Synchronization in Ladder-Coupled Chaotic Circuits
Including Ring Structures
Katsuya Nakabai, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: {nakabai.uwate.nishio} @ee.tokushima-u.ac.jp

Abstract-In this study, we investigate synchronization phenomena in ladder-coupled chaotic circuits including ring structures. We focus on the effect of the position of the ring structuresin the network to the synchronization of the whole network.
In our proposed network model, chaotic circuits are coupled
by resistors. We set the parameters of the circuits to generate
periodic solutions or chaotic solutions. Computer simulations
confirm that the various synchronization phenomena occur indifferent situations.
I. INTROD UCTIO N

Synchronization in a network is one of the interesting topics
in scientific disciplines. Investigation of this is important workin order to clarify nonlinear phenomena in the natural world. It
is observed over the various fields, for example, engineering,
biology, sociology and so on [1]-[3]. Incidentally, the networks
have different topology and characters in each even if they are
similar type [4]-[6]. Therefore, the investigation of dynamics
in a different network is significant and research has proceeded
to analyze the dynamics of synchronization on each topology.
On the other hand, synchronization of a coupled chaotic
system is interesting models to describe various higherdimensional nonlinear phenomena [7], [8]. Recently, studying
chaotic phenomena by using coupled chaotic circuits increases.
On the chaotic circuit with simple elements, circuit experimentand computer simulation are suitable for studying in terms ofthat experiment time is short and repeatability of experiments
is high. Networks by using coupled chaotic circuits canbe modeled natural world or social networks. Moreover, it
is important for future engineering to investigate nonlinear
phenomena like chaotic synchronization. As these reasons, achaotic-generate circuit is counted on application, for instance,
chaotic communication systems, Internet of Things, modelingneurons and so on.
In modern society, the agenda of vegetation exists about
environment or food. The study of plant bioelectric potential
has attracted rising attention recently from the viewpoint ofplant growth [9]. Bioelectric potential is known that it changes
depending on the surrounding environment. Therefore, we
consider the bioelectric potential as the voltage phase difference of chaotic circuits and we model the plants by chaotic
circuits.
In the previous study, we have investigated synchronization
phenomena in our proposed chaotic model with hybrid topol-

978-1-7281-0397-6/19/$31.00 ©2019 IEEEogy by ladder and rings modeling sapling of the plant [10].
In this study, we increa se the types of network and investigate
the synchronization. In our proposed network model, chaotic
circuits are coupled by resistors. These chaotic circuits arelittle difference of each other by the bifurcation parameter. We
set the parameters of the circuits to generate periodic solutions
or chaotic solutions . The number of rings is changed to eachmodel. And we observe the synchronization phenomena in
different network model s.

II. N ETWORK MODEL

In this study, we use chaotic circuits. The model of chaotic
circuit is shown in Fig. 1. This circuit consists of a negative
resistor, two inductors, a capacitor and dual-directional diode s.
This chaotic circuit is called Nishio- Inaba circuit [11].
We propose ladder-coupled chaotic circuits including ring
structures. In this case of Fig . 2, we set five chaotic circuits
in ladder structure and one ring structure coupling to CC 1
by resistors. Likewise, we propose three network models that
coupled ring structure to CC1, CC2 or CC3. We set theparameters of the system as (3 = 3.0 and (j = 470 .0. Next, wechange the condition in terms of the parameter a. a denotesa degree of chaos. The network models are divided into two
types by the value of a . One pattern is to unify the value of
all a (unified system) like shown in Fig. 2, and the other is
to change the value of all a (ununified system) like shown in
Fig. 3. In the case of Fig . 2, the value of a is unified 0.412.
And in the case of Fig. 3, the value of a is not unified. The
parameters a are set from 0.411 to 0.415 with step size 0.001
from CCI to CC5. The a of CC6 and CC7 are added 0.001
from the a of coupled circuit.
Subsequently, we augment the number of rings in the caseof one ring, two rings or three rings . Likewise in ununified
systems, the a of CC6, CC7, CC8 and CC9 are added 0.001
from the a of coupled circuit. We show the models from Fig. 3
to Fig. 7. Figure 4 and 6 are unified systems with two andthree rings and Fig. 5 and 7 are ununified systems with two
and three rings.

