Download Dynamics of two coupled canonical Chua`s circuits

NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS
45
Dynamics of two coupled canonical Chua’s circuits
K. Thamilmaran and D. V. Senthilkumar
Abstract—The chaotic behaviour of the canonical Chua’s circuit is extensively reported in the literature. However, for the first time, two identical
canonical Chua’s circuits have been coupled resistively and the resultant
synchronization of chaos in them has been studied. The method of linear
difference signal has been applied. The corresponding differential equations have been integrated numerically and the synchronization threshold
of the coupling parameter has been identified. Prior to synchronization,
the coupled oscillator is found to exhibit hyperchaotic behaviour for certain
higher strengths of coupling parameter. The Lyapunov exponents of the
system and the bifurcation diagrams of the main synchronization regime
have also been presented. The results have been confirmed by laboratory
experiments.
Keywords—Chaos, Hyperchaos, Synchronization and Mutual coupling.
II. C IRCUIT REALIZATION OF COUPLED CANONICAL
C HUA’ S CIRCUITS AND ITS DYNAMICS : E XPERIMENTAL
S TUDY
The experimental circuit is shown in Fig. 1. It consists of
two canonical Chua’s circuit one being the drive system and the
other being the response system. Individually each canonical
which has
circuit consists of a nonlinear resistive element,
the typical
characteristic, viz., negative resistance around
the origin and positive slopes away from the break point and a
, which can be built using off-the
linear negative conductor,
shelf op-amp.
I. I NTRODUCTION
N the last decade control and synchronization of chaos in coupled systems have received considerable attention [1]. This is
because coupled systems are good models to describe nonlinear
phenomena in natural sciences, chemistry, communication engineering, etc [2]. Theoretically, when two chaotic systems which
are independently excited, interact mutually through a coupling
resistance, the coupled system reaches a state wherein there is
a perfect convergence of the state variables of both the systems
and mutual sharing of different dynamical features [3]. This
state of convergence arising due to adjustment of characteristics
of individual chaotic systems is defined as synchronization.
Recently, Pecora and Carroll have introduced the concept of
synchronized chaos [4] and the method of cascading synchronization [5]. This idea has been in fact successfully tested
in a variety of nonlinear dynamical systems, including Lorenz
equations, the Rossler system, hysteresis circuits, Chua’s circuit, and so on [6] - [10] . Also, synchronization of chaos has
been investigated in two mutually coupled identical nonlinear
systems [11], [12]. Very recently, Kyprianidis and Stouboulos
have studied coupled four-dimensional autonomous canonical
Chua’circuit with an external force [13], by adding one inductor,
one linear resistor and one voltage source to the original canonical Chua’s circuit, and so its dynamics is given by a system of
ten-coupled first-order differential equations.
In this paper we consider the evolution from nonsynchronized self-oscillations to synchronized one with mutual
coupling parameter in a system of two coupled canonical Chua’s
circuits. The most interesting aspect of this coupled oscillator
system is that it is the simplest (six dimension) coupled oscillator reported so far in the canonical Chua’s circuit family that
can exhibit hyperchaotic oscillations.
K. Thamilmaran and D. V. Senthilkumar are with Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirappalli - 620
024, India, e-mail: [email protected],. The work forms part of a Department
of Science and Technology, Government of India research project of Prof. M.
Lakshmanan.
L
R
-
-
-
L
-
+
v’ 1
C1
C2
- G1
+
iL + v
1
v2 +
R
RC
’
+iL
+ v’ 2
C1
NR N
R
-
-
C2
-
- G1
Fig. 1. Circuit realization of the two resistively coupled chaotic circuits (
coupling)
.
The circuit equations for the subsystems (either drive or response system) when they are uncoupled are easily obtained by
applying Kirchoff’s laws to the various branches of the subcircuit of Fig. 1. For example, we get the following set of three
coupled first-order differential equations for the left hand subsystem in Fig. 1,
where
! ! % & "# ! $
! '(*)+# (1)
-, /.103254 6-78-,9;:=< >.@?BA <CD< B? A < E 2 GFIH3
Similarly, one can write the state equations for the right hand
subsystem.
In their original work, Chua and Lin [14] [15] have studied the dynamics of the third-order electrical circuit by varyonly and have kept
ing the control parameter, capacitor
the other parameters fixed. Such a circuit in this case admits
perid-doubling bifurcations, intermittency and crisis-induced intermittency [15] behaviour for certain ranges of circuit parameters. They assumed the control parameter to be the capacitor
only for both numerical and experimental realizations. In
!
