Mathematical Problems in Engineering

Volume 2017, Article ID 6504969, 13 pages

https://doi.org/10.1155/2017/6504969

## Coexisting Oscillation and Extreme Multistability for a Memcapacitor-Based Circuit

^{1}Institute of Modern Circuits and Intelligent Information, Hangzhou Dianzi University, Hangzhou 310018, China^{2}Department of Automation, Shanghai University, Shanghai 200072, China

Correspondence should be addressed to Xiaowei Wang; moc.361@gnaw_324arual

Received 6 September 2016; Revised 7 November 2016; Accepted 6 December 2016; Published 23 January 2017

Academic Editor: Mingshu Peng

Copyright © 2017 Guangyi Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The coexisting oscillations are observed with a memcapacitor-based circuit that consists of two linear inductors, two linear resistors, and an active nonlinear charge-controlled memcapacitor. We analyze the dynamics of this circuit and find that it owns an infinite number of equilibrium points and coexisting attractors, which means extreme multistability arises. Furthermore, we also show the stability of the infinite many equilibria and analyze the coexistence of fix point, limit cycle, and chaotic attractor in detail. Finally, an experimental result of the proposed oscillator via an analog electronic circuit is given.

#### 1. Introduction

Memristor is known as the fourth basic circuit element that was firstly postulated by Chua in 1971 [1] and has attracted worldwide immense attention from both theory and applications even since a solid-state device called the memristor was fabricated by the HP Lab in 2008 [2]. Then, Di Ventra et al. extend the notion of memristor to memcapacitor and meminductor [3], whose properties depend on the state and history of them in circuit. These nanoscale elements, that store information without need for an internal power supply, can be used for nonvolatile memories, nonlinear circuits, neural networks, and so on.

In the field of nonlinear dynamics, memristors can be as fundamental elements for the designs of new nonlinear circuits substituting other nonlinear devices, whereupon many memristor-based chaotic oscillators were presented by using HP memristor and the other memristor models of piecewise linear, quadric, and cubic (flux-charge) functions. Itoh and Chua firstly derived several nonlinear oscillators based on Chua’s oscillators by replacing Chua’s diodes with piecewise linear models of memristor [4]. The authors in [5] designed a periodically forced memristive Chua’s circuit using the flux-controlled memductance model . HP memristor model is a first model of actual memristive device, which was implemented by a memristor emulator in [6]. The paper [7] introduced a complete mathematical model for the HP memristor which takes into consideration all boundary situations providing the interrelation between memristance charge and flux. Based on HP memristor and other memristor models, some chaotic oscillators were presented [8–10]. Recently, a new memristor-based chaotic system was designed based on HP memristor in [11] and was realized using FPGA (Field Programmable Gate Array) technology. A novel digital-analog hybrid chaotic system with generalized memristor was constructed for the production of random number [12]. However, the memcapacitor and meminductor have received a little attention since the actual solid-state devices of them have not been successfully achieved. Because of their potential application values, memcapacitor and meminductor have still attracted more and more attentions.

The first meminductor emulator was designed in [13] whose inductance can be varied by an external current source without employing any memristive system. In [14], a flux-controlled memristive emulator using light-dependent resistor (LDR) was proposed and the mutator for transferring memristor into a flux-controlled meminductor is described. A mathematical model of a flux-controlled meminductor and its equivalent circuit model for exploring the properties of the meminductor in a nonlinear circuit were presented in [15].

It would be worthwhile to prospectively study effective memcapacitor models and its applications. So several memcapacitor models, including piecewise linear, quadric and cubic function models, memristor-based memcapacitor models, and memcapacitor emulators, were proposed in [16–21], and a mathematical memcapacitor model and a corresponding circuit model are established in [22]. Some special phenomena such as hidden attractors, coexistence attractors, and extreme multistability were found in memcapacitor-based chaotic oscillators [23, 24] and memristor-based chaotic oscillators [25–27]. In fact, multistability and coexisting attractors have caught the attention of researcher in general chaotic systems [28–30].

This paper introduces a new chaotic oscillator based on a charge-controlled memcapacitor model and its dynamical behaviors are analyzed. The most important properties of the memcapacitor-based circuit are that it possesses an infinite number of equilibrium points and coexisting attractors and displays the coexisting attractors and stability of the infinite many equilibria, called extreme multistability. The rest of this paper is organized as follows. In Section 2, the chaotic oscillator is described and the typical chaotic attractors are given. In Section 3, the dissipativity and equilibrium stability of the system are studied. In Section 4, coexisting attractors with different initial and are described. In Section 5, an analog circuit is designed to realize the memcapacitor-based oscillator. Finally, some conclusions of this paper are given in Section 6.

#### 2. Memcapacitor-Based Chaotic Oscillator Circuit

The chaotic oscillator circuit based on a charge-controlled memcapacitor is shown in Figure 1, which contains two linear inductors, and , two linear resistors, and , and an active memcapacitor. The active memcapacitor is an active 2-terminal circuit consisting of a linear active resistor and a nonlinear memcapacitor in series, in which the active resistor provides energy for the circuit. Hence, the series circuit, that is, the active 2-terminal circuit as a whole, exhibits an active feature, so called the active memcapacitor.