A preview of this full-text is provided by Springer Nature.
Content available from International Journal of Dynamics and Control
This content is subject to copyright. Terms and conditions apply.
Int. J. Dynam. Control (2017) 5:1114–1121
DOI 10.1007/s40435-016-0258-6
Hybrid function projective synchronization of chaotic systems
via adaptive control
Ayub Khan1·Shikha1
Received: 3 March 2016 / Revised: 8 June 2016 / Accepted: 27 June 2016 / Published online: 11 July 2016
© Springer-Verlag Berlin Heidelberg 2016
Abstract In this manuscript, hybrid function projective
synchronization of Bhalekar–Gejji and Pehlivan chaotic
systems is established by applying adaptive control tech-
nique where the system parameters are unknown. In this
manuscript both the master and slave system are cho-
sen in such a way that none of them can be derived
from the member of the unified chaotic system. We con-
struct an adaptive controller in such a manner that master
and slave system attain global chaos synchronization. The
results derived for the synchronization have been estab-
lished using adaptive control theory and Lyapunov stabil-
ity theory. Fundamental dynamical properties of both the
chaotic systems are also described. The results are vali-
dated by numerical simulation which are performed by using
Matlab.
Keywords Chaotic system ·Fundamental dynamical
properties ·Synchronization ·Adaptive control ·Lyapunov
stability theory
1 Introduction
Chaotic dynamics is a fascinating area in nonlinear sci-
ence which has been thoroughly studied during the last few
decades. Chaotic phenomena is observed in many areas for
instance chemical systems, electrical engineering, biologi-
cal systems, power converters, computer science, celestial
BShikha
sshikha7014@gmail.com
Ayub Khan
akhan12@jmi.ac.in
1Department of Mathematics, Jamia Millia Islamia,
New Delhi 110025, India
mechanics, quantum physics, traffic forecasting, psychology,
secure communication and so on [1]. A chaotic system shows
complex dynamics and some of the special characteristics of
these systems are high sensitive dependence on initial condi-
tions, topological mixing, density of periodic orbits, strange
attractors, broad spectrums of Fourier transform, bounded
and fractal properties of the motion in the phase space etc [2].
In 1990, Pecora and Carroll [3] gave the synchronization of
chaotic systems using the concept of master and slave sys-
tem. Also, in 1990 Ott et al. [4] introduced the OGY method
for controlling chaos. Due to the vast practical applications
of chaotic dynamical systems in fields stated above, many
researches have been done theoretically and experimentally
on controlling chaos and synchronization [5,6].
Specifically, synchronization of nonlinear dynamical sys-
tems gives the capability to gain an accurate and deep
understanding of collective dynamical behavior in physical,
chemical and biological and other systems. The presence of
synchronous behavior has been observed in different mathe-
matical, physical, sociology, physiology, biological and other
systems [32]. In search of better methods for chaos control
and synchronization different types of methods have been
developed for controlling chaos and synchronization of non
identical and identical systems for instance linear feedback
[7], optimal control [8], adaptive control [9], active con-
trol [10], active sliding control [11], passive control [12],
impulsive control [13], backstepping control [14], etc [31].
In these published works, it is essential to know the values
of system’s parameters for the derivation of the controller. In
practical situations, these parameters are unknown. There-
fore, the derivation of an adaptive controller for the control
and synchronization of chaotic systems in the presence of
unknown system parameters is an important issue [15–17].
So far, a variety of synchronization approaches, such as
complete synchronization (CS) [18], generalized synchro-
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
