2014 International Symposium on Nonlinear Theory and its Applications
NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Investigation of Asynchronous States in Chaotic Circuits Coupled by Resistors
as a Ladder
Yasuteru Hosokawa† and Yoshifumi Nishio‡†Department of Media and Information Systems, Shikoku University
Furukawa, Ohjin, Tokushima 771–1192, Japan‡Department of Electrical and Electronic Engineering, Tokushima University
2–1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: hosokawa@keiei.shikoku-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—There are many studies about coupledchaotic circuits. In these studies, various kinds of interesting phenomena are observed. Especially, interesting behaviours are observed in asynchronous parameter regions
near synchronous parameter regions.
In this study, asynchronous states in chaotic circuits coupled by resistors as a ladder are investigated. Especially,
relationships among the phase shift, coupling strengths andthe number of circuits are investigated.

iv

i2-R 1v

f(v ) L 1
11

C1

3

C2

4

L2

ChaoticCircuit

Figure 1: Chaotic circuit applied to the system.

1. Introduction
In the natural world, there are many coupled systems
which have interesting behaviours. For instance, brain andneural networks can process various kinds of informations.
Animal skins which consists of many cells show various
kinds of patterns. glowing pattern of fireflies, human relations, computer networks and so on also show interesting
behaviours. One of interesting behaviours is a synchronization. Especially, chaotic coupled systems show very
interesting behaviours.
There are many studies about coupled chaotic circuits. In
these studies, various kinds of interesting phenomena areobserved. Especially, interesting behaviours are observedin asynchronous parameter regions near synchronous parameter regions. By decreasing a coupling strength, acoupled system does not keep synchronization states andthe system becomes asynchronous states. However, inthese parameter region, clustering phenomena, switching
phenomena between synchronization and asynchronizationand so on are observed. These phenomena can be observed
vast area of the region. By more decreasing a couplingstrength, these phenomena are disappeared and the system
becomes asynchronous states perfectly.
In this study, asynchronous states in chaotic circuits coupled by resistors as a ladder are investigated. Especially,relationships among the phase shift, coupling strengths andthe number of circuits are investigated. In order to investigate about phase shift in asynchronous states, the lack ofan oscillation cycle is defined. By using the definition, thephase shift is treated as the number of the lack.

(a)(b)

Figure 2: Attractor of the circuit as shown in Fig. 1. (a)Circuit experimental result. Each axis is set as 0.5[v/div]
Horizontal axis: v1 . Vertical axis: v3 . (b) Computer simulation result. Horizontal axis: x1 . Vertical axis: x3 .
2. System Setup
The chaotic circuit applied to the system is shown in
Fig. 1. This circuit was proposed and analyzed in [8].
Figure 2 shows a attractor of the circuit. Parameters are
set as C1 = 0.1 [µF], C2 = 0.068 [µF], L1 = 30 [mH],L2 = 100 [mH] and R1 = 720 [Ω] in the circuit experiment,
α = 0.75, β = 1.50, γ = 0.30 and δ = 5 in the computer simulation. Chaotic attractor is observed. Largest
Lyapunov exponent which calculated by the method in [8]is 0.302.
Figure 3 shows an investigated system. Chaotic circuits
are coupled by resistors. Diodes of the chaotic circuit aremodeled as a piecewise linear function shown in Fig. 4
In order to investigate the system, system equations arederived.

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RC
Chaotic
CircuitRC
Chaotic
CircuitRC

x13

Chaotic
Circuitx23

Chaotic
Circuit

x33
x43
x53

Figure 3: Bidirectionally coupled diode model.

f(v)

gf(v)

v-Vth

Figure 5: Computer simulation result. α = 0.75, β = 1.50,
γ = 0.30, δ = 5 and ε = 0.16

1V thwhere√
1vn1dL 1 i2
τ= √= “ · ”, xn1 =, xn2 =,t,dt
VC
L1 C 1th1 Vth√
vn31C1L 1 i4xn3 =, xn4 =, α=, β=,VthC1 VthR1C2√L1L11L1γ= , δ=gand ε =.
L2C1RC C1v

g1
Figure 4: Bidirectionally coupled diode model.

