 Research
 Open Access
Stability and Hopf bifurcation analysis of fractionalorder complexvalued neural networks with time delays
 R Rakkiyappan^{1}Email author,
 K Udhayakumar^{1},
 G Velmurugan^{1},
 Jinde Cao^{2, 3}Email author and
 Ahmed Alsaedi^{3}
https://doi.org/10.1186/s1366201712663
© The Author(s) 2017
 Received: 8 March 2017
 Accepted: 6 July 2017
 Published: 4 August 2017
Abstract
This paper considers a class of fractionalorder complexvalued Hopfield neural networks (CVHNNs) with time delay for analyzing the dynamic behaviors such as local asymptotic stability and Hopf bifurcation. In the case of a neural network with hub and ring structure, the stability of the equilibrium state is investigated by analyzing the eigenvalue of the corresponding characteristic matrix for the hub and ring structured fractionalorder time delay models using a Laplace transformation for the Caputofractional derivatives. Some sufficient conditions are established to guarantee the uniqueness of the equilibrium point. In addition, conditions for the occurrence of a Hopf bifurcation are also presented. Finally, numerical examples are given to demonstrate the effectiveness of the derived results.
Keywords
 Hopfield neural networks
 fractionalorder
 time delays
 hub structure
 ring structure
 stability
 Hopf bifurcation
1 Introduction
The discipline of neural networks, as other fields of science, has a long history of evolution with lots of ups and downs. In 1943 Warren McCulloch and Walter Pitts presented the first model of artificial neurons, named the Threshold Logic Unit (TLU). In the last few decades, the subjective analysis of neural networks (NNs) has received huge attention because of its strong applications in numerous fields such as signal and image processing, associative memories, combinatorial optimization and many others [1–6]. However, such practical applications of NNs are strongly dependent on the qualitative behaviors of NNs. In both biological and physical models, the occurrence of time delays plays an important role. Time delay, which happens usually due to system process and information flow to a particular part of dynamical systems, is unavoidable. Time delays in NNs may cause unexpected dynamical behaviors, like oscillation and poor performance, in networks; see [7–11]. Thus, the analysis on NNs with the effects of time delays has attracted the attention of many researchers and results have been published [12–17]. The stability of neural networks with both leakage delay and a reactiondiffusion term is discussed, and several sufficient conditions were obtained with the help of analysis technique and Lyapunov theory [16]. The problem of fixedtime synchronization of memristive neural networks was studied in [17].
Fractional calculus was introduced many years ago [8, 9, 18, 19], but in recent years the development of fractional calculus was improved step by step in many fields such as engineering mathematics, applied mathematics, and many fields of physics. Thus, recently researchers were strongly attracted towards fractionalorder systems compared with the integerorder one because of their great advantage of infinite memory and hereditary properties [3, 19–22]. In the field of neural networks, since the importance of the memory term is extremely high, the incorporation of fractional terms into neural networks leads to a new class of networks called fractionalorder NNs [23]. Fractionalorder neural networks with parameters such as state vectors, nonlinear activation functions and weight functions in a complex domain \(\mathbb{C}^{n}\) are called fractionalorder complexvalued neural networks (CVNNs) [3, 7, 24–28]. CVNNs has more remarkable applications in various dynamical systems, such as Lyapunov asymptotic stability, global stability and Hopf bifurcations compared with the realvalued NNs [8, 18, 29–32]. In realvalued NNs, continuously differentiable and bounded sigmoid activation functions are to be chosen, but in the case of a complex domain the activation functions are entire bounded, and they will reduce to a constant in the complex domain according to the Liouville theorem [18]. Therefore in the complex domain choosing the appropriate activation function is a great challenge [7, 33].