(l ~O .411

1l ~ O . 4 12

1l ~ O .4 13

1l ~O . 414

1l ~ O. 4 15@
a= 0.412

a=0 .412

Fig . 1: Chaotic circuit.
a = 0.4 12

11= 0.4 12

a = 0.4 12

Fig . 3: Ununified system with one ring.

a=O.4 12

a = 0.4 12

1l ~O . 412(1 '"0.4 12(1 :11 ~ O. 4 12

1l ~O .4 12

11 ~O .4 12

1l ~O . 4 12

0.4 12I1=OA I2

Fig . 2: Unified system with one ring.

a=0.412a=0.4 12I1=OA I2

Fig. 4: Unified system with two rings.
By changing the variables and parameters,
i1 =

. V C
Vrc
~V X n , 12 = ~VYn
L1

a =r

,

V

= V ZnLs = "dVrc~ ' (3 = L 2 ' u , L 2 '1'"Y = R 'V L 1C1

11=O .411

a= 0.412

a=0.413

11= 0.414

11=00415(1)

t = y'L 1 C 2 T .

a=0.41211 ~ O .4 12

R is the resistor and each chaotic circuit is coupled by
the resistor. R is defined as coupling strength for one of the

1l ~O .41

11=0. 4 14

4Fig . 5: Ununified system with two rings.

bifurcation parameters.
The normalized circuit equations of the systems are given
as follows:

dx ,
dT

+ Zi

dYi

.

- - = aXi

a ~O.

-d-T = Zi - j(y)d:dz

a-0.4 12

a-0.4 12

a-0.412a ~ 0 .41

2

(2)

N

= - Xi - (3Y i -4J 2

2..= '"Yi j (Zi - Zj )a-0.4 12

11"' 0.412

a- 0.412

a-OA I2

u""0.412

a-0.4 12

i ,j = l(i , j = 1,2, ' . " N ).

Fig. 6: Unified system with three rings.

In Eq . (2), N is the number of coupled chaotic circuits and'"Y is the coupling strength as '"Y = 0.008 in this study. We set

the parameters of the circuits to generate periodic solutions or
chaotic solutions. When we set the parameter as a = 0.411
to a = 0.415, the circuits generate three-periodic solutions.
And when the parameter is more than a = 0.416, the circuits
generate chaotic solutions.

III . SIMU LATION RESU LTS
We investigate synchronization phenomena in laddercoupled chaotic circuits including ring structures. In thistime , we use attractors and Lissajous figures to research
synchronization. Lissajous figures show the pha se difference11-

0.4 12

a- 0.412

11-

0.414

11-

0.414

a- 0.4\6a- 0.4 16

Fig. 7: Ununified system with three rings.

between each chaotic circuit. When the phase difference isclose to 0, the Lissajous figures become like Fig. 8(a). In
this case, the system causes phase synchronization (perfectsynchronization) and this condition is stable . And when thephase differential increa ses, the Lissajous figures approach theshape of Fig. 8(b). This shows that the system is asynchronousand this condition is unstable. Moreover, the phase difference
is to some extent, the Lissajous figure becomes like Fig . 8(c).
In this case, the system cause s chaotic synchronization. We
compare the synchronization with these three pattern s.

system has multiformity and can adapt to the surrounding
dynamics .

CCI

I

CC2

• Ununified system• Unif ied system

CC3

Couple d Circuit Posit ion t o ring strcture

Fig. 9: The number of chaotic attractors in case of one ring.
(a) Synchronization.

Chaotic synchro(b) A synchronous. (c)nization.
Fig. 8: Lissajous figure.

