Synchronization of Markovian jumping inertial neural networks and its applications in image encryption☆
Introduction
In the recent years, the studies on recurrent neural networks (RNNs), such as Hopfield neural network, cellular neural network, Cohen–Grossberg neural network, bidirectional associative neural network have attracted many researchers due to its potential applications in various fields such as processing of signals, processing of an image, pattern recognition and so on. In particular, the class of two-layer heteroassociative neural networks, known as bidirectional associative memory (BAM) type neural network model which was first introduced and studied by author in Kosko (1988). BAM neural network is a special class of RNNs, described by differential equations. Many new properties have been proposed in the literature because of its special structure in connection weights. Among many dynamical behaviors of the neural networks, stability and synchronization are the two important research topics that received much attention in the recent decade. In this regard, many authors have extensively investigated the dynamical characteristics of BAM neural networks and proposed numerous interesting results, for more details, one can refer Cao and Wang (2002), Feng and Plamondon (2002), Liang and Cao (2004) and Mohamad (2001). Obviously, stability properties are not the unique dynamical behavior, there are many other dynamical behaviors such as periodic oscillation, bifurcation and chaos are also need to be discussed. Especially, studies on periodic solutions are of great interest in secure communications. Also, if a BAM neural network is trained to mimic the human brain wave, then its dynamical behavior is required to depict periodic synchronous vibration or chaos. On the other hand, the synchronization hypothesis for brain activities has been confirmed by modern neurophysiological experiments such as visual cortex, thalamo-cortical systems, olfactory bulb and cortex, for more details refer Freeman (1978) and Gray (1994). Among various factors, the existence of time delays is considered to be one of the main factor, which causes the system to exhibit periodic/chaotic behaviors, refer Arik (2000) and Gao, Lam, and Chen (2006) and it occurs unavoidably owing to the finite switching speed of neurons and amplifiers. In the review of past research, various types of time delays in neural networks have been broadly investigated. RNNs have problems in catching long term dependencies in the input stream and this can be known as information latching problem (Bengio, Frasconi, & Simard, 1993). Information latching problem may be handled by extracting finite state representations (modes) from trained networks (Bollé et al., 1992, Casey, 1996, Cleeremans et al., 1989). Recently, authors in Tiňo, Čerňanskỳ, and Beňušková (2004) investigated the information switching (or jumping) between different modes, that is, jump from one mode to another which can be governed by a Markovian chain. Hence, in this study, BAM neural network with such jumping behavior is analyzed. Further, investigations on BAM neural network with Markovian jumping parameters always have a unique significance while modeling a class of neural networks with network modes. Numerous stability results are reported in the literature based on neural networks. For an instance, the stability and dissipativity have analyzed in Wu, Lam, Su, and Chu (2012), by considering the static delayed neural networks. Moreover the delay-dependent stability criteria were proposed in Hu, Gao, and Shi (2009) for delayed Cohen–Grossberg neural networks.
Studies on synchronization criteria for nonlinear dynamical systems have been attracted by many researchers in recent days due to their validated applications in secure communications and cryptography. For an instance, authors in Sheikhan, Shahnazi, and Garoucy (2013) and Wen, Zeng, Huang, Meng, and Yao (2015) investigated the synchronization of delayed chaotic systems and successfully applied it in the secure communication field (image communications). On the other hand, studies on synchronization problem of neural systems such as, neural networks, gene regulatory networks and memristor-based recurrent neural networks have become a hot research topic among researchers, for more details one can refer Bai, Lonngren, and Uçar (2005), Chandrasekar, Rakkiyappan, Cao, and Lakshmanan (2014), Chandrasekar, Rakkiyappan, and Jinde (2015), Chen and Hsu (2012), Rakkiyappan, Chandrasekar, Park, and Kwon (2014), Yang, Cao, and Lu (2013) and Zhang, Xie, Wang, and Zheng (2007). Especially, authors in Bai et al. (2005) investigated the synchronization criteria for switched neural networks via neural activation function and proposed the image encryption/decryption process through the solutions of the networks.
Observing that dozens of existing studies mainly focused on first derivative of states in neuronal system, whereas it is necessary to study the influence of inductance, into the artificial neural networks, equivalently an inertial term, regarded as a critical tool to produce complicated bifurcation behavior and chaos, see Babcock and Westervelt (1987), Coleman and Renninger (1976) and Wheeler and Schieve (1997). The authors in Liu, Liao, Guo, and Wu (2009) investigated the stability and existence of periodic solutions for inertial BAM neural networks with time delays. Besides that, the stability and synchronization have been investigated by the authors in Cao and Wan (2014) and Tu, Cao, and Hayat (2016) for inertial BAM neural network with time delay based on matrix measure and Halanay inequality. In Zhang and Quan (2015), authors derived the global exponential stability conditions for inertial BAM neural networks with time delays based on inequality technique. Moreover, the authors in Qi, Li, and Huang (2015) and Zhang, Li, Huang, and Tan (2015) derived the stability conditions of inertial BAM neural network under the impulsive and periodically intermittent controls. Based on the above works, it is clear that, very few research reports have produced on inertial type neural networks (see Liu, Liao, Guo et al., 2009; Liu, Liao, Liu, Zhou, & Guo, 2009; Wheeler & Schieve, 1997).