!
46
NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS
!
our present study, we have also verified the same results for inand linear
dividual circuits by considering both capacitor
resistor
as variables. The summarized results such as perioddoubling bifurcations and chaos followed by periodic windows
are shown in the computer generated one parameter bifurcation
plane as in the Fig. 2, from this we can
diagram in the
clearly see the route to chaos via period-doubling bifurcations
while the numerical and
experimental phase portraits of chaotic
attractor at =1050 are shown in Figs. 3(a) and 3(b). When
the two canonical Chua’s circuits are coupled together the circuit becomes six dimensional and the dynamics of the above circuit now depends on five control parameters, namely the capacitances , the inductances , the linear resistance and the
coupling resistance , besides the parameters associated with
and nonlinear element
.
linear negative conductance
)-
1
0.6
)D )
!
)
%
0.4
0.2
0
y
-0.2
-0.4
)
6
;
(a)
0.8
-0.6
-0.8
-1
-2
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
2
x
0
-0.5
-1
-1.5
v
-2
1
(b)
-2.5
-3
-3.5
-4
-4.5
1400
1350
1300
1250
1200
1150
1100
1050
R
Fig. 2. Computer simulated one-parmater bifurcation diagram in the plane of the individual third-order canonical
Chua’s circuit
.
v2
To investigate mutual synchronization, we fix the parameters
of the canonical Chua’s circuit so that the system exhibits a
chaotic attractor, specifically, the so-called double scroll attractor. The following nominal values produce the double scroll:
=100 nF,
=30.5 nF, =107 mH,
=0.68
V,
=-0.105
mS, =7.0 mS, =-0.45 mS and =1050 .
!
,
%
)
7
? A
v1
A. Computer Simulation
For computer simulation study, we normalize the state equations of the coupled canonical Chua’s circuit (Fig. 1) given
, ,
in (1) by appropriate rescaling, , , , ,
, , ,
, ,
, and
, which give the following set of normalized equations,
G ? A&
!
H 7 !
!
? A
! ? A ! ! ! ? A
" , F C)
9G "
. ! /"
!
!
#
!
#
$
G ! /"
! ! #
> ! #
$
! ! ? A
? A& !
! % where
> ;
and
Fig. 3. Numerical(Fig. (a)) and experimental(Fig. (b)) phase portraits of chaotic
attrator of the individual third-order. canonical Chua’s circuit
%
.103254 H"# 9< . F&< D< F< &CH3
. 02 4 H"#
I;9< . F< < F< &I
Now the dynamics of (2) depends on the parameters , ! , ,
H , , ! F ) .
The phase portraits of the system composed of identical os (2)
cillators are shown in Fig. 4. The fine diagonal line in Fig. 4(c)
indicates complete synchronization of the individual oscillators
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 721302, DECEMBER 28-30, 2003
at larger coupling coefficients. Obviously, the system of synchronized chaotic oscillators is not a hyperchaotic one. Visual
inspection of the phase portraits at smaller coupling coefficents
does not allow one to distinguish between simple chaotic and
hyperchaotic states. This can be done by direct calculation of
there are two positive
the Lyapunov Exponents (LEs). At !
the situation is rather
LEs, as expected. Meanwhile at !
& , one obtains hyperchaotic
complicated. At a value of !
attractor shown in Fig. 4(a). Phase space trajectories on this
attractor are characterised by two positive Lyapunov exponents
(
,
, ,
,
& ,
& & & ). At the value of !
& ,
just before the onset of synchronization, the phase space attractor is shown in Fig. 4(b) whose phase space trajectories char&
acterised by two positive Lyapunov exponents (
,
,
,
,
& & , ) and further on increasing the the value of !
(at !
& & ), the attractor shown in Fig. 4(c), which represents the onset of synchronization and the corresponging Lya&
& ,
,
, punov exponents are (
& ,
& ).
,
0
0
02 F 4
30 2 0 F ! 30 2 0 032 0 0
30 2 0 0 0
0 2 0 4 2 0 2
!
1
30 2 0 032 0 0 02 0 0 032 0 0& 4
02 0 0 -F 52 4
032 F 02 0 ! 032 0 0 F 0 032 0 0 02 0 0 032 0 4 - F 2 F 0.8
0.6
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
x’
x’
0
-0.2
-0.4
-0.6
-0.8
x’
0
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-1
1
-0.8
-0.6
-0.4
-0.2
0
0.2
x
x
x
(a)
(b)
(c)
0.4
0.6
0.8
1
Fig. 4. Typical examples of phase portraits of hyperchaotic attractors and sychronized attractor observed from the coupled differential equations (2) at
Vs
prodifferent coupling resistances (mutual coupling): shown in
jection; (a)
,
,
,
,
,
,
, (b)
,
,
,
,
,
and (c)
, (
,
,
,
,
,
.
