Generation of n-scroll attractors by Josephson junctions

Abstract

In this paper Josephson junctions are used in order to generate n-scroll attractors. Instead of synthesiz-ing the nonlinearity of n-scroll attractors via linear compo-nents, we propose this paper to make use of the nonlinearity of the Josephson junction itself.

Full text

Generation of n-scroll attractors by Josephson junctions
M¨
us¸tak E. Yalc¸ın†and Johan A. K. Suykens‡
†Istanbul Technical University,
Faculty of Electrical and Electronics Engineering,
Electronics and Communication Department
80626, Maslak, ˙
Istanbul, Turkey
‡Katholieke Universiteit Leuven
Department of Electrical Eng. ESAT, SCD/SISTA
Kasteelpark Arenberg 10,
B-3001, Leuven, Belgium,
Email: mustak.yalcin@itu.edu.tr, johan.suykens@esat.kuleuven.be
Abstract—In this paper Josephson junctions are used in
order to generate n-scroll attractors. Instead of synthesiz-
ing the nonlinearity of n-scroll attractors via linear compo-
nents, we propose this paper to make use of the nonlinearity
of the Josephson junction itself.
1. Introduction
Since the observation of n-double scroll attractors [9]
from a generalized Chua’s circuit [2], multi-scroll attrac-
tors [13, 5] have received considerable attention. Short af-
ter the presentation of n-double scroll attractors, n-scroll
attractor have been studied by Suykens et al. [8] which
also allows for an odd instead of an even number of scrolls.
Then an experimental confirmation of n-double scroll and
n-scroll attractors were given followed by the circuit real-
ization of 2-double scroll attractor [1] and 5-scroll attrac-
tor [12]. In n-scroll attractors the scrolls are only located
along one state variable direction in state space. The fam-
ily of scroll grid attractors has been introduced where this
is no longer a restriction [11]. These families allow that the
scrolls can be located in any state variable direction. Cur-
rently multi-scroll chaotic attractors is a common umbrella
for the collection of n-scroll, n-double scroll, and families
of scroll grid chaotic attractors.
A design of multi-scroll chaotic attractor is basically a
problem of design of multiple equilibrium points in a scroll
base chaotic attractor by modifications of nonlinear char-
acteristic. In [9] the modification is done by introducing
additional breakpoints in the piecewise nonlinear charac-
teristic of Chua’s circuit in order to obtain n-double scroll
attractors. In [10] the same modification is done by the sine
functions. In [11] and [6] authors used a collection of the
step functions and smooth hyperbolic tangent functions for
the modifications, respectively.
In fact the design and realization of multi-scroll attrac-
tors depends on synthesizing the nonlinearity with an elec-
trical circuit. The question arises whether there exists an
electrical device that can naturally allow to design a multi-
scroll chaotic attractor. In this paper we want to argue that
Josephson junctions [4] are suitable candidate devices that
possess such a nonlinearity. Josephson Junctions are super-
conducting devices that can generate high frequency oscil-
lations. In [3] chaotic dynamics from Josephson junction
have been reported. In this paper, in the first introduced
model we employ the nonlinearity of the junctions in a n-
scroll attractor. Then we use also the dynamics of the junc-
tions in a second model for generating n-scroll attractor.
The phase difference of the junctions will be one of the
state variable of the resulting n-scroll attractor.
This paper is organized as follows. In Section 1 we pro-
pose n-scroll attractors obtained as 1-D scroll grid attrac-
tors using a sine function. In Section 2 two new models are
introduced by making use of the Josephson junction non-
linear characteristic and the phase dynamics together with
its nonlinearity.
2. n-scroll attractors via the sine function
In [10] a sine function was replacing the nonlinear char-
acteristic of Chua’s circuit. With the sine function different
numbers of scrolls can be designed. Here we apply a simi-
lar approach to a simple Mx+Nx+1-scroll attractor
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
˙x=y
˙y=z
˙z=−ax −ay −az +af(x)(1)
where
f(x)=
Mx

i=1
g(−2i+1)
2(x)+
Nx

i=1
g(2i−1)
2(x) (2)
and
gθ(ζ)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1,ζ≥θ, θ > 0
0,ζ<θ,θ>0
0,ζ≥θ, θ < 0
−1,ζ<θ,θ<0.
(3)
x,y,z∈R,ζ∈R[13]. In [6] the authors have replaced
the nonlinear function (3) by a smooth hyperbolic tangent
Bruges, Belgium, October 18-21, 2005
Theory and its Applications (NOLTA2005)
2005 International Symposium on Nonlinear
501

