Papers
International Journal of Bifurcation and Chaos, Vol. 12, No. 1 (2002) 23–41
c
World Scientific Publishing Company
FAMILIES OF SCROLL GRID ATTRACTORS
M
¨
US¸TAK E. YALC¸IN
∗
, JOHAN A. K. SUYKENS
†
and JOOS VANDEWALLE
‡
Katholieke Universiteit Leuven,
Department of Electrical Engineering ESAT-SISTA,
Kardinaal Mercierlaan 94, B-3001 Leuven, Belgium
∗
mey@esat.kuleuven.ac.be
†
johan.suykens@esat.kuleuven.ac.be
‡
joos.vandewalle@esat.kuleuven.ac.be
SERDAR
¨
OZO
˘
GUZ
Istanbul Technical University,
Faculty of Electrical-Electronics Engineering
80626, Maslak, Istanbul, Turkey
serdar@ehb.itu.edu.tr
Received February 13, 2001; Revised April 9, 2001
In this paper a new family of scroll grid attractors is presented. These families are classified into
three called 1D-, 2D- and 3D-grid scroll attractors depending on the location of the equilibrium
points in state space. The scrolls generated from 1D-, 2D- and 3D-grid scroll attractors are
located around the equilibrium points on a line, on a plane or in 3D, respectively. Due to the
generalization of the nonlinear characteristics, it is possible to increase the number of scrolls
in all state variable directions. A number of strange attractors from the scroll grid attractor
families are presented. They have been experimentally verified using current feedback opamps.
Also Lur’e representations are given for the scroll grid attractor families.
1. Introduction
Since the discovery of Chua’s circuit [Chua et al.,
1986; Madan, 1993; Chua, 1994] many scientists
from different disciplines have been studying the
double scroll family. Chua’s circuit is a simple
third-order piecewise-linear (PWL) system which
has become a paradigm for chaos. The realization
of chaotic systems brought chaotic signals into en-
gineering applications. Presently, many researchers
investigate the applications of chaotic signals to
communication systems [Kolumban et al., 1998;
Hasler, 1994]. One open question is how one can
systematically increase the complexity of behavior
while keeping the systems as simple as possible.
The new circuits presented in this paper give an
affirmative answer to that question. Amongst the
many generalizations of Chua’s circuit, a more com-
plicated double scroll family of so-called n-double
scroll attractors has been proposed by Suykens and
Vandewalle [1993] by introducing additional break-
points in the nonlinearity. A more complete fam-
ily of n-scroll instead of n-double scroll attractors
has been obtained from a generalized Chua’s circuit
reported in [Suykens et al., 1997]. Experimental
confirmations of 2-double scroll and 5-scroll attrac-
tors have been given in [Arena et al., 1996] and
[Yal¸cın et al., 2000a], respectively. The basic idea
of generalizing the chaos generators with PWL non-
linearities is to introduce additional breakpoints in
the nonlinearity. These breakpoints create equilib-
rium points which are located on a line in state
space. Here, we will consider a new chaos genera-
tor which has a simple circuit implementation. The
†
Author for correspondence.
23
24 M. E. Yal¸cın et al.
strange attractor families generated from the new
chaos generators will be called scroll grid attractors.
For these families it is possible to cover the whole
state space with scrolls. The new attractor families
are classified into three subfamilies according to the
location of the equilibrium points:
• 1D-grid scroll attractor family: This is also known
as n-scroll attractors [Suykens et al., 1997]. The
equilibrium points of this family are located on
a line and the scrolls generated from the gener-
alized nonlinearity are located around that line
along the x state variable direction in state space.
Furthermore, the x state variable is also the vari-
able on which the nonlinearity operates.
• 2D-grid scroll attractor family: In this family, the
system consists of two nonlinear functions oper-
ating on the x and y state variables. The equilib-
rium points are located in the x − y plane. The
scrolls generated from the generalized nonlineari-
ties can be increased in the x and y state variable
directions.
• 3D-grid scroll attractor family: This is the most
complete class of the presented scroll grid attrac-
tor families. The equilibrium points are located in
3D and the system has three nonlinear functions.
Due to the generalization of each nonlinearity,
the scrolls can be generated in all state variable
directions.
In this paper, the main contribution is to show the
possibility of generating the equilibrium points on
a plane or in 3D instead of on a line. As a re-
sult, it is possible to increase the number of scrolls
into all state variable directions. In the literature, a
quad screw attractor [Kataoka & Saito, 2000] from
a 4D chaotic oscillator with hysteresis [Saito, 1990]
is comparable with a 2 × 2-scroll grid attractor,
which is a member of the 2D-grid scroll attractor
family. However, it should be noted that the sys-
tem presented here is simpler than the other one.
Moreover, the 2 × 2-scroll grid attractor is only one
member of the scroll grid attractor family. It will be
shown that this family can be extended by adding
a simple nonlinearity. Another comparable attrac-
tor family are n-double scroll hypercubes [Suykens
& Chua, 1997] which occur in weak unidirectional
or diffusive coupling of n-double scroll cells within
one-dimensional Cellular Neural Networks [Chua &
Roska, 1993]. However, this family produces hy-
perchaotic behavior and the order of the system is
much higher than the third-order circuit proposed
in this paper. From a system and control theo-
retical point of view, the proposed system can be
represented as a Lur’e system. Hence, many results
concerning stability and synchronization are appli-
cable to it [Vidyasagar, 1993; Khalil, 1993; Suykens
et al., 1999]. From a circuit design point of view, the
new circuit is easily realized by using simple com-
parators. Moreover, it is possible to systematically
increase the complexity of the circuit, by simply
using additional core nonlinearities. From an appli-
cation point of view, this system can produce more
complicated signals. Hence it is promising in many
applications for chaotic systems as communications
and cryptosystems.
