2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Coupled Chaotic System Based on Shinriki-Mori Circuit
Tatsuki Sobue1 , Yasuteru Hosokawa1 and Yoshifumi Nishio2
1

Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Phone:+81-88-665-1300
FAX:+81-88-637-1318
E-mail: s002337018@edu.shikoku-u.ac.jp
E-mail: hosokawa@keiei.shikoku-u.ac.jp

2Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
FAX:+81-88-656-7471
E-mail: nishio@ee.tokushima-u.ac.jp

Abstracti

Shinriki-Mori circuit have been applied to coupled systems
for investigations of a coupled system. In this study, we propose a novel coupled system based on Shinriki-Mori circuit.

R1f th (v - v )
CRCv

CCCv

1

C1

L1

f th (v - v )
C

2

i
R2

1. Introduction

11v

2

C2

2

L2

There are many studies about synchronization phenomenaof coupled chaotic circuits. In these system, some famousf th (v - v )chaotic circuits are applied. One of famous chaotic circuit
is Shinriki-Mori circuit [1] [2]. There are many investigaiNtions about coupled chaotic circuits using this circuit. OneRNof important factors is a parameter mismatch in these invesvN CN LN
tigations. By including a parameter mismatches, some interesting phenomena are observed. In the first place, the circuit
includes parameter mismatches. There are not perfectly sametwo things in the natural world.
Figure 1: Proposed System.
However, it is not easy to change a circuit characteristic
intentionally by changing parameter mismatches. Especially,switching phenomena which are very interesting phenomena
are influenced by a parameter mismatches.
Using this model, the circuit equation is described as folIn this study, a novel coupled system for investigation of lows.
coupled chaotic system is proposed. This system can generate similar two chaotic waveforms intentionally. This system
is also a coupled chaotic system. By investigating this sysvc
dvn
tem, the contribution of the field of coupled chaotic circuits is=− fth (vc − vn ) − in ,Cn
expected.
dtRnN∑dvcvcCc=−{αn fth (vc − vn )} ,(1)2. Circuit Modeldt
Rc n=1din
Figure 1 shows a proposed circuit. This system is based
 Ln dt = vn ,on Shinriki-Mori circuit proposed in [1]. The modified points
are follows. Right parts of the circuit are duplexed. Resistors Rn (n = 1, 2, · · · N ) are added for changing a parameter whereof bidirectionally coupled diodes. In this study, bidirectionally coupled diodes and resistors are modeled as a piecewisef (v) = v + (|v − Vth | − |v + Vth |)/2.
(2)
linear function shown in Fig. 2.
C- 529 -Nv(a) show projections of attractors onto v3 − x1 plane. Figures 3 (b) and (c) show projections of attractors onto v3 − x1
plane and v3 − x1 plane, respectively.
By changing parameter Rc and αc , one periodic orbit (1),twoperiodic orbit (2), chaos (3) (4), double scroll type attrac(a)(b)
tor(5),
window (6) and chaos (7) are observed. In this investii
gation, parameters of right side elements are same. Therefore,
v1 and v2 are synchronized as shown in Fig. 3 (c).
Figure 4-5 shows the case of applying different resistances
1
1of the right side of the system. When differences are small asv
shown in Fig. 4, these are synchronized. However, by increas1
ing the differences, these become asynchronization states as
1shown in Fig. 5. We consider that there are possibility ofapplying this state to some applications because these have
similar waveforms in spite of the asynchronization state.
(c)On the other hand, the case applying same resistance anddifferent capacitances of the right side of the system is shown
Figure 2: Bidirectionally coupled diodes model. (a) Circuit in Fig. 6-7. In these cases, these are not synchronized at all.
schematic. (b) Circuit model. (c) v − i characteristic.
However, in the case of Fig. 6, waveforms and attractors arevery similar forms.

i

By substituting the variables and the parameters,vn
xn =, xN +n =Vth√

4. Conclusion
L1 in,
C1 Vth1d1= “ · ”, τ = √t, αn =dt
Rn
L1 C1√1L1C1C1αc =, βn =, βc =Rc C1
CnCc
L1and γn =, (n = 1, 2, · · · N.)
Ln√

L1,C1equations (1) and (2) are normalized asẋn= βn {αn f (xc − xn ) − xN +n } , ẋN +n = γn xn ,
[]N∑



= βc αc xc −{αn f (xc − xn )} , ẋc(3)In this study, we propose a novel coupled system basedon Shinriki-Mori circuit. The relationships between observed
synchronization phenomena and parameters are investigated.
In the future works, the large number N case will be investigated.
References[1] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations in a Modified van der Pol Oscillator containing
a positive nonlinear conductance,” Proc. IEEE, vol. 69,
no. 3, pp. 394-395, 1981.

