Adaptive Synchronization of Memristor-based Chaotic Neural Systems

Chaotic neural networks consisting of a great number of chaotic neurons are able to reproduce the rich dynamics observed in biological nervous systems. In recent years, the memristor has attracted much interest in the efficient implementation of artificial synapses and neurons. This work addresses adaptive synchronization of a class of memristor-based neural chaotic systems using a novel adaptive backstepping approach. A systematic design procedure is presented. Simulation results have demonstrated the effectiveness of the proposed adaptive synchronization method and its potential in practical application of memristive chaotic oscillators in secure communication. Keywords: Adaptive synchronization, memristors, chaos, chaotic systems, backstepping. __________________________________________________________________________________________ 1. Introduction The memristor, defined by the relationship between the flux and the charge of a device, was theoretically predicted by Leon Chua in 1971 and called the fourth fundamental circuit element after the resistor, the capacitor and the inductor [1]. In 2008, Williams and his team from HP Lab proved the existence of the memristor in nanoscale electronics while developing ultra-high density nonvolatile memory [2]. Afterwards, the research on memristors or memristive systems has gained ever-increasing attention from both academia and industry [3-12]. In particular, many efforts have been devoted into discovery of some important properties of typical memristors [3,4], various memristive devices and materials [5-7], as well as promising application potentials [8-14]. Due to those pioneers’ valuable work, the key features of the memristor can be summarized as follows. (i) The memristor is a kind of nonlinear devices in simple sandwiching structure, featuring hysteretic currentvoltage characteristic under periodic external excitation conditions. (ii) The memristor’s capabilities of nanoscale size, variable resistance and power-off mode storage make it a competitive candidate of the next-generation nonvolatile memory [8,9]. (iii) The conductivity of a memristor depends on the total flux/charge ever passing through it. This property is very similar to the biological synaptic plasticity, that is, the strength of a synaptic weight is in the control of the ionic flowing through the synapse between two adjacent neurons. Thereby, by combining the advantages of tiny scale and simple structure, the memristor naturally becomes the preferred artificial synapses in large-scale and massively-parallel neuromorphic architectures that merge computation and memory [10-11]. (iv) The memristor also has potential in nonlinear circuit design and realization such as chaotic oscillators. A novel implementation scheme for chaotic oscillators using nanoscale memristors might achieve richer dynamic behaviors with much smaller and simpler circuits, compared with the traditional operational-amplifier-based method. In fact, many memristive chaotic systems have been designed and investigated [12-14]. The memristive element used in most of these systems is the generalized memristor or memristive system with an odd-symmetric flux-charge characteristic similar to the current-voltage curve of Chua’s diode [12]. Recently, more attention has been paid on chaotic systems consisting of HP memristors [13,14]. In this paper, the latter will be focused on. Since the chaos synchronization was shown to be possible by Pecora and Carroll [15], synchronization between coupled chaotic systems has been extensively investigated [16-20]. The concept of chaos synchronization refers to making two identical chaotic dynamical systems with different initial conditions oscillate in a synchronized manner [20]. Up to now, various synchronization phenomena have been observed in different chaotic systems, including complete synchronization, generalized synchronization, phase synchronization, lag synchronization and so on [16]. In practical applications, it is well known that synchronization plays an essential role in ______________ * E-mail address: [email protected] ISSN: 1791-2377 © 2015 Kavala Institute of Technology. All rights reserved. Jestr JOURNAL OF Engineering Science and Technology Review www.jestr.org Xiaofang Hu and Shukai Duan /Journal of Engineering Science and Technology Review 8 (2) (2015) 17 – 23 18 chaos secure communication systems. So far, for the canonical chaotic systems such as Lorenz’s system, Chua’s circuits, and Chen’s system, many different synchronizing approaches have been proposed and intensively studied, including conventional linear control schemes and advanced nonlinear control techniques [18]. Recently, Fernando Corinto has also demonstrated the influence of the memristor synapse on the synchronous behaviors of two Hindmarsh-Rose chaotic neurons [21] and two FitzHughNagumo chaotic neurons [22]. In many real applications, the parameters of chaotic systems under study might not be exactly known. In such cases, the so-called adaptive controllers are suitable. As a powerful adaptive control scheme, backstepping has been widely used in applications since it can guarantee tracking, global stability, and transient performance of a broad class of strict-feedback systems [20]. Unfortunately, very few discussions on synchronization of the memristor-based chaotic systems with parametric uncertainties have been reported, which motivates this study. This paper aims at studying adaptive synchronization of two coupled memristor-based chaotic neural systems using a backstepping approach and presenting a systematic design procedure. The rest of the paper is organized as follows. In Section 2, a third-order memristor-based chaotic neural system is firstly introduced. Then, a controller for synchronization of the two coupled memristor-based chaotic neural systems is designed. Adaptive synchronization of the chaotic neural systems with uncertain parameters is studied in Section 3. Simulation results are presented in Section 4 and Section 5 outlines the conclusions. 2. Synchronizing Two Coupled Memristor-Based Chaotic Neural Systems via a Backstepping Design 2.1. The Memristor-based Chaotic System The memristor-based chaotic neural system is described as follows [13], ! x1 = x2 ! x2 = −ax1 + x3 ! x3 = 1+ x2 + bg(− | x1 |) ⎧ ⎨ ⎪