NNs contain several types of structures such as hub structure, ring structure and so on. In NNs, for appropriate neuronal wiring, it is necessary to have coordinated activation of neuronal assemblies. Hypothetically a few highly linked neurons with long ranging connectivity, the ‘hub neurons’, would be the most effective manner to organize networkwide synchronicity. Therefore, it is significant to consider the hub structure in neural networks to realize better performance. Moreover, ring architectures can be found in a number of neural structures from neurons in the mind and other sciences. The factual cortical bonding prototype is tremendously sparse. Almost all connections among adjacent cells and long range links became very rare. Thus, ring NNs can be analyzed to gain better understanding of the mechanisms underlying the performance of NNs. Many authors have discussed global stability, Lyapunov stability for the fractionalorder NNs, see [2, 14, 27, 29, 33–37], and for finitetime stability results are discussed in [26] and bifurcation of a delayed fractional systems is considered in [5, 38–40].
Stability theory is one of the most important and rapidly developing fields of applied mathematics and mechanics. In the design of a dynamical neural network it is always of interest to study the stability properties of the network. The stability studies in NNs have been developed finding the conditions which ensure that each trajectory of the network converges to an equilibrium point depending on the initial conditions. These completely stable NNs have been used as computing and cognitive machines. Moreover, it is well known that the dynamic behaviors such as periodic phenomenon, bifurcation and chaos are also of great interest. In general, in a dynamical system, if a parameter is allowed to vary, then the behavior of the whole dynamical system may change. The value of the parameter at which these changes occur is known as the bifurcation value and the parameter that is varied is known as the bifurcation parameter. In delay differential equations, Hopf bifurcation occurs in systems of differential equations consisting of two or more equations. Even though there is an extensive literature on bifurcation analysis of some special NNs, most of them deal only with the twoequation models and there were only few papers on the bifurcations of the highdimensional models [41–46]. However, all above mentioned works are on the bifurcation analysis of integerorder NNs. Nevertheless, most results on the integerorder NNs cannot be simply extended to the case of the fractionalorder one. To the best of our knowledge, no work on the Hopf bifurcation analysis of fractionalorder CVHNNs with hub structure and ring structure has been proposed in the literature. Therefore, this paper aims at fulfilling such a gap by investigating the stability and Hopf bifurcation of a class of delayed fractionalorder CVHNNs with hub structure and ring structure.
Motivated by the above discussion, in this paper we investigate the stability and Hopf bifurcation of the fractionalorder CVHNNs with time delays in two types of structures named ring and hub structures. First we consider the fractionalorder CVHNNs in the hub structure with time delays. Later the sufficient conditions for stability of the fractional system with respect to the equilibrium point of the system are derived by using some inequality techniques. Further, the bifurcation point of the hub structure with the corresponding critical frequency of the system and for the case similar to the ring structured one are derived. By using the bifurcation point and critical frequency the transversality condition is verified for the point of bifurcation occurring.
The rest of the paper is organized as follows. Necessary basic definitions and a problem description are given in Section 2. In Section 3, the two structures of fractionalorder CVHNNs with time delays to investigate the stability and Hopf bifurcation point of the system are presented and two numerical examples are to validate the efficacy of our theoretical findings in Section 4 and lastly a conclusion is discussed in Section 5.
Notations and preliminaries. The authors have derived the results using the following notations in this paper. \(\mathbb{R}^{n}\) and \(\mathbb{C}^{n}\) are ndimensional euclidean space and ndimensional complex domain, respectively. \(z(t)\) is a complex variable and is denoted \(z(t) = x(t) + i y(t)\), which is in the complex domain. \({}_{0}^{C} D_{t}^{\alpha}z(t)\) denotes the Caputofractional derivative of the complex variable \(z(t)\). There are several definitions of fractionalorder derivatives and fractional integrals are extensively used, named the RiemannLiouville derivative and integrals, the Caputo derivative, and GrunwaldLetnikov derivative etc. In this section we give the definitions of the first two.
