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International Journal of Dynamics and Control (2018) 6:1160–1174
https://doi.org/10.1007/s40435-017-0382-y
Dynamical behavior and reduced-order combination synchronization
of a novel chaotic system
Khan Ayub1·Shikha1
Received: 8 September 2017 / Revised: 20 November 2017 / Accepted: 6 December 2017 / Published online: 16 December 2017
© Springer-Verlag GmbH Germany, part of Springer Nature 2017
Abstract
This manuscript investigates a novel 3D autonomous chaotic system which generates two strange attractors. The Lyapunov
exponent, bifurcation diagram, Poincaré section, Kaplan–Yorke dimension, equilibria and phase portraits are given to justify
the chaotic nature of the system. The novel system displays fixed orbit, periodic orbit, chaotic orbit as the parameter value
varies. The reduced order combination synchronization is also performed by considering three identical 3D novel chaotic
systems in two parts (a) choosing two third order master systems and one second order slave system which is the projection
in the 2D plane. (b) choosing one third order master system and two second order slave systems which are the projection in
the 2D plane. Numerical simulations justify the validity of the theoretical results discussed.
Keywords Chaotic system ·Lyapunov exponent ·Bifurcation diagram ·Poincaré section ·Kaplan–Yorke dimension ·
Reduced order ·Combination synchronization
1 Introduction
In the most recent decades specialists from all fields of natu-
ral sciences have considered wonders that include nonlinear
systems showing chaotic nature [1–8]. The peculiar rea-
son for this is that the chaotic nonlinear dynamics systems
display very rich dynamical behavior. Chaotic systems are
third order or higher order nonlinear differential equations
with at least one positive Lyapunov exponent. In 1976 [9]
Rössler led imperative work that revived the enthusiasm in
3D autonomous chaotic system. Since Lorenz proposed a
astounding chaotic system for portraying atmospheric con-
vention in 1963 [10], hordes of consideration have been
committed to the examination to the autonomous system.
Recently, there has been increasing enthusiasm in creating
chaos particularly since Chen and Ueta [5] found a new
chaotic system, called the Chen system, by adding a simple
state feedback to the second equation of the Lorenz system,
which is not topologically equivalent to the Lorenz system.
BShikha
sshikha7014@gmail.com
Khan Ayub
akhan12@jmi.ac.in
1Department of Mathematics, Jamia Millia Islamia, New Delhi
110 025, India
After that, Lü and Chen [6] further constructed a new chaotic
system, bearing the name of the Lü system, which represents
the transition between the Lorenz system and Chen system.
Thereafter,a very rich family of the so-called unified system
or Lorenz system family [7] was introduced as a connection
of the Lorenz, Chen, and Lu systems.
Analyzing the dynamics of such chaotic systems got a lot
of enthusiasm for the current past because to its applications
in secure communication [8], information processing [11],
biological systems [12], chemical reactions [13], neural net-
work [14] etc. So, more an more chaotic or hyper chaotic
systems showing wide dynamical behavior were found [15]–
[24]. For example Wu and Li [15] presented a new chaotic
system and studied its basic dynamical properties and then
synchronize it using different control techniques. Kingni et.
al. [16] introduced a novel chaotic system with a circular
equilibrium. Analysis, circuit simulation and its fractional
order form is also discussed. Tacha et. al. [17] obtained a
novel nonlinear finance system. Adaptive control is used
by the author to synchronize and circuit emulation is also
being done. Tong [18] describe the forming mechanism of
the chaotic system and analyses its dynamics. Zhang and
Han [19] describes the dynamic analysis of a autonomous
chaotic system with cubic nonlinearity. Çiçek et. al. [20]
gives a 3D chaotic system which presents its application
in secure communication. Deng et. al. [21] introduces a
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