First, we simulate the circuit model with one ring, and theresult is shown in Fig . 9, 12 and 13. In this case, despite ofthat the circuits are set three periodic attractors , both unified
system and ununified system cause chaotic synchronizationand observe chaotic attractors by coupling ring to CC1 orCC2. On the other hand, when we make the ring structure
CC3 , all chaotic circuits are generated three periodic attractors
and cause perfect synchronization in both cases.
Second, we simulate the model with two rings, and the resultis shown in Fig. 10, 14 and 15. In this case , the positionof occurred chaotic synchronization increases more than thesituation of one ring. When we make the ring structure on CC1
and CC3 in the unified system and couple rings to CC1 andCC5 in the ununified system, many positions in the systems are
asynchronous. Though, when we make the ring structures on
CC2 and CC3 or CC2 and CC4, synchronized state of both
systems are difference. The circuits of the ununified systemoccur chaotic synchronization and the circuits of the ununified
system occur perfect synchronization. Furthermore, in the caseof CC2 and CC5 in the ununified system, despite CC8 andCC9 has chaotic parameter as 0: = 0.416 , CC8 and CC9generate three periodic solutions and perfect synchronization
occur on CC8-CC9.
Finally, we simulate the model with three rings , and theresult is shown in Fig. 11, 16 and 17. In this case , on both
systems, the positions of the bridge to the ladder and ringtend to be asynchronous. However, in the case of coupling to
CC1 , CC3 and CC5, both systems obtain synchronization on
bridge. Furthermore, in the case of coupling rings to CC2 , CC3
and CC4 , synchronization is difference in both systems . The
ununified system has asynchronous however unified system
obtains many perfect synchronizations.
From these results, we confirm that when the number ofrings increases, synchronization phenomena are easy to occur
in the ununified system. We consider the reason why thetendency exists is that by changing the characteristic, the

10
9
8
7
6

5

~ I.

• Ununified syste m• Unif ied system<.,,'\.
<...'"
(, ~
(,<"
c..'"
<J' (,">
c.. <''''P"'<."C",.p (,(,.. .P <."(,,,,/..; (5)P (,(,,·.P. <."(,,,\p

Coup led Circu it Position t o ring st rct ure

Fig. 10: The number of chaotic attractors in case of two rings.

• Ununified syst em• Unified syst em

Coupled Circuit Posit ion t o ring st rct ure

Fig. 11: The number of chaotic attractors in case of three rings.
IV. CONCLUSIONSIn this study, we have investigated synchronization phenomena in ladder-coupled chaotic circuits including ring structuresmodeling sapling of the plant. The models are two types
named unified system and ununified system. We have set theparameters of the circuits to generate periodic solutions or
chaotic solutions. And we have propo sed that we change thenumber of ring structure. By the computer simulations, wehave confirmed that when we changed the number of ring s,
we can obtain various synchronization.
In the case of one ring , the large difference of dynamics
does not exist, however, the position s of synchronization arethe differen ce. In the cas e of two rings , the unified systems
tend to have occu rred cha otic syn chronization and ununified
systems tend to have occurred perfect synchro nization. In theca se of three ring s, a synchron ization phen omen on is ea sier tohave occurred in the ununified system than the unified system.
In our futu re work s, we investigate synchro nization phenom ena by using cy as the random value in the ununified
system. Furtherm ore, we carry out circuit experime nt andco nfirm the reproducibility.

C(IJ~dC<,
CCl·CCl
CCl-tC3
CCJ·teltel·CelCCl·CCi
CCI-tC1
CCl·CCi
CCl·CC1
CCl·CCl
CCl·eel
CCJ·CCl
CCJ·eel---'"CCl -CC2

.
.~

c

u

CC2-CC3
CC3-CC4
CC4-CC5
CCl -CC6

-CC7
~ CCl
CC2-CC6'" CC2-CC7
CC3-CC6
CC3-CC7
CC6-CC7

tel-tC3

Coupl ed Circ u it Positio n t o rin g s t rct urc

CCI

CC2

tel·eelCC3

Perf e ct s ync .

As y n c hron o us

Perfect s ync .

Cel·CClPer fec t sy nc .

Chaotic sy nc.

Per fect s ync.

Cel·eel

Perf e ct sy nc .

C hao t ic s yn c .

Pe rfect s yn c.

CCHC1

Perf e ct sy nc .

C ha ot ic s yn c .

Per fect sy nc .

CCl·eel-----------Perfect sy nc.
Pe rfe ct s ync .

CCltel

CC1.CC\

CClW

CCl.CCI

CCl.CC\

ChaJoc sync.

Cholticijr<

CMooc sync.

Asynchronous

PerIKlsync.

CO<ie ,YnC.

PerleclS)'nt.

------ ---- -- --- -- ---- -- ---- ---C h ~ oc s ync.

ChiKI!i:Sj'1lC.

CMooc sync.

Asynchronous

PerIKt S)'1lC.

Ch.J iUsync.

Perleclsync.

ChaJticsync.

Choltkijr<

CMo!i: sync.

Asynchronous

PerIKIS)'nc.

CO<ie,YnC.

Perfeclsync.