Definition 1
[6]
Definition 2
[13]
Definition 3
[47]
Definition 4
[13]
2 Model description
Lemma 1
[3]
If all the roots of the characteristic equation \(\operatorname {Det}(J(s))\) have negative real parts, then the zero equilibrium of system (1) is Lyapunov globally asymptotically stable.
Lemma 2
[3]
If the order of the system (1) lies between 0 and 1 all the characteristic roots of the matrix \(J(s)\) satisfy \(\vert \arg( \lambda) \vert >\frac{\alpha\pi}{2}\) and the characteristic equation \(\operatorname {Det}(J(s))\) has no purely imaginary roots for any \(\gamma_{rm}>0\), \(r=m=1,2,3,\ldots,n\), then the zero equilibrium solution of system (1) is Lyapunov globally asymptotically stable.
 \((\mathbb{H}1)\) :

The nonlinear activation functions \(f_{q}\) (\(q=1,2,3,\ldots,n\)) satisfies the Lipschitz condition, that is, there exist constants \(F_{k} > 0 \) such that$$ \bigl\vert f_{q}(x)f_{q}(y) \bigr\vert \leq F_{k} \vert xy \vert , \quad\mbox{for all } x,y\in \mathbb{R}. $$
 \((\mathbb{H}2)\) :

There exist constants \(\zeta_{p}\) (\(p=1,2,3,\ldots,n\)) such that the following inequality holds:$$ \zeta_{p} c_{p} > \sum_{k=1}^{n} \zeta _{p} \bigl(F_{k} \vert a_{pk} \vert +F_{k} \vert b_{pk} \vert \bigr),\quad p=1,2,3,\ldots, n. $$
Lemma 3
[38]
If hypotheses \((\mathbb{H}1)\) and \((\mathbb{H}2)\) are satisfied then there exists a unique equilibrium point for the system (1).
Remark 1
From Lemma 3, we can conveniently discuss the existence and uniqueness of the equilibrium point for neural networks.
3 Main results
3.1 Fractionalorder CVHNNs with hub structure and time delays
 \((\mathbb{H}3)\) :

\(\Psi_{i} > 0\) (\(i=1,2\)).
Lemma 4
If \(\Psi_{1} > 0\) and \(\Psi_{2} > 0 \) holds, then the zero equilibrium point of the fractionalorder system (7) is asymptotically stable when \(\gamma=0 \).
Remark 2
The conditions \(\Psi_{1} > 0\) and \(\Psi_{2} > 0 \) are sufficient condition for Lemma 4. If the conditions are retrieved by another method which entails that all the roots of equation (13) satisfy \(\vert \arg (\lambda) \vert > \frac{\alpha\pi}{2}\) then Lemma 4 may still hold.
 \((\mathbb{H}4)\) :

\(\operatorname {Re}[ \frac{ds}{d \gamma} ]  _{( \gamma= \gamma_{0} , \omega= \omega_{0})} \neq0\), where \(\gamma_{0}\) and \(\omega_{0}\) are bifurcation point and critical frequency, respectively.
Theorem 1
 (1)
The zero equilibrium point is asymptotically stable for \(\gamma\in[0, \gamma_{0} ) \).
 (2)
The system (7) exhibits a Hopf bifurcation at the origin when \(\gamma=\gamma_{0} \), that is, the system (7) has a branch of periodic solutions bifurcating from the zero equilibrium point near \(\gamma=\gamma_{0} \).
Remark 3
The results on stability and Hopf bifurcation of fractionalorder CVHNNs with hub structure and time delays have not been attained before. The determined conditions on bifurcation are very straightforward, detailed and impressive and simple to be verified in the present work by applying Hopf bifurcation theory. Our work is to develop the study of the theory of nonlinear dynamics.
3.2 Fractionalorder CVHNNs with ring structure and time delays
 \((\mathbb{A}1)\) :

\(f_{p} \in\mathbb{C}^{1}(\mathbb {R}, \mathbb{R})\), \(f_{p}(0)=0\),
We know that \(\sin^{2}\omega\gamma+\cos^{2} \omega\gamma=1\), from this identity we obtain \(d_{1}^{2}(\omega)+d_{2}^{2}(\omega)=1 \).
