Syst. I Fundam. Theory Appl
**2002**, 49, 54–62. [Google Scholar] - Lu, J.; Cao, J. Adaptive synchronization in tree-like dynamical networks. Nonlinear Anal. Real World Appl
**2007**, 8, 1252–1260. [Google Scholar] - Lu, J.; Cao, J. Adaptive synchronization of uncertain dynamical networks with delayed coupling. Nonlinear Dyn
**2008**, 53, 107–115. [Google Scholar] - Li, L.; Ho, D.W.C.; Lu, J. A unified approach to practical consensus with quantized data and time delay. IEEE Trans. Circuits Syst. I Regul. Pap
**2013**, 60, 2668–2678. [Google Scholar] - Cao, J.; Li, L. Cluster synchronization in an array of hybrid coupled neural networks with delay. Neural Netw
**2009**, 22, 335–342. [Google Scholar] - Li, C.; Sun, W.; Kurths, J. Synchronization between two coupled complex networks. Phys. Rev. E
**2007**, 76, 046204. [Google Scholar] [CrossRef] - Li, Z.; Xue, X. Outer synchronization of coupled networks using arbitrary coupling strength. Chaos
**2010**, 20, 023106. [Google Scholar] [CrossRef] - Tang, H.; Chen, L.; Lu, J.; Tse, C.K. Adaptive synchronization between two complex networks with nonidentical topological structures. Physica A
**2008**, 387, 5623–5630. [Google Scholar] - Zheng, S.; Bi, Q.; Cai, G. Adaptive projective synchronization in complex networks with time-varying coupling delay. Phys. Lett. A
**2009**, 373, 1553–1559. [Google Scholar] - Wang, G.; Cao, J.; Lu, J. Outer synchronization between two nonidentical networks with circumstance noise. Physica A
**2010**, 389, 1480–1488. [Google Scholar] - Wu, X.; Zheng, W.; Zhou, J. Generalized outer synchronization between complex dynamical networks. Chaos
**2009**, 19, 013109. [Google Scholar] [CrossRef] - Shang, Y.; Chen, M.; Kurths, J. Generalized synchronization of complex networks. Phys. Rev. E
**2009**, 80, 027201. [Google Scholar] [CrossRef] - Wu, Y.; Li, C.; Wu, Y.; Kurths, J. Generalized synchronization between two different complex networks. Commun. Nonlinear Sci. Numer. Simul
**2012**, 17, 349–355. [Google Scholar] - Wu, Y.; Li, C.; Yang, A.; Song, L.; Wu, Y. Pinning adaptive anti-synchronization between two general complex dynamical networks with non-delayed and delayed coupling. Appl. Math. Comput
**2012**, 218, 7445–7452. [Google Scholar] - Yin, S.; Ding, S.X.; Xie, X.; Luo, H. A review on basic data-driven approaches for industrial process monitoring. IEEE Trans. Ind. Electron
**2014**, 61, 6418–6428. [Google Scholar] - Yin, S.; Li, X.; Gao, H.; Kaynak, O. Data-based techniques focused on modern industry: An overview. IEEE Trans. Ind. Electron
**2015**, 62, 657–667. [Google Scholar] - Yin, S.; Huang, Z. Performance monitoring for vehicle suspension system via fuzzy positivistic C-means clustering based on accelerometer measurements. IEEE/ASME Trans. Mechatron
**2014**. [Google Scholar] [CrossRef] - Yin, S.; Zhu, X.; Kaynak, O. Improved PLS focused on key performance indictor related fault diagnosis. IEEE Trans. Ind. Electron
**2015**, 62, 1651–1658. [Google Scholar] - Yin, S.; Wang, G.; Yang, X. Robust PLS approach for KPI-related prediction and diagnosis against outliers and missing data. Int. J. Syst. Sci
**2014**, 45, 1375–1382. [Google Scholar] - Liu, H.; Lu, J.; Lü, J.; Hill, D.J. Structure identification of uncertain general complex dynamical networks with time delay. Automatica
**2009**, 45, 1799–1807. [Google Scholar] - Xu, Y.; Zhou, W.; Fang, J.; Sun, W.; Pan, L. Topology identification and adaptive synchronization of uncertain complex networks with non-derivative and derivative coupling. J. Frankl. Inst
**2010**, 347, 1566–1576. [Google Scholar] - Zhang, Q.; Lu, J. Exponentially adaptive synchronization of an uncertain delayed dynamical network. J. Syst. Sci. Complex
**2011**, 24, 207–217. [Google Scholar] - Che, Y.; Li, R.; Han, C.; Wang, J.; Cui, S.; Deng, B.; Wei, X. Adaptive lag synchronization based topology identification scheme of uncertain general complex dynamical networks. Eur. Phys. J. B
**2012**, 85. [Google Scholar] [CrossRef] - Sun, W.; Li, S. Generalized outer synchronization between two uncertain dynamical networks. Nonlinear Dyn
**2014**, 77, 481–489. [Google Scholar] - Che, Y.; Li, R.; Han, C.; Cui, S.; Wang, J.; Wei, X.; Deng, B. Topology identification of uncertain nonlinearly coupled complex networks with delays based on anticipatory synchronization. Chaos
**2013**, 23, 013127. [Google Scholar] [CrossRef] - Wu, X.; Lu, H. Outer synchronization of uncertain general complex delayed networks with adaptive coupling. Neurocomputing
**2012**, 82, 157–166. [Google Scholar] - Cheng, C.; Liao, T.; Hwang, C. Exponential synchronization of a class of chaotic neural networks. Chaos Solitons Fractals
**2005**, 24, 197–206. [Google Scholar] - Khalil, H.K. Nonlinear Systems, 2nd ed; Prentice Hall: Upper Saddle River, NJ, USA, 2002. [Google Scholar]
- Hindmarsh, J.L.; Rose, R.M. A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. Ser. B
**1984**, 221, 87–102. [Google Scholar] - Shi, X.; Lu, Q. Firing patterns and complete synchronization of coupled Hindmarsh-Rose neurons. Chin. Phys
**2005**, 14, 77–85. [Google Scholar]

**Figure 2.**Identification of unknown parameters θ

_{i}(i = 1,…, 5) of networks Equations (3) and (4).

**Figure 3.**Evolution of adaptive feedback gains d

_{i}(i = 1,…, 5) of networks Equations (3) and (4).

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Wu, Y.; Liu, L.
Exponential Outer Synchronization between Two Uncertain Time-Varying Complex Networks with Nonlinear Coupling. *Entropy* **2015**, *17*, 3097-3109.
https://doi.org/10.3390/e17053097

**AMA Style**

Wu Y, Liu L.
Exponential Outer Synchronization between Two Uncertain Time-Varying Complex Networks with Nonlinear Coupling. *Entropy*. 2015; 17(5):3097-3109.
https://doi.org/10.3390/e17053097

**Chicago/Turabian Style**

Wu, Yongqing, and Li Liu.
2015. "Exponential Outer Synchronization between Two Uncertain Time-Varying Complex Networks with Nonlinear Coupling" *Entropy* 17, no. 5: 3097-3109.
https://doi.org/10.3390/e17053097