Asynchronous

Asynchronous

CMoocsync.

Asynch ronous

Perleclsync.

CO<ie,YnC.

Perfeclsync.

ChiKI!i:sync.

ChiKlocsyoc.

CMo!i:synt.

Asynchronous

ChaJticsync.

Cholticijr<

Ch.Jooc sync.

Asynchronous---------------- ---- ----------------------------------- ---------- ------------------- ------ ---------- ----

--·

.~
·0
c

CC3-CC4··'"~

CC4-CC5
CCl -CC6
CCl -CC7
CC2-CC6
CC2-CC7
CC3-CC6
CC3-CC7
CC6-CC7

CO·CC2

Cbotic"",Perfect sync .

Perf ect sy nc .

Choltkijr<

Coup led Circ uit Positi on t o ri ng st rc t ure~ottPerfeclSj'llt.

Asynchronous

PerledS)'nc.

Perftttsync.

ChiKI!i:sync.

CIl.loocsync.

CharocS)'llt.

PerIKtsync.

Perlecl S)'llt.

Perleclsync.

Pedecl S)'llt.

Cholticijr<

Chaottsync.

CO<ie,,",PerIKl sync.

Cho<ie ,,",

Perfeclsynt.

Fig. 15: Synchronized pattern in ununified system with two rings.

----

CCl ·CC2

CCl ,CC2,CC4
Chao tic sy nc.

CCl ,CC3,CCS
Perfect sync.

CC2,CC3,CC4
Async hrono us

CC2·CC3

Chaot ic sync.

Chao tic sy nc.

CC3· CC4------------- --- ----- --- - -CC4-CC5

Chaotic sync.

Asy nch rono us

Cha otic sy nc.
Asynch ronousCCl -CC6

Asynch rono us

Asynchr ono us

CCI -CC7

As ynch ronous

Asynch rono us

CC3·CCG------- -- --Coupled Circuit Posi tion to ring s trctu re
CCl .CC2,CC3
Chaot ic sync.

CC3·CC7

Chaotic sync.

Perf ect sync.

Perf ect sync.

CC2·CCa

Asynch rono us

Chao tic sy nc.

Chao tic sy nc.

Perf ect s ync.

Perfect sy nc.

CC2·CC9

Asynch rono us

Chaotic sy nc.

Chao tic s ync.

Pe rfect s ync.

Perfect s ync.

Chaotic sync.

Perf ect sync.

Perf ect sync.

Chaotic sync.

-- -Cha otic s ync.

Perf ect sync.

COlO

CC3-CC9
CC3-CCIO

As ynch ronous

CC3-CCllAs ynch ronous

CC4-CCIO
CC4-CCl l

Perf ect sync.

CC5·CCIO

Perfect sy nc.

Perf ect sync.

Perf ect sync.

Perf ect sync.

CiCII~Positioolomgstrctl re

CCltelCClCCS

CClCO

CCl.tel

CClCCS

ChaJoc sflt,

CboticSJ'"
C1wocsync.

CIo<ieSJ'"
CMm:sync.

ChaJlic sync.
ChaJocsync.

Chalti:sync.

CIo<ieSJ'"
CMtfr sync.

ChaJti:sync.
ChaJocsync.

=s

CboticSJ't.

tel-CC5

Chaoti:sroc.

Myncllrms

CboticSJ't.
Chalti: sflt,

ChaJti:sync.
ChaJocsync.

CO-CC5
CO-tC1

Chaoocwrc.
ChaJti:sync.

Chatfrs)'llC.

ChalOC5yrc..

AsrnunortOOs

CMm:s)'1lC.

CboticSJ't.

CboticSJ'"~ ndlrms

CC3-CCa

Perf ect sync.

~nchrm------ ---------------------- --------- ------- ---- ------------- ------- ------ ------- ------==---------- ---------=-----= ----- -------- ------ --- ------- ------ ------------CC2·CCi
CC2-tC1

CboticSJ'"

Chaoti:sync.
Chaooc sync.

CIo<ieSJ'"
Chatfr sync.

Chaoocsync.
ChaJticsync.

Cbotic"",ChaJticS)'llC.

CC5·CCll
CCG·CC7

ccs -ccs
ClO-CCl--

CCl ·CC2
CC2-CC3

Chaliil: sync.

Cham:sfl\C.

Chaltic sflt,
Chaott syTt

Chaoocs)'llC.