 \((\mathbb{A}2)\) :

\(\chi_{i} > 0\) (\(i=1,2\)).
Lemma 5
If \(\chi_{1} > 0\) and \(\chi_{2} > 0 \), then the equilibrium point of the fractionalorder CVHNNs with ring structured and time delay system (30) is stable when \(\gamma=0 \).
 \((\mathbb{A}3)\) :

\(\operatorname {Re}[ \frac{ds}{d \gamma} ]  _{( \gamma= \gamma_{0} , \omega= \omega_{0})} \neq0\),
Theorem 2
 (1)
The zero equilibrium point is locally asymptotically stable for \(\gamma\in[0, \gamma_{0} ) \).
 (2)
The fractionalorder system (30) exhibits a Hopf bifurcation at the origin when \(\gamma=\gamma_{0} \), that is, the system (30) has a branch of periodic solutions bifurcating from the zero equilibrium point near \(\gamma=\gamma_{0} \).
4 Numerical examples
In this section, numerical examples are given to verify the effectiveness of the analytical results. The simulation results are based on the AdamsBashforthMoulton predictorcorrector scheme for considered examples. The step length is taken in all the examples as \(h=0.01\).
Example 1
Example 2
5 Conclusions
In this paper, the class of fractionalorder CVHNNs with hub and ring structured system is considered in a time delay sense. In this paper we use the Laplace transforms to the system of linearized equation to get the characteristic matrix for further stability analysis of the equilibrium which has been completely characterized. The critical frequency of the fractionalorder α for which the corresponding bifurcation point may occur has been identified and the transversality condition is verified for the corresponding \(\omega_{0}\) and \(\gamma_{0}\). Moreover, the obtained conditions are simple, precise, effective and easy to verify. Numerical examples are given to verify the effectiveness of the theoretical results. Future work will focus on the analysis of stability and Hopf bifurcation of highdimensional fractionalorder ring structure CVHNNs with multiple delays.
Declarations
Acknowledgements
This work was supported by CSIR research project No.25(0237)/14/EMRII.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Cochocki, A, Unbehauen, R: Neural Networks for Optimization and Signal Processing. Wiley, New York (1993) Google Scholar
 Gopalsamy, K, He, X: Stability in asymmetric Hopfield nets with transmission delays. Phys. D: Nonlinear Phenom. 76, 344358 (1994) MathSciNetView ArticleMATHGoogle Scholar
 Huang, C, Cao, J, Xiao, M, Alsaedi, A, Hayat, T: Bifurcations in a delayed fractional complexvalued neural network. Appl. Math. Comput. 292, 210227 (2017) MathSciNetGoogle Scholar
 Hirose, A: ComplexValued Neural Networks. Springer, Berlin (2006) View ArticleMATHGoogle Scholar
 Kim, T, Adali, T: Fully complex multi layer perceptron network for nonlinear signal processing. J. VLSI Signal Process. Syst. Signal Image Video Technol. 32, 2943 (2002) View ArticleMATHGoogle Scholar
 Lin, S, Lu, C: Laplace transform for solving some families of fractional differential equations and its applications. Adv. Differ. Equ. 2013, 137 (2013) MathSciNetView ArticleGoogle Scholar
 Li, X, Rakkiyappan, R, Velmurugan, G: Dissipativity analysis of memristorbased complexvalued neural networks with timevarying delays. Inf. Sci. 294, 645665 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Rakkiyappan, R, Cao, J, Velmurugan, G: Existence and uniform stability analysis of fractionalorder complexvalued neural networks with time delays. IEEE Trans. Neural Netw. Learn. Syst. 26, 8497 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Velmurugan, G, Rakkiyappan, R, Vembarasan, V, Cao, J, Alsaedi, A: Dissipativity and stability analysis of fractionalorder complexvalued neural networks with time delay. Neural Netw. 86, 4253 (2016) View ArticleGoogle Scholar
