Analysis, synchronization and circuit design of a novel butterfly attractor

Highlights

• We introduce a novel chaotic system whose dynamics support butterfly attractors.

• Qualitative properties of the novel chaotic system are discussed in detail.

• Adaptive control laws are derived to achieve global chaos synchronization.

• A novel electronic circuit the new chaotic system is designed and realized.

• We show a good qualitative agreement between the simulations and the experiments.

Abstract

This research paper introduces a novel three-dimensional autonomous system, whose dynamics support periodic and chaotic butterfly attractors as certain parameters vary. A special case of this system, exhibiting reflectional symmetry, is amenable to analytical and numerical analysis. Qualitative properties of the new chaotic system are discussed in detail. Adaptive control laws are derived to achieve global chaotic synchronization of the new chaotic system with unknown parameters. Furthermore, a novel electronic circuit realization of the new chaotic system is presented, examined and realized using Orcad-PSpice program and physical components. The proposed novel butterfly chaotic attractor is very useful for the deliberate generation of chaos in applications.

Introduction

Many scientists were amazed when Lorenz [1] discovered chaos in a simple system of three-dimensional (3D) autonomous ordinary differential equations with quadratic nonlinearities. Since then many such simple examples of chaotic behaviour been found and studied within the context of 3D quadratic autonomous systems.

In 1976, Ro¨ssler introduced another canonical low dimensional dissipative dynamical system [2]. Sprott [3] embarked upon an extensive search for autonomous 3D chaotic systems. Chen [4], [5] constructed another chaotic system which was not topologically equivalent to the Lorenz system. Lü and Chen [6] found an interesting new chaotic system, which represented the transition between the Lorenz and Chen attractors.

Some 3D autonomous chaotic systems have three equilibrium points: for example, a saddle and two unstable saddle-foci, at typical values for the existence of a chaotic attractor [1], [4], [5], [7]. Other 3D chaotic systems, such as the Ro¨ssler system [2], the diffusionless Lorenz System (DLS) [8] and the Burke-Shaw system [9], have two unstable saddle-foci. Recently, Yang et al. [10] and Pehlivan and Uyaroglu [11] introduced and analyzed new 3D chaotic systems with six terms, having only two quadratic terms in a form very similar to the Lorenz and Chen systems, but with two very different fixed points: two stable node-foci. Therefore, the study of such dynamics is of great interest.

For a set of low-dimensional nonlinear Faraday disk dynamo models, Moroz [7], [12], [13] extracted the lowest order unstable periodic orbits from long time series integrations in the chaotic régimes. The arrangement of these unstable periodic orbits on the branched manifold of the chaotic attractor was found to be topologically equivalent to that of the Lorenz system [1] at its classic parameter values.

There has been increasing interest in chaos-based engineering applications: effectively creating chaos via simple physical systems, such as electronic circuits. Some important analysis of chaotic electronic circuits was carried out by Cuomo and Oppenheim [14], Elwakil and Kennedy [15], Cam [16], Pehlivan [17] and Sundarapandian and Pehlivan [18].

Recently, designing circuits to produce chaotic attractors has become a focus for electronics engineers, because of the potential in real-world applications in various chaos-based technologies and information systems [11], [19], [20], [21], [22], [23], [24]. Motivated by such previous works, this paper introduces another novel chaotic system, a butterfly attractor. A detailed qualitative analysis is presented for the novel chaotic system and its properties.

In this research work, we introduce a novel 3-D autonomous chaotic system whose dynamics support periodic and chaotic butterfly attractors as certain parameters vary. Our novel chaotic system has four quadratic nonlinearities, which are more than necessary for generating chaos. Hence, our novel chaotic system is not a simple chaotic flow [3]. Our main interest in this research work is to study the chaos generated with the four quadratic nonlinearities in the system and to discuss in detail the qualitative properties of the system.

Since the seminal paper by Ott et al. [25], the control of chaotic systems has been recognized as an important problem in the literature. The idea behind the control of a chaotic system is to devise a state feedback control law to stabilize the system around its unstable equilibrium points [26], [27].

Synchronization of chaotic systems is said to occur when a chaotic attractor drives another chaotic attractor. In the last two decades, there has been considerable interest devoted to the synchronization of chaotic and hyperchaotic systems. Active control methods are used for the global control of a system when the system parameters are known. Adaptive control methods are used for global control of a system when the system parameters are unknown.

In their seminal paper in 1990, Pecora and Carroll [28] proposed a method to synchronize two identical chaotic systems and showed that it was possible for some chaotic systems to become globally synchronized.

In the last two decades, chaos control and synchronization have been applied to a wide variety of fields including biology [29], ecology [30], cardiology [31], robotics [32], physics [33], chemistry [34], and secure communications [35].

Some common methods applied to the chaos synchronization problem are active control [36], [37], [38], [39], [40], adaptive control [41], [42], [43], sampled-data feedback methods [44], [45], time-delay feedback methods [46], [47], backstepping methods [48], [49], [50], [51], [52], and sliding mode control methods [53], [54], [55], [56].

In this paper, an adaptive control law has been designed for the global chaotic synchronization of the novel butterfly attractor, when the system parameters are unknown. The synchronization result has been established using Lyapunov stability theory [57]. Moreover, a novel electronic circuit realization of the new chaotic system is presented and examined using the $\mathrm{OrCad}\mathrm{-}{\mathrm{PSpice}}^{\circledR }$ program.

This paper is organized as follows. In Section 2, we introduce the model and techniques to be used to explore this system. The most general formulation has four free parameters, one of which can always be removed by rescaling time. We set one of the remaining parameters to zero. This yields the simplest tractable problem in which the equations become invariant under reflectional symmetry. Section 3 contains the linear stability and bifurcation analysis, while in Section 4 we report on some numerical integrations that were performed. We include bifurcation transition diagrams, a selection of phase portraits and introduce a new limit cycle. We also extract the lowest order upos, using the method of close returns on a Poincaré section for some typical examples from the chaotic régime.

This paper also discusses the adaptive synchronization of the new chaotic system exhibiting. In Section 5, an adaptive synchronization law has been designed to synchronize identical chaotic systems with unknown parameters, and in Section 6, a new circuit realization of the proposed chaotic system is presented and examined using the $\mathrm{OrCad}\mathrm{-}{\mathrm{PSpice}}^{\circledR }$ program for one of the parameter choices. Finally, conclusions and discussions are given in Section 7.