"!$# % !'&(! ) * "+$#
"1*&$ "2!(-# ,8. 3 ".+3"! "!$# 0/49 3 . 5": 3,-2 "" 6(# .;" % ""7!!"+ 3! "/0" *) < =
) 73 "!" ,9 * "6 /3A >".6 3 ": :" ?29 " !$B# " %C#'<6 @ "3
)
>>
In Fig. 5 the computer generated bifurcation diagrams, is shown for
versus the coupling coefficient, !
.
F C) 9
F 0&4C0
B. Experimental Synchronization of chaos in the identical oscillators
In the present case, the oscillators have been built using the
same circuit parameters as given in Fig. 1 and only the coupled
linear resistor is varied. When =0, the two canonical circuits are uncoupled and each will follow its own dynamics. Due
to sensitive dependence on initial conditions, the trajectories of
the two systems evolve independent of each other, i.e. there is
no synchronization. When the oscillators are coupled to each
), it has been
other and the coupling is strong enough ( shown numerically that the oscillators ”forget” their own initial conditions and after a short transient the two oscillators will
synchronize completly, i.e. the trajectories of the two canonical
Chua’ circuits follow each other. The experimental illustrations
are presented in Fig. 6. In the experiments it is convenient to
plane, i.e. the output
observe the phase portraits in the
signal from one generator against the output signal from another
one.
In the figure, when two systems synchronize, on the
screen of oscilloscope the relation between
and is a
line through the origin as shown in Fig. 6(c). When the
two systems do not synchronize, the relation between
and
deviates from this line. The greater the coupling resistor whose value is larger than its critical value, the greater is the
deviation: that is, the more the systems do not synchronize.
When is large enough, the two systems do not synchronize
and the relation between and becomes chaotic as shown in
Fig. 6(a) and 6(b). In the case of unsynchronized chaotic oscillations the phase portrait (or Lissajous figure) is a very complicated one, indicating ”random” mutual phase and amplitudes
of the oscillations (Fig. 6(a)). This chaotic status is called hyperchaotic attractor characterised by more than one postive Lyapunov exponents, which occur in the four or higher dimensional
system. Previously, hyperchaos was observed in a network of
(both open and closed) five unidirectionally coupled Chua’s circuits [16] [17]. However, in our study we have observed the hyperchaotic behaviour in a coupled third-order canonical Chua’s
circuit. Meanwhile the fine diagonal line in Fig. 6(c) proves that
the momentary amplitudes and phases of the oscillations from
the two oscillators do coincide with each other, i.e. the oscillators are fully synchronized.
)
)
)
0
4 J
)
-1
-1
-1
0
-0.2
47
3
)
2
1
1
1.5
0.8
0.8
0.6
0.6
0.4
1
0.4
0.2
0.2
0.5
0
v’
1
0
-0.2
0
-0.4
-0.4
’
x 1− x 1
v’
1
v’
1
-0.2
-0.6
-0.6
-0.5
-0.8
-0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1
v1
-1.5
(a)
v1
v1
(c)
(b)
-2
1
2
3
4
5
6
Rc
7
H<".I
8
9
10
D E Vs GF , in the
Fig. 5. Computer simulated bifurcation diagram of case of mutual coupling for Fig. 6. Typical example of experimental phase portraits of hyperchaotic attractors and sychronized attractor observed from the two coupled identical canonical Chua’s oscillators at different coupling resistances: (a)
, (b) and (c) Veritcal scale
=0.5 V/division; horizontal scale. =0.5 V/division
I
F KC#
5I
F L&' 0I %
F M 0I
48
NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS
III. C ONCLUSION
In this paper numerical and experimental investigations of a
set of coupled canonical Chua’s circuits are carried out. The
advantage of our study is that we have observed hyperchaotic
oscillations in simplest coupled canonical Chua’s cirucits (maximum of six dimensional phase space), incontrast to which previous studies involving coupled canonical circuits of higher order (more than six) circuits. Yet it exhibits hyperchaos and synchronization of chaotic orbits. This feature it is hoped will be
convenient for application in secure communications and spatio
temporal chaos.
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[14]
[15]
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