Synchronization in an array of coupled neural networks with delayed impulses: Average impulsive delay method
Introduction
Synchronization of coupled neural networks has been widely studied during the past decade due to the fact that it may contribute to the parallel image processing and pattern storage and retrieval (Arenas et al., 2008, Ding et al., 2017, Lu and Chen, 2006, Lu et al., 2012, Watts and Strogatz, 1998, Yang, Wang et al., 2018). Moreover, some interesting work on dynamical coupled networks can be seen in Huang et al., 2017, Huang et al., 2013, Li, Liu et al., 2019, Niu et al., 2018, Rakkiyappan et al., 2017, Xu et al., 2017, Yang, Song et al., 2018, Zhang and Zeng, 2019 and Zhu and Cao (2012). In practice, the states of nodes may have abrupt jump at certain instants, which is known as impulsive effects (Benchohra et al., 2006, Lakshmikantham and Simeonov, 1989, Nesic and Teel, 2004). It is known that impulsive effects extensively emerge in biological networks, and these networks can be modeled by a class of impulsive differential equations (Hu et al., 2019, Kan et al., 2019, Liu et al., 2018, Liu et al., 2019, Zhu, 2014). Because impulses may have effect on the performance of the networks, some significant work has been accomplished on synchronization problem of impulsive dynamical networks (Li, 2017, Li, Lou et al., 2018, Lu et al., 2011, Wang et al., 2019, Yang et al., 2017, Yang, Lu et al., 2018).
Recently, many investigators devote to investigating impulsive systems with delay-dependent impulses (see Chen et al., 2014, Chen and Zheng, 2009, Fu et al., 2019, Li and Song, 2016). To some degree, this may result from the fact that the time delay is inevitable in digital sampling and output of pulse signal. Specifically, certain interesting work on synchronization problem of coupled neural networks with delayed impulses have been obtained (Chen et al., 2014, Liu and Zhang, 2016, Wang, Liu et al., 2018, Yang et al., 2015, Yang, Wang et al., 2018). It is known that the time delay in continuous dynamics can bring either positive or negative effect on the certain dynamical behavior. For example, if the time delay becomes larger in continuous systems, then the system may become unstable, while on the other hand, it may bring certain stabilizing effect and good properties to a class of unstable systems (Gu, Chen, & Kharitonov, 2003). From the aspect of stability, the time delay in impulses can also play double roles, that is, positive or/and negative roles (Li, Song & Wu, 2019). Although there exists some work on the synchronization problem concerning delayed impulses, they only consider the negative impacts of the delay in impulses (Chen et al., 2014, Liu and Zhang, 2016, Wang, Liu et al., 2018, Yang et al., 2015, Yang, Wang et al., 2018).
Recall the positive effect of the time delay in impulses, it is natural to ask under what condition such time delay can bring synchronizing impact? It is crucial and challenging to derive synchronization criteria for dynamical networks with delayed impulses and meanwhile the time delay can bring synchronizing impact. To the best of our knowledge, there exists no available criterion for synchronization of such networks with delayed impulses, in which the time delay can bring synchronizing impact to the networks.
It is worth noting that in previous results, the time delays in impulses are confined by some strong constraints, e.g., the time delays are required to be fixed or their upper bound should be small enough in Chen et al., 2014, Chen and Zheng, 2009 and Li and Song (2016), and thus in a certain sense, the corresponding criteria may be conservative. In real-world cases, time delays in impulses are not always fixed or small enough, however, they may be flexible, beyond the impulsive interval and even unbounded. Unfortunately, there exists less work on synchronization problem with delayed impulses, in which the time delays are allowed to be flexible and larger than the length of impulsive interval. Additionally, it is known that the case of unbounded time delay in continuous dynamics has been well considered (Li and Cao, 2017, Li, Song and Wu, 2018), but to the best of our knowledge, there is no available work on the case of unbounded time delay in discrete dynamics, i.e., in impulses. Thus, the synchronization problem of impulsive networks with delayed impulses has not been fully studied so far. Therefore, it still remains as a significant and open problem.
