Complexity

Volume 2018, Article ID 3286070, 12 pages

https://doi.org/10.1155/2018/3286070

## Multistability Analysis and Function Projective Synchronization in Relay Coupled Oscillators

Correspondence should be addressed to Ahmad Taher Azar; gro.eeei@raza_t_damha

Received 22 July 2017; Revised 1 November 2017; Accepted 9 November 2017; Published 18 January 2018

Academic Editor: Sajad Jafari

Copyright © 2018 Ahmad Taher Azar 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

Regions of stability phases discovered in a general class of Genesio−Tesi chaotic oscillators are proposed. In a relatively large region of two-parameter space, the system has coexisting point attractors and limit cycles. The variation of two parameters is used to characterize the multistability by plotting the isospike diagrams for two nonsymmetric initial conditions. The parameters window in which the jerk system exhibits the unusual and striking feature of multiple attractors (e.g., coexistence of six disconnected periodic chaotic attractors and three-point attraction) is investigated. The second aspect of this study presents the synchronization of systems that act as mediators between two dynamical units that, in turn, show function projective synchronization (FPS) with each other. These are the so-called relay systems. In a wide range of operating parameters; this setup leads to synchronization between the outer circuits, while the relaying element remains unsynchronized. The results show that the coupled systems can achieve function projective synchronization in a determined time despite the unpredictability of the scaling function. In the coupling path, the outer dynamical systems show finite-time synchronization of their outputs, that is, displaying the same dynamics at exactly the same moment. Further, this effect is rather general and it has a wide range of applications where sustained oscillations should be retained for proper functioning of the systems.

#### 1. Introduction

Multistability, meaning the coexistence of many different kinds of attractors, is an intrinsic property of many nonlinear dynamical systems and has become very important research topic and received much attention recently [1, 2]. Multistability poses a threat for engineering systems because the system may unpredictably switch into an undesirable state. Multistability exhibits a rich diversity of stable states of a nonlinear dynamical system and makes the system offer a great flexibility. Particularly, when the number of coexisting attractors generating from a dynamical system tends to be infinite, the coexistence of many attractors depending on the initial condition of a certain state variable is alleged to be extreme multistability [3]. The occurrence of multiple attractors, which implies multiple stability and thus hysteretic dynamics, is one of the most important phenomena encountered in nonlinear dynamical systems. Such type of behavior has been reported in a wide range of systems including electronic circuits [4], laser [5], biological systems [6], Lorenz system [7], Josephson junction [8], and chemical reactions [9]. Multiple attractor bifurcations are said to occur when multiple coexisting attractors are simultaneously created at a bifurcation point [10]. It has been shown earlier that in some cases border collision bifurcations may lead to multiple attractor bifurcations [11]. More recently, Bao and collaborators [12] developed hidden extreme multistability in memristive hyperchaotic system. In that paper, they established a novel memristive hyperchaotic system with no equilibrium based on the newly proposed circuit realization scheme and investigated the phenomenon of extreme multistability with hidden oscillation that reveals the coexistence of infinitely many hidden attractors in the proposed memristive hyperchaotic system. Kengne et al. [13] presented the basic dynamical properties of a simple autonomous jerk system including equilibria and stability, phase portraits, frequency spectra, bifurcation diagrams, and Lyapunov exponent plots. It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry-restoring crisis scenarios. One of the key contributions presented in their work was that the jerk system experiences the striking feature of multiple attractors (e.g., coexistence of four disconnected periodic and chaotic attractors) [13, 14]. It is important to note that the results obtained revealed that there are some unexplored parameters’ regions of this circuit where four disconnected nonstatic attractors coexist.

The interaction of two nonlinear systems via a third parameter-matched circuit typically leads to a variety of significant behaviors, among which the most intriguing is probably synchronization* (known usually as relay synchronization)*, that is, the coordination of a particular dynamical property of their motion [15]. The interaction between two chaotic systems has been deeply studied during the past decade, focusing on the ability of synchronization even in the presence of noise or delay. In [16], Wagemakers et al. examined the robustness of isochronous synchronization in simple arrays of bidirectionally coupled systems. The results of the study showed experimentally that the relaying unit does not need to be identical to the outer systems which are the ones to be synchronized. Sharma et al. in [17] proposed the dynamics of nonlinear oscillators indirectly coupled through a dynamical environment. The results of the study showed that this form of indirect coupling leads to synchronization and phase-flip transition in periodic as well as chaotic regime of oscillators. The phase-flip transition in the case of relay coupled system was investigated by Sharma et al. in [18]. In that paper, the authors show that, in the absence of time delay, relay coupling through conjugate variables has the same effect as when the interactions involve a time delay. However, this phase-flip transition does not occur abruptly at a certain critical value of the coupling parameter. Relay synchronization (RS) has been used with electronic circuits, as a technique for transmitting and recovering encrypted messages, which can be sent bidirectionally and simultaneously [19]. Apart from its technological applications, RS has also been proposed as a possible mechanism at the basis of isochronous synchronization between distant areas of the brain [20]. Nana and Woafo proposed a theoretical and experimental synchronization of three oscillators coupled as emitter-relay-receiver system [21]. They proposed an experimental setup and showed that it is impossible to achieve a zero synchronization error due to the tolerances of the electrical components. Some demonstrations of chaotic masking of communication as well as selected secure communication lines were observed. Gutiérrez et al. in [22] showed that a generalized synchronization (GS) in relay systems with instantaneous coupling could be obtained. The authors proved the existence of GS in unidirectional coupled units (drive system → response system) by checking the ability of the response system to react identically to different initial conditions of the same driver system, which can be quantified by evaluating the mutual false nearest neighbors or by measuring the conditional Lyapunov exponents. Despite such evidence of RS, there are still open questions of a fundamental nature. The main issue is to characterize properly the relationship established in RS between the dynamics of the relay system and that of the synchronized systems. From the previously mentioned references, the literature needs a strict analysis of the performance of the RS using a scaling function. Besides, the projective synchronization (PS) has been used in the research of secure communication because of the unpredictability of the scaling function which may be a useful element [23]. So the development of the function projective synchronization in relay systems is important challenging research point. This motivates the present study.

