818 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 61, NO. 10, OCTOBER 2014
High-Order Grid Multiscroll Chaotic Attractors
Generated by the Second-Generation
Current Conveyor Circuit
Ting Zuo, Kehui Sun, Xingxing Ai, and Huihai Wang
Abstract—The multidirectional multiscroll chaotic attractors
generated by second-generation current conveyor (CCII) are
investigated. A fourth-order and a fifth-order grid multiscroll
chaotic system are presented by designing staircase function and
shifted voltage sign functions in the high-order Jerk system. Using
the CCIIs, a chaotic circuit is designed and implemented, which
can generate 4 × 2 × 2 × 2– and 4 × 2 × 2 × 2 × 2–scroll
chaotic attractors, respectively, via switching control. Compared
with the voltage operational-amplifier-based circuit, the proposed
circuit has better performance in terms of circuit structure and
bandwidth. Numerical simulations and experimental results are
accordant.
Index Terms—Chaos, grid multiscroll, high-order Jerk system,
second-generation current conveyor (CCII).
I. INTRODUCTION
C
HAOTIC attractors with a more complicated topological
structure such as multiscroll attractors have been inten-
sively investigated for their potential engineering applications
[1]–[12]. Since Yalcin et al. proposed a family of grid attractors
in 2002 [5], many nonlinear functions have been proposed to
generate grid multiscroll attractors, such as hyperbolic function
[6], step wave [7], and saturated sequence [8]. However, the
presented approaches are mainly confined to the third-order
system. There are very few reports on generating grid multi-
scroll attractors in higher order systems, since the higher the
order is, the more difficult to implement. In 1997, Suykens and
Chua first extended multiscroll to higher order systems [14].
In 2006, Chlouverakis and Sprott proposed a high-order Jerk
system [15] described by
a
n
d
n
x
dτ
n
+ a
n−1
d
n−1
x
dτ
n−1
+ ···+ a
1
dx
dτ
+ a
0
x = f(x) (1)
Manuscript received July 7, 2014; accepted July 30, 2014. Date of publica-
tion August 7, 2014; date of current version October 1, 2014. This work was
supported in part by the National Natural Science Foundation of China under
Grants 61161006 and 61073187, and in part by the Foundational Research Fund
for the Central Universities, China, under Grant 72150050651. This brief was
recommended by Associate Editor E. Tlelo-Cuautle.
T. Zuo, X. Ai, and H. Wang are with the School of Physics and Electronics,
Central South University, Changsha 410083, China.
K. Sun is with the School of Physics and Electronics, Central South Uni-
versity, Changsha 410083, China, and also with the School of Physics Science
and Technology, Xinjiang University, Urumqi 830046, China (e-mail: kehui@
csu.edu.cn).
Color versions of one or more of the figures in this brief are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCSII.2014.2345307
where n =4, 5. A multiscroll high-order circuit on the basis of
(1) was proposed by Liu and Yu [16], and f (x) is a staircase
function given by
f(x)=kp
sgn(x)+
N
i=1
sgn(x+2ip)+
N
i=1
sgn(x−2ip)
(2)
f(x)=kp
M
i=1
sgn(x+(2i−1)p)+
M
i=1
sgn(x−(2i−1)p)
(3)
where p is a control parameter, k is a constant, and N and M are
positive integers. Equations (2) and (3) can generate 2N +2–
and 2M +1–scroll chaotic attractors, respectively. By adjust-
ing the parameter p, we can control the width and the amplitude
of the step wave; thus, the distance between the equilibrium
points of each scroll and the size of scroll can be controlled.
However, system (1) just shows multiscroll attractors in one
direction. In [19], hyperchaotic multiscroll attractors are pre-
sented in n-dimensional systems (n ≥ 4), but there is no circuit
implementation.
On the other hand, an operational amplifier (op amp)-based
chaotic circuit requires a tradeoff between gain and bandwidth,
which leads to the limit on operating frequency. Nonlinear
functions, for instance, sign function, staircase function, and
saturated nonlinear function series (SNFS) are usually modeled
by using the piecewise linear (PWL) approach, but PWL mod-
els designed with op amps can only be used at low frequency
[11], [12]. To obtain signals with higher frequency, a current
conveyor (CC) is introduced, and it is an analog building block
in a current-mode circuit with high slew rate and wide band-
width [20]–[22], which can completely replace the voltage op
amp. An RC chaos generator utilizing the second-generation
CC (CCII) is presented in [23], but the design method lacks
versatility and universality. A five-scroll chaos generator circuit
is designed by second-generation dual-output CCs [25], but it
is verified only by PSpice simulation. In [27], a third-order
system along with SNFS is complemented with CCII+s, and
the chaotic frequency spectrum is centered to 43 kHz.
In this brief, high-order grid multiscroll chaotic systems are
constructed first by employing nonlinear functions into the
high-order Jerk system in Section II. In Section III, a high-order
chaotic circuit based on CCII is designed and realized, which
can generate four- and five-directional multiscroll chaotic at-
tractors, respectively. In the end, the conclusion is drawn.
1549-7747 © 2014
IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.