Inspired by the above existing problems and discussions, the method of average impulsive delay (AID) is proposed to describe the delays in impulses overall. Based on the ideas of AII and AID, which will be introduced in detail in the next section, some effective synchronization criteria are derived. We illustrate that the time delay in impulses can play double roles, namely, it may desynchronize a synchronous network or synchronize a nonsynchronized network, which will be systematically and fully investigated in this paper. Moreover, the case of unbounded delays in impulses is further studied. The main contributions of this paper are listed as follows.
- (1)
In this paper, the method of AID is proposed and further, synchronization in an array of coupled neural networks with delayed impulses is fully considered. By the ideas of AII and AID, certain globally exponential synchronization criteria are derived.
- (2)
It is shown that our results possess stronger robustness and less conservativeness than existing results, because our results do not require that the time delays in impulses are fixed or the upper bound should be small enough. Conversely, the time delays can be flexible and even larger than the length of impulsive interval.
- (3)
Further, a unified relationship is established among the AII, AID and rate coefficients of the impulsive dynamical network such that the network is globally exponentially synchronized (GES). We prove that the time delay in impulses can play double roles, concretely, it may desynchronize a synchronous network or synchronize a nonsynchronized network.
- (4)
The case of average impulsive delay , which implies that the time delays in impulses are unbounded, is also considered. Moreover, a synchronization criterion is derived for such networks with delay-unbounded impulses.
The remainder of the paper is organized as follows. In Section 2, we formulate the problem, and necessary notations and definitions are given. In Section 3, the main results are presented. In Section 4, two examples are proposed and summaries are given in Section 5.
Notations In what follows, we assume that matrices are with compatible dimensions if not explicitly stated. Let denote the set of real numbers, the set of positive real numbers, the set of positive integer numbers, and the -dimensional real-valued vectors and -dimensional real matrices, respectively. Let the notation denote the Euclidean norm of vectors, i.e., . Additionally, let or denote that the matrix is a positive or negative definite matrix, and denote the identity matrix with dimensions. represents the largest eigenvalue of the corresponding matrix. The notation “T” denotes the transpose of a matrix or a vector. Let represent the maximum integer of no more than . denotes the Kronecker-product.
Section snippets
Model description and preliminaries
Consider an array of coupled neural networks composed of coupled nodes and each of the nodes is an -dimensional neural network. We describe the th node as following system: where presents the state vector of the th neural network; and . Assume that neural networks are coupled together and impulsive effects are considered, and then it yields the impulsive dynamical network as
Main results
For convenience, we denote and . Then, we rewrite the impulsive dynamical networks in the following Kronecker-product form: where .
Lemma 2 Suppose . If then it yields that: (a) Assume that is the left eigenvector of Lu & Chen, 2006
Examples
Two examples are presented to show the validity of derived results in this section. Consider a chaotic system as the single node of the network, which is proposed as follows (Lu et al., 2010): where presents the state vector, and the parameters are , and function . Hence, the Lipschitz constants can be set as . System (43) has a chaotic
Conclusion
In this paper, synchronization criteria were investigated for nonlinear impulsive dynamical networks with delayed impulses. Specifically, we considered two cases, namely, (1) average impulsive delay ; (2) average impulsive delay , which implies that the time delays in impulses are unbounded. In order to describe the delays in impulses overall, the method of average impulsive delay (AID) was proposed. By the ideas of average impulsive interval (AII) and AID, some criteria for exponential
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant Nos. 61973078, the Natural Science Foundation of Jiangsu Province of China under Grant No. BK20170019, and the Fundamental Research Funds for the Central Universities under Grant No. 2242019k1G013, ”333 Engineering” Foundation of Jiangsu Province of China under Grant No. BRA2019260, and Jiangsu Province Six Talent Peaks Project, China under Grant No. 2015-ZNDW-002.
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