Complexity

Volume 2018 (2018), Article ID 4658785, 16 pages

https://doi.org/10.1155/2018/4658785

## Asymmetric Double Strange Attractors in a Simple Autonomous Jerk Circuit

^{1}Laboratoire d’Automatique et Informatique Appliquée (LAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, P.O. Box 134, Bandjoun, Cameroon^{2}Groupe de Recherche sur les Technologies Médicales Adaptées aux Tropiques (GRETMAT), Laboratoire d’Electronique et de Traitement du Signal (LETS), ENSP, University of Yaoundé I, P.O. Box 8390, Yaounde, Cameroon

Correspondence should be addressed to G. H. Kom

Received 19 June 2017; Accepted 17 August 2017; Published 8 February 2018

Academic Editor: Mohamed Belhaq

Copyright © 2018 G. H. Kom 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 dynamics of a simple autonomous jerk circuit previously introduced by Sprott in 2011 are investigated. In this paper, the model is described by a three-time continuous dimensional autonomous system with an exponential nonlinearity. Using standard nonlinear techniques such as time series, bifurcation diagrams, Lyapunov exponent plots, and Poincaré sections, the dynamics of the system are characterized with respect to its parameters. Period-doubling bifurcations, periodic windows, and coexisting bifurcations are reported. As a major result of this work, it is found that the system experiences the unusual phenomenon of asymmetric bistability marked by the presence of two different attractors (e.g., screw-like Shilnikov attractor with a spiralling-like Feigenbaum attractor) for the same parameters setting, depending solely on the choice of initial states. Among few cases of lower dimensional systems capable of such type of behavior reported to date (e.g., Colpitts oscillator, Newton–Leipnik system, and hyperchaotic oscillator with gyrators), the jerk circuit/system considered in this work represents the simplest prototype. Results of theoretical analysis are perfectly reproduced by laboratory experimental measurements.

#### 1. Introduction

The phenomenon of multistability (i.e., the occurrence of multiple attractors for the same parameters setting depending solely on the choice of initial conditions) has captivated the attention of most researchers in recent years. They have done many works in various fields of science and engineering such as electrical circuits [1–5], laser systems [6, 7], biological systems [8], and chemical reactions [9]. Systems with only one attractor are called monostable systems. In such systems, the basin of attraction (i.e., the set of initial conditions for which the asymptotic dynamics converge to the underlined attractor) is the whole state space. In contrast, in a multistable system, each attractor has its own basin of attraction. Correspondingly, the basin boundaries can have a simple structure (simple demarcation) or a very complex structure (i.e., nontrivial or fractal basin boundaries). A physical implication of fractal basin boundaries is random jump between coexisting attractors in experiment. Various types of attractors can coexist such as fixed points, period-n limit cycles, toruses, and strange attractors. Multistability makes a system offer a great flexibility [10]. In particular, the coexistence of infinitely many attractors is called extreme multistability and has been reported in two unidirectionally coupled Lorenz systems [11], two bidirectionally coupled Rossler oscillators [12, 13], and very recently a memristor oscillator [14]. Multistability can be advantageously exploited for image processing [10] or taken as an additional source of randomness which is particularly suited for information engineering applications [15]. In general, the phenomenon of multiple attractors is mostly observed in symmetric dynamical systems [16]. Such systems exhibit pairs of mutually symmetric attractors that merge to form a single symmetric one via the well-known attractor merging crisis as a parameter is varied. However, asymmetric multistability (i.e., coexistence of nonsymmetric attractors) is also reported in systems without any symmetry such as Colpitts oscillator [3], Newton–Leipnik system [17], and hyperchaotic oscillator with gyrators [4].

In the present contribution, we consider the dynamics of an extremely simple chaotic jerk circuit recently introduced by Sprott [18] with particular attention on the chaos mechanism as well as the possibility of multiple coexisting attractors. In this particular jerk circuit, the exponential nonlinearity is implemented by a single semiconductor diode. Therefore, the model is nonsymmetric and thus generic. Furthermore, the circuit cannot support symmetric attractors. However, as mentioned above, the possibility of multiple coexisting attractors is not excluded. To begin, considering results of [18–20], we recall that jerk systems are third-order differential equations of the form . The term is the nonlinear function and designated the “jerk.” It indicates the third-time derivative of which corresponds to the first-time derivative of acceleration in mechanical system. In the studies undertaken by Eichhorn et al. [21] on multistability behavior of simple asymmetric jerk systems, the authors explore the dynamics of two simplest polynomial jerky dynamics (JD1: and JD2: ) known to experience chaotic behavior in some parameter ranges. The authors, by numerical estimation of Lyapunov spectra, also establish dependence of long-time dynamical behavior on the system parameters. Forward and backward bifurcation diagrams have been used to study some parameters of the dependence on initial conditions (e.g., coexistence of two stable attractors, hysteresis). In both (JD1 and JD2) cases not more than two coexisting attractors are found due to the absence of any symmetry. Very recently, a series of works concerning the issue of coexisting multiple attractors in simple jerk dynamical systems were carried out by Kengne et al. [2, 22]. Motivated by the outcomes we have mentioned above, this paper studies the dynamics of the simple jerk circuit previously introduced by Sprott [18] with the following key objectives: (a) to carry out a systematic analysis of the novel jerk circuit and explain the chaos mechanism; (b) to precise the region in parameter space, in which the proposed model exhibits multiple coexisting attractors and hysteretic dynamics; (c) to realize an experimental study of the system to support the theoretical predictions. More importantly, we provide some design tools (i.e., bifurcation diagrams) that are of precious utility for a practical circuit design of this type of oscillators in relevant engineering applications.

The rest of the paper is organised as follows. Section 2 deals with modeling process. The electronic structure of the novel jerk circuit is described and an appropriate mathematical model is derived. In Section 3 the complex dynamics of the oscillator are investigated from normalized mathematical model. Basic properties of the model are also discussed. The stability of the single equilibrium point is analyzed and conditions for the occurrence of Hopf bifurcation are obtained. In Section 4, the bifurcation structures of the system are investigated numerically showing period-doubling, periodic windows, and coexisting bifurcations. Some windows (in the parameter space) corresponding to the occurrence of multiple coexisting attractors (for the same parameters settings) are uncovered. Correspondingly, basins of attraction of various competing attractors are depicted showing nontrivial basin boundaries. Section 5 is devoted to the laboratory experimental study. In this section, an appropriate analog computer is proposed for the investigation of the dynamic behavior of the jerk system. The physical implementation of the oscillator is carried out using electronic components. Laboratory experimental measurements show a very good agreement with the theoretical analysis. Finally some concluding remarks and proposal for future work are presented in Section 6.

#### 2. Circuit Description and State Equation

##### 2.1. Circuit Description

Figure 1 shows the electronic structure of the jerk circuit under consideration [18].