This paper is organized as follows. In Sec. 2 we
present a generalized chaos generator which pro-
duces 1D-grid scroll attractors. 2D- and 3D-grid
scroll attractors are presented in Secs. 3 and 4, re-
spectively. In Sec. 5 Lur’e representations for the
systems are given. Finally, in Sec. 6 the realization
of some of 1D-, 2D- and 3D-grid scroll attractors is
given.
2. A New Family of n-Scroll
Attractors
A simple chaos generator model has been recently
proposed by Elwakil et al. [2000] which is described
by
˙x = Ax + BΦ(x)(1)
with
A =
010
001
−a −a −a
, B =
000
000
00a
,
Φ=
0
0
f
1
(x)
where
f
1
(ζ)=
(
1,ζ≥ 0
−1,ζ<0 ,
(2)
and x =[x; y; z] ∈
R
3
, ζ ∈ R. In [Elwakil
et al., 2000], it has been reported that the model is
extremely simple and produces a double scroll-like
attractor for a =0.8. A generalization of this origi-
nal model for generating n-scrolls has been recently
Families of Scroll Grid Attractors 25
−1 −0.5 0 0.5 1 1.5
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
x
y
Fig. 1. Double scroll attractor; f
1
(x) − g
0.5
(x), x
0
− [0.4565; 0.0185; 0.8214].
0011
01
01
00
00
11
11
0
0
1
1
01
00
00
11
11
000
000
000
000
000
111
111
111
111
111
00
00
00
00
00
11
11
11
11
11
00
00
00
00
11
11
11
11
00
00
00
00
11
11
11
11
00
00
11
11
00
00
11
11
00
00
11
11
0
0
1
1
0
0
1
1
x
3/2
5/2
1/2
-5/2
-3/2
-1/2
-2
-1
-3
1
2
3
f(x)
x
Fig. 2. 1D-grid scroll attractors: the equilibrium points (brown box) are shown at the intersection of x (red solid) and f(x)
(blue solid).
26 M. E. Yal¸cın et al.
given by Yal¸cın et al. [2000c]. Here, we consider
a minor modification of the latter model by taking
the nonlinearity
f
1
(x)=
M
x
X
i=1
g
(−2i+1)
2
(x)+
N
x
X
i=1
g
(2i−1)
2
(x)(3)
where
g
θ
(ζ)=
1,ζ≥ θθ>0
0,ζ<θ θ>0
0,ζ≥ θθ<0
−1,ζ<θθ<0 .
(4)
A computer simulation for the double scroll attrac-
tor is shown in Fig. 1 corresponding to M
x
=0,
N
x
=1anda =0.8. A generalization of the system
Eq. (1) can be systematically obtained by introduc-
ing additional breakpoints in the nonlinearity where
each breakpoint can be implemented by Eq. (4).
Therefore, we call Eq. (4) the core function. The
equilibrium points can be found from the following
set of equations
x = f
1
(x)
y =0
z =0.
The equilibrium points are located at the intersec-
tion of the nonlinear function f
1
(x)andx drawn in
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
y
(a)
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
−1
−0.5
0
0.5
1
x
y
(b)
−5 −4 −3 −2 −1 0 1 2 3 4 5 6
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x
y
(c)
Fig. 3. (a) M =1,N = 1, 3-scroll attractor, x
0
=[0.30460; 18970; 1934]; (b) M =0,N = 4, 5-scroll attractor, x
0
=
[0.3529; 0.8132; 0.0099]; (c) M =4,N = 5, 10-scroll attractor, x
0
=[0.6721; 0.8381; 0.0196].
Families of Scroll Grid Attractors 27
−6 −4 −2 0 2 4 6 8 10 12 14
−3
−2
−1
0
1
2
3
x
y
a=0.1
(a)
−2 −1 0 1 2 3 4 5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
y
a=.23
(b)
−1 0 1 2 3 4 5
−1.5
−1
−0.5
0
0.5
1
x
y
a=.34
(c)
−1 0 1 2 3 4 5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
y
a=.37
(d)
−1 0 1 2 3 4 5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
a=.41
x
y
(e)
−1 0 1 2 3 4 5
−1
−0.5
0
0.5
1
1.5
x
y
a=.47
(f)
−1 0 1 2 3 4 5
−1
−0.5
0
0.5
1
1.5
x
y
a=.50
(g)
−1 0 1 2 3 4 5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x
y
a=.61
(h)
−1 0 1 2 3 4 5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
y
x
a=.7
(i)
−1 0 1 2 3 4 5
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x
y
a=.81
(j)
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
x
y
a=1
(k)
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
x
y
a=2
(l)
Fig. 4. Bifurcations related to 5-scroll attractors with respect to parameter value a:(a)a =0.1, for five different initial
conditions which are close to the equilibrium points; (b) a =0.23; (c) a =0.34; (d) a =0.37; (e) a =0.41; (f) a =0.47;
(g) a =0.5; (h) a =0.61; (i) a =0.7; (j) a =0.81, for five different initial conditions which are close to the equilibrium points;
(k) a =1;(1)a =2.
