# Stability Analysis and Synchronization for a Class of Fractional-Order Neural Networks

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- By utilizing Laplace transform techniques, The boundness and convergence of solution for FONN are investigated.
- A linear controller is designed for synchronizing fractional chaotic networks. Integration of the sign function is utilized in our control methods, so chattering phenomenon can be avoided.
- A simple auxiliary function is constructed, which may be helpful for stability analysis of fractional-order systems.

## 2. Preliminaries

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Lemma**

**4.**

**Lemma**

**5.**

**Lemma**

**6.**

## 3. Main Results

#### 3.1. System Description

**Definition**

**1.**

**Assumption**

**1.**

**Assumption**

**2.**

**Remark**

**1.**

#### 3.2. Stability Analysis

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

**Remark**

**4.**

**Theorem**

**2.**

**Proof.**

#### 3.3. Synchronization

**Theorem**

**3.**

**Proof.**

## 4. Simulation Results

**Example**

**1.**

**Figure 1.**Boundedness of the solution $x\left(t\right)$ for fractional-order neural network (44) with ${c}_{1}=0.3$ and ${c}_{2}=0.35$.

**Figure 2.**Convergence of the solution $x\left(t\right)$ for fractional-order neural network (44) with ${c}_{1}=2$ and ${c}_{2}=3$.