Chaoocsync.

Pededsync.

CboticSJ'"

Pededsync.

Pofl<lSJ'"
Pfrlfdsync.

Pededsync.
~ ""' SJ'"

Fig . 14: Synchronized pattern in unified system with two rings.

As ynchronous

Per fect sync.

Async hrono us

Pe rfect sync.

Asynchrono us

Pe rfect sync.

Async hrono us
Async hrono us

Perfec t sync.

Chaot ic sync.

Perfect sync .

Perfect sync.

Pe rfe ct sy nc.

Perfec t sy nc.

Chaotic sy nc.

Perfec t sync.

Pe rfe ct sy nc.

Perfec t sy nc.

Pe rfe ct sync .

Pe rfect sync.

Pe rfe ct sy nc.

Coupled Circuit Position to ring strctu re
CCl ,CC2,CC3
Chaotic s ync.

CCl ,CC2,CC4
Chaotic sync.

Chaotic sy nc.

CCl,CC 3,CC5
Pe rfect sync.

CC2,CC3,CC4
Perfec t sync.

---- --- ---- ----- --- -- ---- -- -- -Perfect sync.

Perfec t sync.

Pe rfect sync.

Perfe ct sync.

Chaotic sync.

Perfect sync.

Perfect sync.

Async hronous

Async hronousCCl -CC7

Async hronous

Async hronous

CC2·CC6CC2·CC7
CC3-CCG
CC3-CC7

Pe rfe ct sync.
Perfect sync.

CC2·CCa

Async hronous

Async hronous

CC2·CC9

Asynchro nous

Async hronous

CC3·CC9

ChaJti:sync.

As ynchro nous

Chaotic sync.

Chalti:sync.

CboticSJ'"
Chalti:sync.

Chaotic s ync.

Asynchrono us

Chaotic sync.

CIo<ieSJ'"

CC5CClPe rfect sync.

Chaotic s ync.

Chaot ic sy nc.
Chaotic sy nc.

CC3-CCa

Chatfr sync.
Chalttsync.

Async h ronous

Per fect sync.

CC4·CC5
CCI -CCG

CC3-CC4

Chaoticsync.

Chaolic syoc.

Asy nchro nous

Pe rfect sync.

Fig. 16: Synchronized pattern in unified system with three rings.

Chaliil: s)'llC.

Chaltic sflt,

ChJJticwrc.
PerleaS)'llC.

Ch.JiD: synt.

Asynchronous

CC3

Isyndvm,= 1
CCl-CCl

ChaID: sync.

sync.

CC2

ChaJocwrc.

CC3-CCl

Perleclsync.

C1iaoocsync.

CCI

Cbotic"",

CC2-m
CC2·CCl

Perleclsync.

PerIKl sync.

ChiKIocsync.

CC2-CC7

ChaJocsync.
ChaJOCS)1lC.

CC2·CCJ

PerleclS)'nc.

ChaID:sync.

C h ~ ocs )'l t.

Pe rfect s ync .

Ccu~

ctun

Ch.JID:synt.

Perlecl sync.

P erfe ct sy nc.

Fig. 13: Synchronized pattern in ununified system with one ring.

<,

Perleclsync.

I

ChaJocS)'llt.

CC2-CCG

CC2-CC3

toringstn:!lJle

CO.CCJ

C ha o tic sy n c .

Fig. 12: Synchronized pattern in unified system with one ring.

CCl -CC2

rcu ~PllSilicn

CC1CClCC3·CClO
CC3-CCll
CC4-CClO
CC4·CCll
CC5·CCI O
CC5-CCll
CC6-CC7
cca·CCg
CIO-CCIPe rfect sync.
Pe rfect sync.

Perfe ct sync.
Perfe ct sync.

Chaotic sy nc.

Chaotic sy nc.

Chaot ic sync.
Chao tic sync.

Chaot ic sy nc.
Chao tic sy nc.

Chaotic sync.
Chaotic sync.

Perf ect sy nc.

Perfect sy nc.

Perf ect sy nc.

Perfect sy nc.

Perfect sync.
Perfec t sync.

Perfec t sync.

Pe rfect sync.

Chaot ic sy nc.

Chaot ic sync.

Per fect sync.

Pe rfe ct sync.

Fig. 17: Synchronized pattern in ununified system with three rings.

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