(n = 1, 2, . . . , N)
The normalized system equations are described as follows.
· First circuitẋ11ẋ 12ẋ13
 ẋ

=αn x11 − x12 − δx11 − δ(|x11 − 1| − |x11 + 1|)/2,=

x11 − x13 ,(4)Parameter mismatches pn is added. This is corresponding
to parameter mismatches of C1 .
3. Interaction of circuits in asynchronous states
3.1. Lack of the cycle

In order to investigate interactions between coupled twocircuits
in asynchronous states, a following expression is= pn β {x12 − x14 + ε(x23 − x13 )} ,applied. Figure 5 shows a one of computer simulation
results. In this simulation, backgrounds are painted into
14 = γxn3 .
(1) two colors. Namely, when the value is increasing, a back· Middle circuits
ground becomes yellow and when the value is decreasing,a background becomes green. In Fig. 5, green and yel
low appear repeatedly as a stripe pattern. Although someẋn1 = αn xn1 − xn2 − δxn1 − δ(|xn1 − 1| − |xn1 + 1|)/2,

gaps occur, all waveforms oscillate with almost same cycle.
This stripe pattern is observed in a vast parameter region ofẋ= xn1 − xn3 , n2
asynchronous states. However, some times green or yellow{}
is omitted from the cycle. In x43 of Fig. 5, one green isẋ=pβx−x+ε(x−2x+x),n3
nn2n4(n−1)3n3(n+1)3omitted, Namely the lack of the cycle is observed.
 ẋ
= γxn3 .
n4
In this study, relationship among the number of the lack,(2) a coupling strength and the number of circuits are inves· Last circuit
tigated. Figure 6 shows a procedure of counting the lack.
The case of two waveforms A and B, numbers of the slopeof a tangent are counted as a and b. When a = 1 and b = 1,ẋ=αx−x−
δx−
δ(|x−1|−|x+1|)/2,
N1
N N1
N2
N1
N1N1



these are reseted as a = 0 and b = 0. When a = 2 or b = 2,thelack is counted. Then, a and b are set as 0.
ẋ
=x−x,
N2N1N3


{}


ẋN3 = pn β xN2 − xN4 + ε(x(N−1)3 − xN3 ) ,
3.2. Relationship the lack and coupling strength ẋ
Figure 7 shows a relationship between a number of the= γx .
N4N3(3)lack and coupling strengths in the case of N = 2. In this- 281 -A
Ba 0b 01010110 1 20001

1001

100Figure 6: Procedure of counting the lack.
Figure 8: Relationship between a number of the lack anda coupling strength. Horizontal axis: coupling strength
ε. Vertical axis: number of the lack. Iteration time is50000000[τ] in each value ε, Step is 0.001[τ], pn = 1 +0.01(n − 1), N = 6, α = 0.75, β = 1.50, γ = 0.30 and δ = 5.
x13x23x33x43x53x63
Figure 7: Relationship between the number of the lack anda coupling strength. Horizontal axis: coupling strength
ε. Vertical axis: number of the lack. Iteration time is50000000[τ] in each value ε, Step is 0.001[τ], pn = 1 +0.01(n − 1), N = 2, α = 0.75, β = 1.50, γ = 0.30 and δ = 5.
parameter region, two circuits are not synchronized at all.
The lack is increased by decreasing coupling strength fromε = 0.01.
The case of N = 6 is shown in Fig. 8. The lack is observed from ε = 0.28 to 0.20 and from 0.18 to 0. By
increasing the number of the circuit, the lack can be observed in a high coupling strength. The number of the lackalso becomes high. Especially, the number around ε = 0.1
is higher than the case of ε = 0. In this area, clustering
phenomena as shown in Fig. 9 are observed. Upper twooscillators and lower three oscillators have almost samewaveforms respectively. However, two groups have diﬀerent waveforms. Therefore, the number of lacks of 2-3 and3-4 becomes higher than the others in this area.
The parameter mismatches between both ends of the system are increased by increasing the number of circuits.
In order to investigate the case of keeping parameter mismatches between both ends, Table 1 cases are investigated.
Figures 10- 12 show relationships between a number of thelack and coupling strengths. In Figure 10, the lack is observed from ε = 0.02 to ε = 0. In the case of N = 5 asshown in Fig. 11, the lack can be observed from ε = 0.2 toε = 0. In the case of N = 9 as shown in Fig. 12, the lack