(4)[2] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena ina Circuit with a Negative Resistance and an Ideal Switchof Diodes,” IEICE Trans., vol. E-70, pp. 744–754, 1987.

n=1

(5)

[3] H. Sekiya, S. Moro, S. Mori, and I. Sasase, “Synchronization Phenomena on Four Simple Chaotic Oscillators
Full-Coupled by Capacitors,” IEICE Trans. Fundamentals, vol. J82-A, No. 3, pp.375–385, 1999.

3. Experimental Results and Computer Simulation Results[4] M. Miyamura, Y. Nishio and A. Ushida, “Clustering in
Globally Coupled System of Chaotic Circuits,” IEEE
Proc. ISCAS’02, vol. 3, pp. 57–60, 2002.

wheref (x) = x + (|x − 1| − |x + 1|)/2.

At first, the case of N = 2 is investigated. Figure 3 shows [5] L. M. Pecora and T. L. Carrol, “Synchronization incircuit experimental results (a) and computer simulation reChaotic Systems,” Physical Review Letters, vol. 64, pp.
sults (b)-(c). Parameters are set as R1 = R2 = 120 [Ω],821–824, 1990.
C1 = C2 = 0.015[µF], L1 = L2 = 50[mH] in circuit experiments. α1 = α2 = 9, β1 = β2 = 1, βc = 0.32 andγ1 = γ2 = 1. Control parameters are selected as Rc . Figure 3- 530 -(1)(1)
Rc = 1.94k[Ω]αc = 0.40(2)(2)
Rc = 1.70k[Ω]αc = 0.60

(3)

(3)

Rc = 1.55k[Ω]αc = 0.80

(a)

(b)

(c)

Figure 4: Different capacitances case. C1 = 1.0 and C2 =1.0. (a) x3 − x1 . (b) x3 − x2 . (c) x1 − x2 . (1) αc = 1.0. (2)αc = 1.1. (3) αc = 1.5.

(4)

Rc = 1.44k[Ω]αc = 1.0

Rc = 1.39k[Ω]αc = 1.05

Rc = 1.28k[Ω]αc = 1.33

(5)

(1)

(6)
(2)

(7)
(3)
Rc = 1.01k[Ω](a)αc = 1.40(b)

(c)
(a)

(b)

(c)

Figure 3: Circuit experimental results (left) and computer calculation results (center and right) (N = 2). R1 = R2 =[Ω], Figure 5: Different resistance case. α1 = 4.0, α2 = 9.0,R3 =[Ω], C1 = C2 =[µF], Cc =[µF], α1 = 4, α2 = 9, β2 = β3 = 0.32 and γ1 = γ2 = 1. (1) αc = 1.00. (2) αc =β2 = β3 = 0.32, γ1 = γ2 = 1, (a) Projection of attractors 1.05. (3) αc = 1.60. (a) x3 − x1 . (b) x3 − x2 . (c) x1 − x2 .
onto v3 − x1 plane. (b) Projection of attractors onto x3 − x1
plane. (c) Poincaré sections. x1 − x2 plane.

- 531 -

(1)

(2)

(3)

(a)

(b)

(c)

Figure 6: Different capacitances case. C1 = 1.1 and C2 = 1.
(a) x3 − x1 . (b) x3 − x2 . (c) x1 − x2 . (1) αc = 0.9. (2)αc = 1.1. (3) αc = 1.5.

(1)

(2)

(3)

(a)

(b)

(c)

Figure 7: Different capacitances case. C1 = 0.33 and C2 =1.0. (a) x3 − x1 . (b) x3 − x2 . (c) x1 − x2 . (1) αc = 1.0. (2)αc = 1.2. (3) αc = 1.6.

- 532 -