Advances in Mathematical Physics

Volume 2018, Article ID 7273531, 12 pages

https://doi.org/10.1155/2018/7273531

## Autonomous Jerk Oscillator with Cosine Hyperbolic Nonlinearity: Analysis, FPGA Implementation, and Synchronization

^{1}Center for Nonlinear Dynamics, Defense University, Ethiopia^{2}Department of Mechanical, Petroleum and Gas Engineering, Faculty of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon^{3}Department of Physics, Higher Teacher Training College, The University of Bamenda, P.O. Box 39 Bamenda, Cameroon^{4}Laboratory of Electronics and Signal Processing (LETS), Department of Physics, Faculty of Science, University of Dschang, P. O. Box 67, Dschang, Cameroon^{5}School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam

Correspondence should be addressed to Gaetan Fautso Kuiate; moc.oohay@etaiuk_ostuaf

Received 22 May 2018; Revised 17 July 2018; Accepted 30 July 2018; Published 17 September 2018

Academic Editor: Dimitrios J. Frantzeskakis

Copyright © 2018 Karthikeyan Rajagopal 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

A two-parameter autonomous jerk oscillator with a cosine hyperbolic nonlinearity is proposed in this paper. Firstly, the stability of equilibrium points of proposed autonomous jerk oscillator is investigated by analyzing the characteristic equation and the existence of Hopf bifurcation is verified using one of the two parameters as a bifurcation parameter. By tuning its two parameters, various dynamical behaviors are found in the proposed autonomous jerk oscillator including periodic attractor, one-scroll chaotic attractor, and coexistence between chaotic and periodic attractors. The proposed autonomous jerk oscillator has period-doubling route to chaos with the variation of one of its parameters and reverse period-doubling route to chaos with the variation of its other parameter. The proposed autonomous jerk oscillator is modelled on Field Programmable Gate Array (FPGA) and the FPGA chip statistics and phase portraits are derived. The chaotic and coexistence of attractors generated in the proposed autonomous jerk oscillator are confirmed by FPGA implementation of the proposed autonomous jerk oscillator. A good qualitative agreement is illustrated between the numerical and FPGA results. Finally synchronization of unidirectional coupled identical proposed autonomous jerk oscillators is achieved using adaptive sliding mode control method.

#### 1. Introduction

There has been a significant increase in studies of chaotic oscillators over the past decades because of their complex behaviors and their promising applications** [1–9]**. There are increasing works on chaos in jerk oscillators** [10–12]**. It is worth noting that if the scalar is denoted as a physical variable at time* t*, the third derivative presents the jerk** [13]**. Different jerk oscillators with chaos were summarized by Sprott** [14]**. By using Josephson junctions, Yalcin constructed a general jerk circuit with multiscroll and hypercube attractors** [15]**. Multiscroll attractors in jerk circuits were presented by Ma et al.** [16]**. An elegant chaotic oscillators based on jerk equation was implemented in a circuit where the nonlinearity was provided by a single diode** [17]**. A simple chaotic jerk circuit was used in a sound encryption scheme** [18]**. Especially, a three-dimensional novel jerk chaotic oscillator with two hyperbolic sinusoidal nonlinearities was reported in** [19]**.

Researchers have shown an increased interest in multistability** [20–26]**. Multistability is associated with the presence of coexisting attractors with different initial conditions** [20]**. Jerk oscillators can exhibit multistability despite their simplicity** [27–31]**. Recently, Njitacke et al. discovered the coexistence of multiple attractors and crisis route to chaos in a chaotic jerk circuit** [27]** and a simple autonomous jerk oscillator with multiple attractors was introduced in** [28]**. Kengne et al. performed an analysis of a jerk circuit including a pair of semiconductor diodes connected in antiparallel and investigated its multiple attractors** [29]**. It is interesting that, by applying the diode bridge memristor into the original jerk circuit, authors proposed a novel hybrid diode based jerk circuit** [30]**. This work revealed that the hybrid diode based jerk circuit exhibits rich dynamical behaviors including multiple coexisting self-excited attractors. Derived from the autonomous jerk circuit, an autonomous memristor-based jerk circuit was constructed** [31]**. Interestingly, for the same values of system parameters, the coexistence of four different attractors was obtained in such a memristor-based jerk circuit.

Motived by published results related to jerk oscillators, some questions arose as to know, e.g., if a jerk oscillator with cosine hyperbolic nonlinearity can exhibit multistability or how such oscillator can synchronize. The aim of our work is to explore aspects of the unanswered questions. Our paper is structured as follows. Section 2 is devoted to the theoretical analysis of proposed autonomous jerk oscillator with a cosine hyperbolic term. FPGA implementation of proposed autonomous jerk oscillator is presented in Section 3. In Section 4, synchronization of unidirectional coupled jerk oscillators is studied by applying adaptive sliding mode control method. Finally, Section 5 concludes our work.

#### 2. Theoretical Analysis of Proposed Autonomous Jerk Oscillator

In chapter three of “Elegant Chaos: Algebraically Simple flow” book published in 2010, Sprott proposed a list of sixteen autonomous chaotic jerk oscillators with different nonlinearities called memory oscillators** [14]**. The nonlinearities of memory oscillators include quadratic, cubic, quintic, absolute, maximum, sign, exponential, sine, and tangent hyperbolic functions. Inspired by** [14]**, in this work we introduce an autonomous jerk oscillator with a cosine hyperbolic nonlinearity described bywhere are state variables of the oscillator and two positive parameters. System (1a), (1b), and (1c) can be converted to a jerk oscillator, as follows:System (1a), (1b), and (1c) is dissipative because . The equilibrium points of system (1a), (1b), and (1c) are obtained by solving , , , which givesEquation (3b) cannot be solved analytically. We thus use the Newton-Raphson method to find the value of . Depending on the value of parameter , (3b) presents no roots for and one root () for . Therefore, for , system (1a), (1b), and (1c) has no equilibrium points while, for , it has two equilibrium points and with (see Figure 1). The characteristic equation evaluated at the equilibrium point isAccording to the Routh-Hurwitz criterion, the real parts of all the roots of (4) are negative if and only ifThe stability analysis of equilibrium point as function of the parameter of system (1a), (1b), and (1c) is depicted in Figure 1.