**Example**

**2.**

**Figure 3.**Time responses of $x\left(t\right)$ of system (16) with initial value ${[-0.3,0.4,0.3]}^{T}.$

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Cao, J.; Liang, J. Boundedness and stability for Cohen–Grossberg neural network with time-varying delays. J. Math. Anal. Appl.
**2004**, 296, 665–685. [Google Scholar] [CrossRef] - Cao, J.; Wang, J. Global asymptotic and robust stability of recurrent neural networks with time delays. IEEE Trans. Circuits Syst. I Regul. Pap.
**2005**, 52, 417–426. [Google Scholar] - Zhang, H.; Wang, Z.; Liu, D. Global asymptotic stability and robust stability of a class of Cohen–Grossberg neural networks with mixed delays. IEEE Trans. Circuits Syst. I Regul. Pap.
**2009**, 56, 616–629. [Google Scholar] [CrossRef] - Song, C.; Cao, J. Dynamics in fractional-order neural networks. Neurocomputing
**2014**, 142, 494–498. [Google Scholar] [CrossRef] - Wang, H.; Yu, Y.; Wen, G. Stability analysis of fractional-order hopfield neural networks with time delays. Neural Netw.
**2014**, 55, 98–109. [Google Scholar] [CrossRef] [PubMed] - Lazarević, M. Stability and stabilization of fractional order time delay systems. Sci. Tech. Rev.
**2011**, 61, 31–45. [Google Scholar] - Liu, H.; Li, S.; Wang, H.; Huo, Y.; Luo, J. Adaptive synchronization for a class of uncertain fractional-order neural networks. Entropy
**2015**, 17, 7185–7200. [Google Scholar] [CrossRef] - Anastassiou, G.A. Fractional neural network approximation. Comput. Math. Appl.
**2012**, 64, 1655–1676. [Google Scholar] [CrossRef] - Liu, H.; Li, S.; Sun, Y.; Wang, H. Prescribed performance synchronization for fractional-order chaotic systems. Chin. Phys. B
**2015**, 24, 153–160. [Google Scholar] [CrossRef] - Chen, L.; Chai, Y.; Wu, R.; Ma, T.; Zhai, H. Dynamic analysis of a class of fractional-order neural networks with delay. Neurocomputing
**2013**, 111, 190–194. [Google Scholar] [CrossRef] - Liu, H.; Li, S.; Sun, Y.; Wang, H. Adaptive fuzzy synchronization for uncertain fractional-order chaotic systems with unknown non-symmetrical control gain. Acta Phys. Sinaca
**2015**, 64, 70503. [Google Scholar] - Kaslik, E.; Sivasundaram, S. Nonlinear dynamics and chaos in fractional-order neural networks. Neural Netw.
**2012**, 32, 245–256. [Google Scholar] [CrossRef] [PubMed] - Arena, P.; Caponetto, R.; Fortuna, L.; Porto, D. Bifurcation and chaos in noninteger order cellular neural networks. Int. J. Bifurc. Chaos
**1998**, 8, 1527–1539. [Google Scholar] [CrossRef] - Huang, X.; Zhao, Z.; Wang, Z.; Li, Y. Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing
**2012**, 94, 13–21. [Google Scholar] [CrossRef] - Boroomand, A.; Menhaj, M.B. Fractional-order hopfield neural networks. In Advances in Neuro-Information Processing; Springer-Verlag: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Chen, J.; Zeng, Z.; Jiang, P. Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw.
**2014**, 51, 1–8. [Google Scholar] [CrossRef] [PubMed] - Wen, G.; Hu, G.; Yu, W.; Cao, J.; Chen, G. Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs. Sys. Control Lett.
**2013**, 62, 1151–1158. [Google Scholar] [CrossRef] - Wen, G.; Hu, G.; Yu, W.; Chen, G. Distributed consensus of higher order multiagent systems with switching topologies. IEEE Trans. Circuits Syst. II Express Br.
**2014**, 61, 359–363. [Google Scholar] - Chen, L.; Chai, Y.; Wu, R.; Yang, J. Stability and stabilization of a class of nonlinear fractional-order systems with caputo derivative. IEEE Trans. Circuits Syst. II Express Br.
**2012**, 59, 602–606. [Google Scholar] [CrossRef] - Ahn, H.S.; Chen, Y. Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica
**2008**, 44, 2985–2988. [Google Scholar] [CrossRef] - Shen, J.; Lam, J. Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica
**2014**, 50, 547–551. [Google Scholar] [CrossRef] - Zheng, C.; Li, N.; Cao, J. Matrix measure based stability criteria for high-order neural networks with proportional delay. Neurocomputing
**2015**, 149, 1149–1154. [Google Scholar] [CrossRef] - Nie, X.; Zheng, W.; Cao, J. Multistability of memristive Cohen–Grossberg neural networks with non-monotonic piecewise linear activation functions and time-varying delays. Neural Netw.
**2015**, 71, 27–36. [Google Scholar] [CrossRef] [PubMed] - Matignon, D. Stability results for fractional differential equations with applications to control processing. Comput. Eng. Syst. Appl.
**1996**, 2, 963–968. [Google Scholar] - Farges, C.; Moze, M.; Sabatier, J. Pseudo-state feedback stabilization of commensurate fractional order systems. Automatica
**2010**, 46, 1730–1734. [Google Scholar] [CrossRef] - Ren, F.; Cao, F.; Cao, J. Mittag–Leffler stability and generalized Mittag–Leffler stability of fractional-order gene regulatory networks. Neurocompu.
**2015**, 160, 185–190. [Google Scholar] [CrossRef] - Rakkiyappan, R.; Velmurugan, G.; Cao, J. Stability analysis of fractional-order complex-valued neural networks with time delays. Chaos Solitons Fractals
**2015**, 78, 297–316. [Google Scholar] [CrossRef] - Trigeassou, J.C.; Maamri, N.; Sabatier, J.; Oustaloup, A. A lyapunov approach to the stability of fractional differential equations. Signal Process.
**2011**, 91, 437–445. [Google Scholar] [CrossRef] - Yu, J.; Hu, C.; Jiang, H. α-stability and α-synchronization for fractional-order neural networks. Neural Netw.
**2012**, 35, 82–87. [Google Scholar] [CrossRef] [PubMed] - Li, K.; Peng, J.; Gao, J. A comment on α-stability and α-synchronization for fractional-order neural networks. Neural Netw.
**2013**, 48, 207–208. [Google Scholar] - Wang, K.; Teng, Z.; Jiang, H. Adaptive synchronization in an array of linearly coupled neural networks with reaction–diffusion terms and time delays. Commun. Nonlinear Sci. Numer. Simul.
**2012**, 17, 3866–3875. [Google Scholar] [CrossRef] - Zhang, D.; Xu, J. Projective synchronization of different chaotic time-delayed neural networks based on integral sliding mode controller. Appl. Math. Comput.
**2010**, 217, 164–174. [Google Scholar] [CrossRef] - Yang, X.; Cao, J. Stochastic synchronization of coupled neural networks with intermittent control. Phys. Lett. A
**2009**, 373, 3259–3272. [Google Scholar] [CrossRef] - Zhang, G.; Shen, Y.; Wang, L. Global anti-synchronization of a class of chaotic memristive neural networks with time-varying delays. Neural netw.
**2013**, 46, 1–8. [Google Scholar] [CrossRef] [PubMed] - Lu, J.; Ho, D.; Cao, J. A unified synchronization criterion for impulsive dynamical networks. Automatica
**2010**, 4, 1215–1221. [Google Scholar] [CrossRef] - Podlubny, I. Fractional differential equations. Soc. Ind. Appl. Math.
**2000**, 42, 766–768. [Google Scholar] - Luo, J.; Li, G.; Liu, H. Linear control of fractional-order financial chaotic systems with input saturation. Discret. Dyn. Nature Soc.
**2014**. [Google Scholar] [CrossRef] - Li, Y.; Chen, Y.; Podlubny, I. Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica
**2009**, 45, 1965–1969. [Google Scholar] [CrossRef] - Aguila-Camacho, N.; Duarte-Mermoud, M.A.; Gallegos, J.A. Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul.
**2014**, 19, 2951–2957. [Google Scholar] [CrossRef]

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Li, G.; Liu, H.
Stability Analysis and Synchronization for a Class of Fractional-Order Neural Networks. *Entropy* **2016**, *18*, 55.
https://doi.org/10.3390/e18020055

**AMA Style**

Li G, Liu H.
Stability Analysis and Synchronization for a Class of Fractional-Order Neural Networks. *Entropy*. 2016; 18(2):55.
https://doi.org/10.3390/e18020055

**Chicago/Turabian Style**

Li, Guanjun, and Heng Liu.
2016. "Stability Analysis and Synchronization for a Class of Fractional-Order Neural Networks" *Entropy* 18, no. 2: 55.
https://doi.org/10.3390/e18020055