Complexity

Volume 2017 (2017), Article ID 6820492, 13 pages

https://doi.org/10.1155/2017/6820492

## A New Simple Chaotic Lorenz-Type System and Its Digital Realization Using a TFT Touch-Screen Display Embedded System

^{1}Electronics and Telecommunications Department, Scientific Research and Advanced Studies Center of Ensenada, Ensenada, BC, Mexico^{2}CONACYT-Autonomous Baja California University (UABC), Ensenada, BC, Mexico

Correspondence should be addressed to César Cruz-Hernández

Received 14 February 2017; Revised 5 May 2017; Accepted 31 May 2017; Published 26 July 2017

Academic Editor: Giacomo Innocenti

Copyright © 2017 Rodrigo Méndez-Ramírez 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

This paper presents a new three-dimensional autonomous chaotic system. The proposed system generates a chaotic attractor with the variation of two parameters. Analytical and numerical studies of the dynamic properties to generate chaos, for continuous version (CV) and discretized version (DV), for the new chaotic system (NCS) were conducted. The CV of the NCS was implemented by using an electronic circuit with operational amplifiers (OAs). In addition, the presence of chaos for DV of the NCS was proved by using the analytical and numerical degradation tests; the time series was calculated to determine the behavior of Lyapunov exponents (LEs). Finally, the DV of NCS was implemented, in real-time, by using a novel embedded system (ES) Mikromedia Plus for PIC32MX7 that includes one microcontroller PIC32 and one thin film transistor touch-screen display (TFTTSD), together with external digital-to-analog converters (DACs).

#### 1. Introduction

In recent years, chaotic systems have attracted the attention of the scientific community due to their potential applications in several areas of science and engineering, as an interesting nonlinear phenomenon with different applications in biology, secure communication, complex networks, experimental network synchronization, fingerprint encryption, among others [1–15]. In 1963, Lorenz proposed a three-dimensional system of two scrolls; this is recognized as the first reported chaotic model [16]. Since then, many chaotic systems like Lorenz—and their chaotic behavior—have been reported in the literature, for example, [17–23]. Currently, we can mention some new chaotic systems reported in the literature [24–33].

The chaotic systems usually are implemented by using electronic circuits in continuous (CV) and discretized versions (DV). The CVs usually are represented by using OAs [31, 34]. For DVs, Matlab or Labview allows simulating the dynamical behaviors of discretized chaotic systems to desirably obtain the less degradation with respect to CVs, and their implementations are reproduced by using ESs as FPGAs [35], DSPs [36], or microcontrollers [37–39].

In this paper, we propose a new simple chaotic system, which is derived from the Lorenz system [16]. The novelty of the proposed chaotic system is the combination of different characteristics that it presents: two critical parameters, only two nonlinearities, low complexity time, low iterations per second, and a larger step size for the discretized version where chaos is preserved, low-cost electronic implementation; also it is flexible and robust with respect to some recent attractors reported in the literature; see, for example, [22, 23]. As a consequence, all these features result in a high ease of implementation that may be of great interest in engineering applications, for example, in cryptography, biometric systems, telemedicine, and secure communications; see, for example, [10, 12, 13, 37]. To the best of our knowledge the electronical implementation in a portable TFTTSD device of DV of chaotic systems, for the reproduction of their nonlinear dynamics in real-time, is new.

The paper is organized as follows: Section 2 reports basic analytical proof and extensive numerical tests to verify the existence of chaos in the proposed NCS. Section 3 presents two electronic implementations to reproduce the NCS: the first by means of OAs and the second of a novel proposed ES by using one TFTTSD and external DACs. In addition, a degradation study taking into account the preservation of chaos is conducted; a comparison among the NCS and some other chaotic systems reported in the literature is made. In Section 4, we give a complete description of digital implementation process of the DV of NSC with the corresponding robustness diagram to determine regions in which the existence of chaos is guaranteed. Finally, in Section 5 we draw some concluding remarks.

#### 2. Basic Analysis and Characterization of NCS

This section presents the state equations of NCS and some numerical and analytical tests to verify the chaos existence in the proposed system. The NCS is built starting from inspecting and modification of Lorenz system [16]. The proposed NCS is described by

The proposed system has seven terms, two quadratic nonlinearities, and four parameters , where and are characterized as the bifurcation parameters. The nonlinear NCS (1) is chaotic with , , , and .

System (1) is symmetrical about the -axis due to its invariance under the coordinate transformation () (). The symmetry is not associated with the* a, b, c,* and* d* parameters.

The divergence for a 3-dimensional flow of dynamical system is defined by

Therefore, the above analysis proves that our system is dissipative. The exponential contraction rate is calculated as follows:where each volume containing the system trajectory shrinks to zero as at an exponential rate of . System orbits are ultimately confined into a specific limit set of zero volume, and the asymptotic motion settles onto an attractor. Thereby, the existence on attractor is proved.

The boundness of the chaotic trajectories of system (1) is proved by means of the following theorem. The boundness of a NCS by using similar approach was reported in [40, 41].

Theorem 1. *Suppose that the parameters , , , and of system (1) are positive. Then, the orbits of system (1) including chaotic orbits are confined in a bounded region.*

*Proof. *Consider the candidate Lyapunov functionand the time derivative of along the trajectories of the NCS (1) is given byLet be the sufficiently large region so that for all trajectories () satisfy () = for with the conditionConsequently on the surface . Since we can write , or the set is a confined region for all the trajectories of chaotic system (1).

The number of the equilibrium points and their stabilities determine the behavior of system (1); it can be found by setting and . The proposed system (1) has five fixed points: , . The Jacobean matrix of CV-system (1) is given byand the characteristic polynomial of (7) is as follows:Evaluating with parameters ,* b* = 2,* c* = 0.5, and* d* = 4 in (8), the stability in the equilibrium points was studied. Table 1 shows the stability results of equilibria where all points of NCS are saddle-focus unstable nodes.