Elsevier

Neurocomputing

Volume 117, 6 October 2013, Pages 33-39
Neurocomputing

A new chaotic Hopfield neural network and its synthesis via parameter switchings

https://doi.org/10.1016/j.neucom.2012.11.022Get rights and content

Abstract

In this paper, we present a new chaotic attractor in Hopfield neural network. Numerical experiments show that the presented Hopfield neural network can display complex dynamics by changing the self-connection weight. Surprisingly, coexistence of a chaotic attractor and a limit cycle is found in this system, which means, the system can exhibit a chaotic attractor or a limit cycle according to different initial values, and this phenomenon is never reported before. We give a rigorous verification of existence of horseshoe chaos by virtue of topological horseshoes theory and estimates of topological entropy in the derived Poincaré maps. Finally, synthesis of the chaotic attractor is studied via parameter switching and a numerical example illustrates the effectiveness of this method.

Introduction

Gaining a deep insight of the dynamical behavior of Hopfield neural networks (HNNs) is of significant importance in the study of the information processing and engineering applications [1]. The HNN [2] is an artificial model abstracted from brain dynamics and is a pivotal model in artificial neurocomputing, which is able to store certain memories or patterns in a manner rather similar to the brain—the full pattern can be recovered if the network is provided with only partial information [1]. In fact, substantial evidences have been found in biological studies to support the presence of chaos in natural neuronal systems. In the past decade, the dynamics on HNNs has drawn a lot of interest [3], [4], [5]. Recently, many researchers suggested that chaos plays a central role in memory storage and retrieval [6], [7], [8]. From this point of view, many artificial neural networks have been proposed in order to realize complex dynamical behavior such as chaos and hyperchaos [9], [10], [11], [12], [13], [14], [15]. Inspired by Refs. [10], [11], [13], [14], [15], we present a new 3-neuron chaotic HNN and it is different from Refs. [10], [11], [13], [14], [15] in its connection weight parameters and connection architecture.

In practical cases, the parameters in some systems are subjected to fluctuations due to uncertain factors such as noises, malfunction of certain components, etc. Many chaotic systems are not robust when they are subjected to such changes in parameters such as those presented by Lorenz [16], Chen [17], Liu [18], several Yang's chaotic attractors in HNN [11], [15] and the new presented chaotic attractor in this paper. Hence, we adopt a parameter switching synthesis method through which the new chaotic attractor can be synthesized. The parameters chosen are not lied chaotic region, as is often the case in natural world. Compared to non-chaotic regions, the chaotic region of the parameter is very small, that is why chaos is seldom observed in the natural world and it is also the reason why autonomous chaos is very hard to find in low dimensional system [19]. From a control point of view, synthesis of chaos is also a way of anti-control of chaos [20], [21], [22], [23], [24], which means generating or enhancing chaocity when it is beneficial or useful. Many attractors have been synthesized recently, like Lü chaotic system and Lotka–Volterra like system [25], [26], but the synthesis method has not been applied to chaotic neural networks. We employ this method to achieve the same effect with the original parameter when it is hard to implement due to uncertainties or limits of the system.

The rest of the paper is structured as follows: in Section 2, a new chaotic HNN model is presented and its complex dynamics is illustrated. In Section 3, we give rigorous proof of the existence of horseshoe chaos by virtue of topological horseshoe theory. In Section 4, synthesis of the newly presented chaotic attractor via parameter switching is studied and numerical result shows the effectiveness of the proposed method. Finally, some concluding remarks are stated in Section 5.

Section snippets

Network model

The neural network is described as x˙i=cixi+j=1nwij·f(xj),xR,i=1,2,3,,n,where f(x) is a monotone continuous function which is bounded above and below, W=(wij) is a matrix, called connection matrix describing the strength of connections between neurons, and n is the number of neurons. Chaos in three dimensional continuous time neural networks with sigmoidal response functions has been numerically observed and investigated in several papers [10], [11], [13], [14], [15]. Here we also consider

Topological horseshoes

In this section, we first recall a result on topological horseshoe, then give a computer-assisted verification of the existence of chaos in the HNN for p=0 by virtue of the topological horseshoe theory.

Synthesis via parameter switchings

In this part we investigate numerically the possibility to synthesize the chaotic attractor proposed in the paper depending on a real parameter by periodic parameter switchings. Synthesis of chaos by parameter switchings is first proposed by Danca [31], since then it has aroused a lot of attention [25], [26], [32], [33], [34], [35]. The parameters in a certain system that can generate chaos are always, if not all, located in small regions. What is more, perturbation of these parameters may lead

Conclusions

In this paper, a new chaotic attractor in HNN is presented, and it can demonstrate very complex dynamical behavior with the variation of the parameter p. The coexistence of a limit cycle and a chaotic attractor is first reported in this paper, which is a very unusual case in non-linear system and is never reported before. We hope that it can arouse enough attention to the study of coexistence of limit cycle and chaotic attractor. We give a rigorous verification of the existence of horseshoe in

Juan Li received her BS degree and MS degree in Mathematics from Hubei University, Wuhan, China, in 2004 and 2007 respectively. She is currently working toward the PhD degree in Control Science and Engineering at Huazhong University of Science and Technology, Wuhan, China. Her research interests include complex networks, consensus and cooperative control of multi-agent systems, chaos dynamics and control.

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    Juan Li received her BS degree and MS degree in Mathematics from Hubei University, Wuhan, China, in 2004 and 2007 respectively. She is currently working toward the PhD degree in Control Science and Engineering at Huazhong University of Science and Technology, Wuhan, China. Her research interests include complex networks, consensus and cooperative control of multi-agent systems, chaos dynamics and control.

    Feng Liu received the BS degree in Automatic Control from Qingdao University, in 2010. He is currently working toward the MS degree in Huazhong University of Science and Technology. His research interests include chaos dynamics and control, genetic regulatory network, complex networks, hybrid system and smart grid.

    Zhi-Hong Guan received PhD degree in Automatic Control Theory and Applications from the South China University of Technology, Guangzhou, China, in 1994. He was a Full Professor of Mathematics and Automatic Control with the Jianghan Petroleum Institute, Jingzhou, China, in 1994. Since December 1997, he has been Full Professor of the Department of Control Science and Engineering, Executive Associate Director of the Center for Non-linear and Complex Systems and Director of the Control and Information Technology in the Huazhong University of Science and Technology (HUST), Wuhan, China. His research interests include complex systems and complex networks, impulsive and hybrid control systems, chaos dynamics and bifurcation, networked control systems, multi-agent systems, genetic regulatory networks, and intelligent electric grid.

    Tao Li received the PhD degree in Huazhong University of Science and Technology, Wuhan, China, in 2008. He is currently an Associate Professor in the School of Electronics and Information, Yangtze University, Jingzhou, China. His research interests include non-linearity complex network systems, complex network theory & application, complex networks spreading dynamics.

    This work was supported by the National Natural Science Foundation of China under Grants 61073025, 61073026, 61170031, 61170024 and 61272069.

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