Hybrid projective combination–combination synchronization in non-identical hyperchaotic systems using adaptive control

Abstract

In this paper, we investigate a hybrid projective combination-combination synchronization scheme among four non-identical hyperchaotic systems via adaptive control method. Based on Lyapunov stability theory, the considered approach identifies the unknown parameters and determines the asymptotic stability globally. It is observed that various synchronization techniques, for instance, chaos control problem, combination synchronization, projective synchronization, etc. turn into particular cases of combination-combination synchronization. The proposed scheme is applicable to secure communication and information processing. Finally, numerical simulations are performed to demonstrate the effectivity and correctness of the considered technique by using MATLAB. Mathematics Subject Classification 34K23 • 34K35 • 37B25 • 37N35 1 Introduction Chaos is described as a nonlinear extremely complex phenomenon found in nature featuring the high sensitivity to the initial conditions. These features are defined as butterfly effect, the term coined by Lorenz in 1963. Chaos theory is widely applicable to various fields of applied sciences and engineering, for instance, secure communication [1], biomedical engineering [2], machine learning, circuits [3], finance models [4], jerk systems [5], weather models [6], neural networks [7], oscillations [8], robotics [9], chemical reactions [10], encryption [11], ecological models [12], etc. Consequently, chaos theory has become one of the most appealing fields for researchers and engineers in recent times. Historically, chaos theory dates back to the remarkable work of Poincare [13] established in 1890s while dealing with three-body problem to stabilize the solar system. In fact, he advocated the qualitatively behaviour, using geometric quantitatively to display the universal configuration of all solutions. Despite the observations made by Poincare [13], the first formal introduction of chaos in a deterministic system was proposed by Lorenz [14]. Lorenz observed that simple meteorological models depicted sensitive dependence on the initial conditions, known as butterfly effect, which established that chaos theory is widely applicable in other interdisciplinary areas also. Though the fundamental work on chaos synchronization had been carried by [15], the rapid growth in the field was seen after the phenomenal work of Ott et al. [16]. However, the concept of chaos synchronization was first introduced by Pecora and Carroll [17], wherein they performed synchronization of two identical chaotic systems based on master-slave composition which was unknown for the past 3 decades. Later on, many researchers carried forward the pioneered work of Pecora and Carroll and it has been established that chaos synchronization is also achievable for non-identical chaotic systems possessing absolutely different initial conditions.

A Comprehensive Analysis on Controlling and Hybrid Synchronization in Identical Chaotic Systems via Active Control Method

“…In past few decades, the existent literature exhibits an important interest in introducing the novel chaotic systems, for instance, Rossler system, Vaidyananthan, Lu system, Chen system, Liu system, Zhu system, Elhadj system etc. In recent times, several schemes to achieve synchronization as well as control among chaotic systems have been reported and discussed such as nonlinear control [3,4,5,6], feedback [7,8,9], back-stepping design [10,11], OYG method [12], sliding mode control [13], passive control [14], impulsive [15], adaptive control [3,7,16], fuzzy control, projective synchronization [17], global synchronization, complete [18], hybrid projective [16,19], anti [20,21], combination synchronization [19], combination difference [3,20,22], phase [23], lag [24], combination-combination [16], compound [25], etc. Keeping these discussions in mind, our immediate aim is to design a methodology to investigate hybrid synchronization (HS) in identical chaotic chemical reactor systems using active control method (ACM).…”

Controlling Chaos in a Newly Designed Chaotic Hamiltonian System Based on Hénon-Heiles Model using Active Controlled Hybrid Projective Synchronization

“…In recent times, chaos synchronization in chaotic systems utilizing various control methods has become an engaging and a fascinating topic of study for the scientists and researchers. In the last three decades, several techniques of significance are initiated and studied to control [14][15][16][17][18] and synchronization [19][20][21][22][23][24][25][26][27] of chaos existing in dynamical systems.…”

Stability Analysis of Chaotic New Hamiltonian System Based on Henon-Heiles Model Using Adaptive Controlled Hybrid Projective Synchronization

“…In recent years, chaos synchronization of chaotic systems using various control techniques has become a fascinating and engaging area of study for the researchers and scientists. Many significant techniques are introduced and studied to control [22,2,8] and synchronization [24,12,25,9,10,17,3,7,11] of chaos occurring in dynamical systems.…”