 Arik, S: Stability analysis of delayed neural networks. IEEE Trans. Circuits Syst. I 47, 10891092 (2000) MathSciNetView ArticleMATHGoogle Scholar
 Wang, H, Yu, Y, Wen, G, Zhang, S: Stability analysis of fractionalorder neural networks with time delay. Neural Process. Lett. 42, 479500 (2015) View ArticleGoogle Scholar
 Hopfield, JJ, Tank, DW: Neural computation of decisions in optimization problems. Biol. Cybern. 52, 141152 (1985) MATHGoogle Scholar
 Ding, Z, Shen, Y: Global dissipativity of fractionalorder neural networks with time delays and discontinuous activations. Neurocomputing 196, 159166 (2016) View ArticleGoogle Scholar
 Feng, C, Plamondon, R: On the stability analysis of delayed neural networks systems. Neural Netw. 14, 11811188 (2001) View ArticleGoogle Scholar
 Cao, J, Rakkiyappan, R, Maheswari, K, Chandrasekar, A: Exponential \(H_{\infty}\) filtering analysis for discretetime switched neural networks with random delays using sojourn probabilities. Sci. China, Technol. Sci. 59, 387402 (2016) View ArticleGoogle Scholar
 Li, R, Cao, J: Stability analysis of reactiondiffusion uncertain memristive neural networks with timevarying delays and leakage term. Appl. Math. Comput. 278, 5469 (2016) MathSciNetGoogle Scholar
 Cao, J, Li, R: Fixedtime synchronization of delayed memristorbased recurrent neural networks. Sci. China Inf. Sci. 60, 032201 (2017) View ArticleGoogle Scholar
 Chen, L, Qu, J, Chai, Y, Wu, R, Qi, G: Synchronization of a class of fractionalorder chaotic neural networks. Entropy 15, 32653276 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Kriesel, D: A Brief Introduction on Neural Networks. Citeseer (2007) Google Scholar
 Wang, Y, Li, T: Stability analysis of fractionalorder nonlinear systems with delay. Math. Probl. Eng. 2014, Article ID 301235 (2014) MathSciNetGoogle Scholar
 Xia, YS, Wang, J: A general projection neural network for solving monotone variational inequalities and related optimization problems. IEEE Trans. Neural Netw. 15, 318328 (2004) View ArticleGoogle Scholar
 Huang, C, Cao, J: Active control strategy for synchronization and antisynchronization of a fractional chaotic financial system. Phys. A, Stat. Mech. Appl. 473, 262275 (2017) MathSciNetView ArticleGoogle Scholar
 Kaslik, E, Sivasundaram, S: Nonlinear dynamics and chaos in fractionalorder neural networks. Neural Netw. 32, 245256 (2012) View ArticleMATHGoogle Scholar
 Huang, C, Cao, J, Xiao, M: Hybrid control on bifurcation for a delayed fractional gene regulatory network. Chaos Solitons Fractals 87, 1929 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Nitta, T: Orthogonality of decision boundaries in complexvalued neural networks. Neural Comput. 16, 7397 (2004) View ArticleMATHGoogle Scholar
 Rakkiyappan, R, Velmurugan, G, Cao, J: Finitetime stability analysis of fractionalorder complexvalued memristorbased neural networks with time delays. Nonlinear Dyn. 78, 28232836 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Rakkiyappan, R, Sivaranjani, K, Velmurugan, G: Passivity and passification of memristorbased complexvalued recurrent neural networks with interval timevarying delays. Neurocomputing 144, 391407 (2014) View ArticleGoogle Scholar