Motivated by the above discussions and facts, this paper is intended to investigate the synchronization of inertial neural networks with Markovian jumping parameters and time varying delays. Moreover, the applications of obtained solutions are utilized in image secure communications. The main contributions of the present works are listed as follows: (1) inertial type neural network with time varying delays is considered; (2) dynamical behaviors are studied via Lyapunov–Krasovskii (L–K) functional and linear matrix inequalities (LMIs); (3) solutions of the proposed problem are applied in the image secure communications. The rest of the paper is organized as follows. In Section 2, model description and preliminaries are briefly outlined. Section 3 contains the derivation of synchronization criteria based on L–K functional. In Section 4, numerical simulations are provided to validate the theoretically derived results and their applications in secure communications are demonstrated. Finally, the proposed result is concluded in Section 5.
Section snippets
Model description and preliminaries
Throughout the manuscript, denotes the -dimensional Euclidean space and denotes the set of all real matrices. The superscript denotes the transposition and the notation (similarly, ), where and are symmetric matrices, means that is positive semi-definite (similarly, positive definite). , represent the sample space, is the - algebra of subsets of the sample space and is the probability measure on . stands for the mathematical expectation.
Main results
In this section, the synchronization problem of the error system is discussed. The analysis is based on the L–K functional and linear matrix inequality (LMI) framework. The above analysis can be summarized in the following theorem. Theorem 1 Given scalars and scalars , master system (1) and slave system (9) are synchronous if there exist , diagonal matrices , real matrices such that the following LMIs hold
Numerical illustration
In this section, a numerical illustration is provided to verify the effectiveness of the derived theoretical results. Example 1 System (2) is considered with the following matrices Now, fix the parameters , and
Conclusion
In this paper, problem on synchronization criteria between the uncontrolled and controlled neural networks has been investigated along with time-varying delay and Markovian jumping parameters. Employing the L–K functional method and the integral inequality, delay-dependent conditions have been derived for synchronization of drive-response systems, which ensured that the error system is globally asymptotically stable through the designed feedback control. Moreover, two numerical illustrations
References (40)
- et al.
Dynamics of simple electronic neural networks
Physica D: Nonlinear Phenomena
(1987) - et al.
Secure communication via multiple parameter modulation in a delayed chaotic system
Chaos, Solitons & Fractals
(2005) - et al.
Delay-dependent stability of neutral systems with time-varying delays using delay-decomposition approach
Applied Mathematical Modelling
(2012) - et al.
Matrix measure strategies for stability and synchronization of inertial bam neural network with time delays
Neural Networks
(2014) - et al.
Synchronization of memristor-based recurrent neural networks with two delay components based on second-order reciprocally convex approach
Neural Networks
(2014) - et al.
Impulsive synchronization of Markovian jumping randomly coupled neural networks with partly unknown transition probabilities via multiple integral approach
Neural Networks
(2015) Spatial properties of an eeg event in the olfactory bulb and cortex
Electroencephalography and Clinical Neurophysiology
(1978)- et al.
New criteria for synchronization stability of general complex dynamical networks with coupling delays
Physics Letters A
(2006) - et al.
Global asymptotic stability of bi-directional associative memory networks with distributed delays
Applied Mathematics and Computation
(2004) - et al.
Stability of bifurcating periodic solutions for a single delayed inertial neuron model under periodic excitation
Nonlinear Analysis. Real World Applications
(2009)
Stability of stochastic differential equations with Markovian switching
Stochastic Processes and their Applications
Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks
Physica D: Nonlinear Phenomena
Stability of inertial bam neural network with time-varying delay via impulsive control
Neurocomputing
Exponential synchronization criteria for Markovian jumping neural networks with time-varying delays and sampled-data control
Nonlinear Analysis. Hybrid Systems
Matrix measure based dissipativity analysis for inertial delayed uncertain neural networks
Neural Networks
Stability and chaos in an inertial two-neuron system
Physica D: Nonlinear Phenomena
Existence and global exponential stability of anti-periodic solutions for bam neural networks with inertial term and delay
Neurocomputing
Global exponential stability via inequality technique for inertial bam neural networks with time delays
Neurocomputing
H filtering for linear neutral systems with mixed time-varying delays and nonlinear perturbations
Journal of the Franklin Institute
Stability analysis of delayed neural networks
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications
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The work of authors are supported by National Board for Higher Mathematics, Department of Atomic Energy, Mumbai under the Grant No. 2/48(3)/2012/NBHM/R&D-II/11020.