−25 −20 −15 −10 −5 0 5 10 15 2
0
−1.5
−1
−0.5
0
0.5
1
1.5
x
y
Figure 1: 11-scroll attractor from (4) (a=0.3, b=0.25,
initial conditions (.1,.2,.3))and numerical simulations are
performed in Matlab (ode23) until t=6000.
function. Here we replace the nonlinearity by a sine func-
tion. Hence the new model is described by
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
˙x=y
˙y=z
˙z=−ay −az +ag(x)(4)
where g(x)=sin(2πbx) (5)
and b∈R.Figure 1. shows 11-scroll attractors obtained
from the system (4) for a=0.3, b=0.25, initial condi-
tions (.1,.2,.3) and numerical simulations are performed in
Matlab using a Runge-Kutta integration rule (ode23) un-
til t=6000. Figure 2 shows 15-scroll attractors obtained
from the same system with the same parameter when the
numerical simulations are performed until t=9000. The
reason to obtain different number of scrolls for the differ-
ent simulation time is indeed that the system is an n-scroll
attractor for the given parameters and the number of scrolls
(n) is defined by the system itself. The sine function in
[10], which is defined by
h(x)=⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
bπ
2s(x−2ac),if x≥2ac
−bsinxπ
2a+d,if −2ac <x<2ac
bπ
2s(x+2ac),if x≤−2ac.
,(6)
is used for a given interval of the variable (x) of the non-
linearity and the scrolls are obtained within this interval.
There is no restriction here to locate the scrolls within an
interval, therefore the number of scrolls can not be known.
3. Josephson Junctions and n-scroll attractors
Josephson junctions are highly nonlinear superconduct-
ing electronic devices. It is also well-known that Joseph-
−30 −20 −10 0 10 20 30 4
0
−1.5
−1
−0.5
0
0.5
1
1.5
x
y
Figure 2: 15-scroll attractor from (4) (a=0.3, b=0.25,
initial conditions (.1,.2,.3)) and t=9000.
son Junctions are superconducting devices that can gener-
ate high frequency oscillations [7]. The aim here is basi-
cally to use the nonlinearity of a Josephson junction in the
model 5. The current in a Josephson junction is described
by I=Icsinφ(7)
where ˙
φ=kV.(8)
Here φis the phase difference and Vis the voltage across
the junction. In a superconducting Josephson junction kis
defined by the fundamental constants k=2e
h(his Planck’s
constant divided by 2πand eis the charge on the electron).
Instead of designing a sine function for the n-scroll at-
tractor which was discussed in the previous section here
the nonlinearity of the Josephson junction (7) is applied to
the model (4). For that reason the current in the Josephson
junction is chosen as g(x) and the voltage across the junc-
tion set to ystate variable. Hence the new model with the
Josephson junction is given by
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙x=y
˙y=z
˙z=−ay −az +a
2IcI
˙
φ=ky
(9)
where Iis given in Eq. (7). Figure 3. shows 10-scroll at-
tractors from the system (9) for a=0.1 and k=1. Further-
more in Figure 4 n-scroll attractors from the same system
(9) are shown for a=0.1 and k=2. A zoom of Figure 4.
is shown in Figure 5.
The model (9) can also be described by
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
˙
φ=ky
˙y=z
˙z=−ay −az +a
2IcI.
(10)
In this case the Josephson junction is integrated to the sys-
tem and phase difference (φ) is one of the state variable of
502

−40 −30 −20 −10 0 10 20 3
0
−1.5
−1
−0.5
0
0.5
1
1.5
x
y
Figure 3: n-scroll attractor from the model (9) with k=1,
a=0.1.
−40 −30 −20 −10 0 10 20 30 4
0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
y
Figure 4: n-scroll attractor from the model (9) with k=2,
a=0.1.
the n-scroll attractor (10). Figure 6 and 7 show the phase
portrait of an n-scroll attractor obtained from this model
with k=1 and k=3, respectively (a=0.1 in both).
4. Conclusions
Based on a model with sine function nonlinearity, a new
model for generating n-scroll attractors has been proposed
that makes use of Josephson junction nonlinearity. The
Josephson junction is integrated into the model itself such
that the phase difference of the junctions is one of the state
variables of the model. The results have been illustrated
with computer simulations.
Acknowledgments
This research work was partially carried out at the
ESAT laboratory and the Interdisciplinary Center of Neu-
−16 −14 −12 −10 −8 −6 −4 −2 0
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
x
y
Figure 5: Zoom of Figure 4.
−70 −60 −50 −40 −30 −20 −10 0 10 2
0
−1.5
−1
−0.5
0
0.5
1
1.5
φ
y
Figure 6: n-scroll attractor from the model (10) with k=1,
a=0.1.
ral Networks ICNN of the Katholieke Universiteit Leu-
ven, in the framework of the Belgian Programme on In-
teruniversity Poles of Attraction, initiated by the Bel-
gian State, Prime Minister’s Office for Science, Technol-
ogy and Culture (IUAP P4-02, IUAP P4-24, IUAP-V),
the Concerted Action Project Ambiorics of the Flemish
Community and the FWO projects G.0226.06, G.0211.05,
G.0499.04, G.0407.02. JS is an associate professor with
K.U. Leuven.
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