 Zhao, H: Global asymptotic stability of Hopfield neural network involving distributed delays. Neural Netw. 17, 4753 (2014) View ArticleMATHGoogle Scholar
 O’Kelly, ME: Routing traffic at hub facilities. Netw. Spat. Econ. 10, 173191 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Liu, C, Li, C, Huang, T, Li, C: Stability of Hopfield neural networks with time delays and variabletime impulses. Neural Comput. Appl. 22, 195202 (2013) View ArticleGoogle Scholar
 Song, C, Cao, J, Liu, Y: Robust consensus of fractionalorder multiagent systems with positive real uncertainty via secondorder neighbors information. Neurocomputing 165, 293299 (2015) View ArticleGoogle Scholar
 Song, C, Cao, J: Dynamics in fractionalorder neural networks. Neurocomputing 142, 494498 (2014) View ArticleGoogle Scholar
 Velmurugan, G, Rakkiyappan, R, Cao, J: Further analysis of global μstability of complexvalued neural networks with unbounded timevarying delays. Neural Netw. 67, 1427 (2015) View ArticleGoogle Scholar
 Chen, T: Global exponential stability of delayed Hopfield neural networks. Neural Netw. 14, 977980 (2001) View ArticleGoogle Scholar
 Fang, Y, Kincaid, TG: Stability analysis of dynamical neural networks. IEEE Trans. Neural Netw. 7, 9961006 (1996) View ArticleGoogle Scholar
 Liao, XF, Chen, G, Sanchez, EN: LMIbased approach for asymptotically stability analysis of delayed neural networks. IEEE Trans. Circuits Syst. I 49, 10331039 (2002) MathSciNetView ArticleGoogle Scholar
 Yu, W, Cao, J: Stability and Hopf bifurcation analysis on a fourneuron BAM neural network with time delays. Phys. Lett. A 351, 6478 (2006) View ArticleMATHGoogle Scholar
 Huang, C, Cao, J, Ma, Z: Delayinduced bifurcation in a trineuron fractional neural network. Int. J. Inf. Syst. Sci. 47, 36683677 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Xiao, M, Zheng, WX, Jiang, G, Cao, J: Undamped oscillations generated by Hopf bifurcations in fractionalorder recurrent neural networks with Caputo derivative. IEEE Trans. Neural Netw. Learn. Syst. 12, 32013214 (2015) MathSciNetView ArticleGoogle Scholar
 Zhang, J, Jin, X: Global stability analysis in delayed Hopfield neural network models. Neural Netw. 13, 745753 (2000) View ArticleGoogle Scholar
 Cao, J, Xiao, M: Stability and Hopf bifurcation in a simplified BAM neural network with two time delays. IEEE Trans. Neural Netw. 18, 416430 (2007) View ArticleGoogle Scholar
 Hu, H, Huang, L: Stability and Hopf bifurcation analysis on a ring of four neurons with delays. Appl. Math. Comput. 213, 587599 (2009) MathSciNetMATHGoogle Scholar
 Li, X, Wei, J: On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays. Chaos Solitons Fractals 26, 519526 (2005) MathSciNetView ArticleMATHGoogle Scholar
 Wei, J, Yuan, Y: Synchronized Hopf bifurcation analysis in a neural network model with delays. J. Math. Anal. Appl. 312, 205229 (2005) MathSciNetView ArticleMATHGoogle Scholar
 Xu, W, Hayat, T, Cao, J, Xiao, M: Hopf bifurcation control for a fluid flow model of Internet congestion control systems via state feedback. IMA J. Math. Control Inf. 33, 6993 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Huang, C, Meng, Y, Cao, J, Alsaedi, A, Alsaadi, FE: New bifurcation results for fractional BAM neural network with leakage delay. Chaos Solitons Fractals 100, 3144 (2017) MathSciNetView ArticleGoogle Scholar
 Rakkiyappan, R, Velmurugan, G, Cao, J: Stability analysis of fractionalorder complexvalued neural networks with time delays. Chaos Solitons Fractals 78, 297316 (2015) MathSciNetView ArticleMATHGoogle Scholar