Complexity

Volume 2018, Article ID 4140762, 15 pages

https://doi.org/10.1155/2018/4140762

## Chaos and Symbol Complexity in a Conformable Fractional-Order Memcapacitor System

^{1}School of Physics and Electronics, Central South University, Changsha 410083, China^{2}School of Computer Science and Technology, Hunan University of Arts and Science, Changde 415000, China^{3}Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Selangor, Malaysia^{4}Malaysia-Italy Centre of Excellence for Mathematical Science, Universiti Putra Malaysia, Serdang, Selangor, Malaysia

Correspondence should be addressed to Santo Banerjee; moc.liamg@nabotnas

Received 27 April 2018; Accepted 5 July 2018; Published 5 August 2018

Academic Editor: Viet-Thanh Pham

Copyright © 2018 Shaobo He 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

Application of conformable fractional calculus in nonlinear dynamics is a new topic, and it has received increasing interests in recent years. In this paper, numerical solution of a conformable fractional nonlinear system is obtained based on the conformable differential transform method. Dynamics of a conformable fractional memcapacitor (CFM) system is analyzed by means of bifurcation diagram and Lyapunov characteristic exponents (LCEs). Rich dynamics is found, and coexisting attractors and transient state are observed. Symbol complexity of the CFM system is estimated by employing the symbolic entropy (SybEn) algorithm, symbolic spectral entropy (SybSEn) algorithm, and symbolic C_{0} (SybC_{0}) algorithm. It shows that pseudorandom sequences generated by the system have high complexity and pass the rigorous NIST test. Results demonstrate that the conformable memcapacitor nonlinear system can also be a good model for real applications.

#### 1. Introduction

In 1971, Chua postulated the concept of memristor that describes a relationship between flux and charge [1]. In 2008, researchers in Hewlett-Packard announced that a solid-state implementation of memristor has been successfully fabricated [2]. Since then, designing memory circuits has received significant attention of researchers, and many different kinds of memristor-based circuits have been designed [3–5]. In 2009, Ventra et al*.* [6] reported memcapacitors and meminductors. Compared with memristors, memcapacitors and meminductors have received much less attention. Currently, memcapacitor and meminductor can be designed based on the memristor. For example, Biolek and Biolkova [7] designed a memcapacitor model based on memristor by means of off-the-shelf circuits. As a matter of fact, memory electronic elements are usually designed nonlinearly. Thus, chaos can be easily found in those memory electronic element-based circuits [8–16]. Bao et al*.* [8–11] presented many valuable works on chaotic memristor circuits. For instance, their most recent work reported quasiperiodic behavior and chaotic busting in a third-order autonomous memristive oscillator [11]. Moreover, Mou et al*.* [12] designed a memory circuit with two memcapacitors that exhibited complex phenomena of state transition and transient chaos accompanied with time evolution and coexisting states. Fractional calculus has been studied for about 300 years, and there are a large number of literatures reporting chaos in the fractional-order nonlinear systems [17–20]. Moreover, fractional-order memory electronic element-based systems increasingly attracted attention of scholars [21, 22]. Since not much research exists about the fractional-order memcapacitor system, a fractional-order system with two memcapacitors is considered in this paper.

All of the abovementioned systems are integer-order systems or fractional-order chaotic systems under Caputo definition or Riemann-Liouville (R-L) definition [23]. In 2014, Khalil et al. [24] proposed a new fractional derivative, and it is called the conformable fractional (CF) derivative. Since the CF definition is prominently compatible with the integer-order derivative, it has been widely studied in different research fields [25–28]. For example, İskender Eroğlu et al*.* [26] proposed an optimal boundary temperature control for a time-conformable fractional heat conduction equation. However, to our best knowledge, there are only two literatures reporting numerical analysis of CF chaotic systems. He et al*.* [29] firstly solved the nonlinear CF equations by the conformable Adomian decomposition method (CADM) and found chaos in the CF simplified Lorenz system. Later, Ruan et al*.* [30] investigated dynamics of a CF memristor system based on CADM, and rich dynamical behaviors were found. It shows that the CF nonlinear systems also generate chaos, and it is an interesting topic to explore complexity in these systems. Recently, Ünal and Gökdoğan [31] modified the differential transform method (DTM) and applied this method to solve CF nonlinear equations. But it has not been used to solve CF chaotic systems. Thus, in this paper, we will use conformable DTM to solve the CF memcapacitor system and analyze this system numerically.

Meanwhile, measuring complexity is also an important method to analyze dynamics of chaotic systems. It reflects the security of the system to some extent. When a system has higher complexity, it means that the time series generated by the system is more random. Currently, there are several methods to measure complexity of time series, such as the permutation entropy (PE) [32], sample entropy (SampEn) [33], spectral entropy (SE) [34], and C_{0} algorithms [35]. It should be noted out that complexity of chaotic systems is mainly estimated based on the original time series, and complexity analysis of nonlinear symbol sequence has aroused interests of researchers [36, 37]. Meanwhile, there are many kinds of pseudorandom sequence generation algorithms. How complexity and dynamics of a chaotic system are determined by the pseudorandom quantization algorithms should be investigated. And whether the CFM system can be actually used in real applications should be verified.

The rest of the article is organized as follows. In Section 2, definitions of conformable fractional derivative and a numerical solution algorithm are proposed. Solution of the CFM system is obtained. In Section 3, dynamics of the CFM system is analyzed by means of Lyapunov characteristic exponents (LCEs), bifurcation diagram, and phase portraits. In Section 4, three different symbol complexity measuring algorithms are designed and the complexity of the CFM system is analyzed. Meanwhile, the NIST test is carried out. Finally, we summarize the results in Section 5.

#### 2. Definitions and Numerical Solution Algorithm

In this section, the system model and definitions about conformable fractional derivative are presented. A numerical solution algorithm for conformable fractional nonlinear systems is designed based on the differential transform method.

##### 2.1. The Conformable Fractional Memcapacitor System

Mou et al. [12] proposed a circuit with memcapacitor, and it is denoted by where , , and are the system parameters, , , and are the state variables, and and represent the two memcapacitors in the circuit in which , , , and are the intrinsic parameters of the two memcapacitors. In [12], , , and is the bifurcation parameter. Moreover, there are three different sets of intrinsic parameters for different types of attractors. The three sets of intrinsic parameters are shown in Table 1.