The aim of this work is to make some dynamical analysis details of complex systems that can exhibit many major features of the regular and chaotic motion which allows a better understanding of its behavior and providing a generic route of function synchronization in relay coupled jerk oscillators.

The remainder of the paper is organized as follows. In Section 2, the nonlinear system is presented and the dynamical behaviors of the circuit are identified with the help of a numerical two-parameter Lyapunov exponent diagram. The finite-time synchronization issue is formulated in Section 3 in which synchronization and numerical simulations are presented. Finally, conclusions and remarks are given in Section 4.

#### 2. The Model and Its Behavior

##### 2.1. Preliminaries

We consider the following chaotic system:where is the system state vector, the constants, and the nonlinear smooth function. Actually, through topological transformation, many existing chaotic systems, such as Chen systems, Lorenz systems, Lu systems, can be transformed as in the form of system (1). More recently, a growing interest is in the analysis of the LEs on Lyapunov diagrams, where we associate colors for the largest and the second largest exponent varying simultaneously two system’s parameters [24, 25]

*Remark 1. *In real world, the order of chaotic system (1) usually will not go beyond fourth order. Therefore, the subscripts is less than or equal to 4 (i.e., ).

If we set , the general class of Genesio–Tesi system is obtained. This system is one of paradigms of chaos since it captures many features of chaotic systems. It includes a simple square part and three simple ordinary differential equations that depend on three positive real parameters. Let us consider for this study the simple autonomous jerk system with multiple attractors presented recently by Kengne et al. and described by the following dynamics equations [13, 14]:where and are the positive constants and the polynomial smooth function. For instance, the system is chaotic for the parameters , , . Equation (2) represents a reliable and palpable resource for generating a wide variety of nonlinear phenomena including the multiple stability behavior. This system is capable of displaying many disconnected attractors (for some suitable sets of parameters) depending solely on the choice of initial conditions [13]. The following section underlines some unexplored parameter’s regions of systems proposed in (2) which shows that many attractors coexist.

##### 2.2. Stability Analysis of the Attractors

This section presents in two complementary ways (described below) phase diagrams characterizing the far-reaching regular organization induced by the set of stable oscillations of the circuit. Although obtained using two very distinct algorithms, the boundaries between chaotic and periodic regions match perfectly by plotting on a fine parameter grid the largest nonzero Lyapunov exponent. Such exponents are familiar indicators that allow one to discriminate chaos (positive exponents) from periodic oscillations (negative exponents). Figures 1 and 2 depict the behavior of MO5 oscillator in the plane for a mesh of parameters points. The results are obtained by using the standard fourth-order Runge-Kutta algorithm with fixed time step . Figures 1 and 2 are obtained by adopting the initial values as and , respectively. As usual the first integration step disregarded as a transient time is considered to approach the attractor. The discrimination of the solutions and the account of the number of peaks within a period of are detailed in [25]. Figures 1(a) and 2(a) display the Lyapunov stability diagram, obtained by plotting in two dimensions (in the plane ) the largest nonzero Lyapunov exponent for the same parameters. The initial conditions are adopted as and , respectively. The orange shadings mark periodic oscillations (negative exponents); the yellow and black colors denote the chaotic behaviors (positive exponent). It is worth noting that the diagrams plotted for the same values of parameters and the different initial conditions should be identical in the case where the circuit depicts no multistability fashion. This aspect is not observed in these two figures. The Lyapunov exponent points out this difference. This method is limited because it can only bring out the regions of coexistence between chaos and regularity, when plotting the isospike diagrams to complete the analysis. Figures 1(b) and 1(c) display the isospike diagrams in the plane for the same values of the parameters for the following initial conditions fixed as and , respectively. We use a palette of 17 colors to represent the number of spikes contained in one period of the oscillate state as indicated by the color dots. Within the parameter range chosen, we obtain 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, and 17 spikes in a single period of . The black color denotes the chaotic behavior. These two figures are used to study the 2D multistability in MO5 oscillators. They consistency shows all coexistences between the regular and nonregular oscillations.