CHIN. PHYS. LETT. Vol. 32, No. 1 (2015) 010503
An Analytical Study on the Synchronization of Murali–Lakshmanan–Chua
Circuits
G. Sivaganesh
**
Alagappa Chettiar College of Engineering and Technology, Karaikudi Tamilnadu-630 004, India
(Received 19 August 2014)
An explicit analytical solution is presented for unidirectionally coupled two Murali–Lakshmanan–Chua circuits
exhibiting chaos synchronization in their dynamics. The transition of the system from an unsynchronized state
to a state of complete synchronization under the influence of the coupling parameter is observed through phase
portraits obtained from the analytical solutions of the circuit equations characterizing the system.
PACS: 05.45.Gg, 05.45.Xt DOI: 10.1088/0256-307X/32/1/010503
The phenomenon of chaos synchronization in cou-
pled chaotic systems has been studied extensively for
the past two decades with a motivation to under-
stand the coherent dynamical behavior of coupled sys-
tems. Chaos synchronization in nonlinear electronic
circuits has been observed in a variety of circuits
[1−7]
and potential applications have been found in se-
cure communication. Several synchronization phe-
nomena such as complete, phase and lag synchro-
nization have been identified in identical and non-
identical chaotic systems and studied experimentally
and numerically. A detailed study on characteriz-
ing the above-mentioned synchronization phenomena
was carried out numerically.
[12]
Complete synchroniza-
tion, being the strongest among the other types of
synchronization, occurs when the states of the cou-
pled systems converge, irrespective of the mismatch
under initial conditions. Complete synchronization
phenomena exhibited by coupled second order dissi-
pative electronic circuits were studied experimentally
and numerically.
[5,6]
The synchronization of coupled
Chua circuits and coupled Murali–Lakshmanan–Chua
(MLC) circuits through a compound chaotic signal
was studied numerically.
[7]
Some of the synchroniza-
tion phenomena exhibited by coupled systems were
studied analytically
[8−11]
and numerically.
[12,13]
The
MLC circuit is a simple second order dissipative non-
linear electronic circuit with Chua’s diode as the only
nonlinear element was suggested by Murali et al.
[14]
The period doubling dynamics of the circuit lead-
ing to chaotic motion was studied experimentally and
numerically.
[5,6,15,16]
An explicit analytical solution to
the normalized circuit equations of the MLC circuit
was presented.
[6,16,17]
The MLC circuit has the piece-
wise linear nonlinear element, Chua’s diode which is
linear within the three regions. Since the circuit equa-
tions describing the MLC circuit is a second order
differential equation, the equation is solved for each
linear regions of Chua’s diode. Similar solutions were
given to some second order nonlinear chaotic circuits
with piecewise nonlinear element as the active cir-
cuit component.
[18−21]
The analytical solutions thus
obtained were used to explain the dynamics of the
circuit through phase portraits of the state variables.
However, this method has not been applied to coupled
chaotic circuits. Here we present an explicit analytical
solution for coupled MLC circuits and study the phe-
nomena of complete synchronization through phase
portraits obtained from the solutions. The Murali–
Lakshmanan–Chua circuit is a simple forced series
LCR circuit with Chua’s diode as the only nonlinear
element, connected parallel to the capacitor . The
normalized state equations of the circuit is
˙ = - (), (1a)
˙ = - - - +
1
sin(
1
). (1b)
The piecewise linear function () representing Chua’s
diode is given by
()=
⎧
⎨
⎩
+( - ), if ≥ 1,
, if ||≤ 1,
- ( - ), if ≤-1,
(2)
where =(/
2
), =
s
, =
a
/, =
b
/,
1
= (
1
/
p
),
1
= (Ω
1
/) and = 1/.
The circuit parameters take the values = 10 nF,
= 18 mH, = 1340 Ω,
s
= 20Ω and the param-
eters of Chua’s diode are chosen as
a
= -0.76 mS,
b
= -0.41 mS and
p
=1 V. The frequency of the
external periodic force
1
is fixed at
1
= Ω /2 =
8.9 kHz. The solutions of the normalized equations in
terms of the state variables () and () obtained for
each of the three piecewise linear regions
0
,
±
are
summarized as follows. Since the roots
1,2
in the
0
region are real and distinct, the fixed point (0, 0) cor-
responding to the
0
region is a saddle or hyperbolic
fixed point. The state variables () and () are
()=
1
m1t
+
2
m2t
+
1
+
2
sin(
1
)+
3
cos(
1
), (3a)
()=
1
[- - ˙ +
1
sin(
1
)]. (3b)
The roots
1,2
in the
±
region are a pair of
complex conjugates. Hence the fixed points
1
=
±(( - )/ + ),
2
= ±(( - )/ + ) cor-
responding to the
±
region is a stable focus fixed
**
Corresponding author. Email: sivaganesh.nld@gmail.com
© 2015 Chinese Physical Society and IOP Publishing Ltd
010503-1