Figure 9: Clustering phenomena. N = 6, pn = 1 + 0.01(n −1), α = 0.75, β = 1.50, γ = 0.30 and δ = 5.
can be observed in vast area. These results show that increasing the number of circuits means increasing the lack.
Additionally, clustering phenomena disappear in very low
coupling strengths area ( lower ε = 0.01 ).
4. Conclusion
In this study, asynchronous state in chaotic circuits coupled by resistors as a ladder has been investigated. Waveforms of each circuit were distinguished by an increaseor decrease of values. On this distinction, an increase or
decrease of values were observed as stripe patterns. By
decreasing a coupling strength, lack of patterns were observed.

- 282 -Table 1: Settings of parameter mismatches
Number of CircuitN
359Parameter mismatches
pk
0.01
0.005
0.0025

Relationships among this lack, coupling strengths andthe number of circuits were investigated. As results, following behaviours were revealed.
• By decreasing coupling strengths, the number of thelack is increased.
• Clustering phenomena are observed in over four circuits.
• By increasing the number of circuits, the number ofthe lack is increased.
Figure 12: Relationship between the number of the lackand a coupling strength. pn = 1 + 0.0025(n − 1), N = 9,
Horizontal axis: coupling strength ε. Vertical axis: numberof the lack. Iteration time is 50000000[τ] in each value ε,
Step is 0.001[τ], α = 0.75, β = 1.50, γ = 0.30 and δ = 5.
[2] T. Endo, and S. Mori, “Mode Analysis of a Ring of aLarge Number of Mutually Coupled van der Pol Oscillators,” IEEE Trans. Circuit and Systems, vol. CAS–25,
no. 5, pp. 7–18, 1978.
Figure 10: Relationship between the number of the lackand a coupling strength. pn = 1 + 0.01(n − 1), N = 3,
Horizontal axis: coupling strength ε. Vertical axis: numberof the lack. Iteration time is 50000000[τ] in each value ε,
Step is 0.001[τ], α = 0.75, β = 1.50, γ = 0.30 and δ = 5.

[3] D. A. Linkens, “Analytical solution of large numbers ofmutually-coupled nearly-sinusoidal oscillators,” IEEE
Trans. Circuits and Systems, vol. CAS-21, pp.294–
300, Mar, 1974.
[4] 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.
[5] 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.
[6] Y. Hosokawa, R. Tsujioka and Y. Nishio, “Relationbetween Synchronous Rate and Small Variations on
an Asymmetrical Coupled Chaotic System,” Proc.
NOLTA’05, pp. 166–169, 2005.

Figure 11: Relationship between the number of the lackand a coupling strength. pn = 1 + 0.005(n − 1), N = 5,
Horizontal axis: coupling strength ε. Vertical axis: numberof the lack. Iteration time is 50000000[τ] in each value ε,Step is 0.001[τ], α = 0.75, β = 1.50, γ = 0.30 and δ = 5.

[7] Y. Hosokawa and Y. Nishio, “Asymmetrical Chaotic
Coupled System Using Two Diﬀerent Parameter Sets,”
Proc. NOLTA’06, pp. 543–546, Sep. 2006.
[8] Y. Hosokawa and Y. Nishio, “Chaotic Circuit Generating Two Main Frequency Band Oscillations,” Proc.
of RISP International Workshop on Nonlinear Circuits,Communications and Processing, pp.697–700, 2014.

References[1] L. M. Pecora and T. L. Carrol, “Synchronization in
Chaotic Systems,” Physical Review Letters, vol. 64, pp.
821–824, 1990.

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