2015 International Symposium on Nonlinear Theory and its Applications
NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015

Synchronous States of Coupled Chaotic Circuits
Measured by Oscillation Frequency
Misa Sekimoto† , Yoko Uwate† and Yoshifumi Nishio††Departmaent of Electrical and Electronic Enginnering, Tokushima University
2-1 Minamijosanjima, Tokushima, 770-8506, Japan
Email: {misa, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate relationship between synchronous state and oscillation frequency in thecoupled chaotic circuit. We consider that oscillation frequency moves closer to steady value when the chaotic circuits become synchronization. From the computer simulation, we confirm synchronous state in the coupled chaoticcircuit is lowly dependent to oscillation frequency. Furthermore, we pay attention to timing of synchronization ofamong the coupled chaotic circuits.

chaotic circuits synchronize every partial circuit. We investigate whether there is relationship in the timing of synchronization as we determine synchronous state of circuitby using oscillation frequency or phase diﬀerence.

2. Circuit ModelFigure 1 shows the model of the chaotic circuit.
i1i2v

1. Introduction
L2L1

The synchronous phenomena are observed as not only
fields of natural science but also various fields. For example, we can confirm the flashing of fireflies (a fireflyis able to match frequency of other fireflies), metronome,
heartbeat of the human, and so on. The synchronous phenomena have been researched extensively in physics [1]and biology [2]∼[3]. In addition, applying synchronous
phenomena to medical technology are developed. Thesesynchronous phenomena are known as one of the nonlinear phenomenon. For the future engineering application,
we consider it is important to investigate synchronous phenomena of coupled chaotic circuits.
Synchronous Discrimination of the chaotic circuit uses
the phase diﬀerence generally. Not only the phase difference but also period and oscillation frequency exist in
the coupled chaotic circuit. We pay attention to oscillation frequency in the coupled chaotic circuit. We can consider that entire circuit synchronizes if and when oscillation frequency included in a certain individual match. In
this paper, we compare synchronous discrimination by using phase diﬀerence and oscillation frequency. We consider
that oscillation frequency converges on a steady value when
the coupled chaotic circuits synchronize.
As simulation result, relationship is uncommon between
synchronization of the chaotic circuit and oscillation frequency, although oscillation frequency converges on asteady value conclusively. Next, we pay attention to timingof synchronization of among the coupled chaotic circuits.
Chaotic circuits which we use in this paper do not synchronize at the same time when the coupling strength exceeds
a certain threshold value. Chaotic circuits synchronize every partial circuit. From these, we focus on timing whenC-rvdFigure 1: The model of the chaotic circuit.

This chaotic circuit is called Nishio-Inaba circuit[4]∼[7]. This circuit consists of single linear negative resistance, single nonlinear resistance consisting of two diodes,
two inductors and single capacitor. The linear negative resistance is realized by using the negative impedance converter made of an operational amplifier [4]. The circuit
dynamics is described by the following piecewise-linear
third-order ordinary diﬀerential equation,L1 didt1 = v + ri1 , L2 di2 = v − vd (i2 ),dt

dv
 C dt = −i1 − i2 .

(1)

We approximate the I − V characteristic of the nonlinear
resistance by the following function,)(VVrd
i2 +− i2 −.
(2)vd (i2 ) =2rdrd
The parameter rd is the slope of the nonlinear resistance.
Equation (1) is normalized by changing the variables according to- 213 -√√CL1 C
di1 =V x ; i2 =Vy ; v = Vz ; “ · ” =,L1L2dτ√√√CL1L1 C=α ;= β ; rd= γ ; t = L1Cτ. (3)rL1L2L2

coupled chaotic circuit are fixed α1 = 0.40 and αn = 0.48
to make it easier to compare transition of oscillation frequency as shown in Fig. 3. Five chaotic circuits is used inthis paper.

Equation (1) is normalized asSetting condition of parameter αn ( n = 5,α2 =α3 =α4 =the variable ).

Figure 3:ẋ = αx + z, ẏ = z − f (y),




 ż = −x − βy.
(4)The nonlinear function f (y) corresponds to the I − V characteristics of the nonlinear resistors consisting of the diodes
is assumed to be described as follow,)(γ11f (y) =y+ − y−.
(5)2γγ

The coupling strength σn is chosen as a control parameter. All σn are set same value. Equation of couplingstrength is shown as√1 Ln1σn =(7).
R Cn
There is inverse relationship between the couplingstrength σ and resistor R.
3.2. Oscillation Frequency

3. Simulation Method

Equation of resonance frequency is described as

3.1. Circuit Model1.
(8)√
2π Ln1Cn
We can not obtain exact value of oscillation frequency
by using Eq. (8) because the proposed chaotic circuit includes a linear negative resistance and a nonlinear resistance consisting of two diodes. New measuring procedure
is proposed to obtain exact value of oscillation frequency.
Equation of new measuring procedure of oscillation frequency is described as,
f =We consider a ladder network using chaotic circuits asshown in Fig. 2. Ladder network is the simple network.
Namely, ladder network is easier to understand the flow ofelectric signal. Each circuit is coupled via resistor R.

Figure 2: Ladder network.

1.
(9)TThe cycle T is obtained from the chaotic attractor. The
chaotic attractor of proposed model is shown in Fig. 4.
F=Normalized equation of coupled chaotic circuits is described as follow,

Zx˙n = αxn + zn .
y˙n = zn − f (yn ).
z˙n = −xn − βyn + σ(z( n + 1) − zn ),(n = 1)z˙=−x−βy+σ(zn−1) − zn ),nnn(
(n= the maximum value)




z˙n = −xn − βyn + σ(z( n + 1) + z( n − 1) − 2zn ),(otherwise)

X

Figure 4: The chaotic attractor.

(6)
For this simulation, the parameters are set as follows,
βn = 3.0 and γn = 470.0. Own oscillation frequency andphase diﬀerence of the chaotic circuits depend on the parameter α. Each circuit is assigned α to investigate transition of oscillation frequency when the chaotic circuits aresynchronized. Parameter α is fixed value 0.40 ≤ α ≤ 0.48.
In addition, the parameter α1 and αn located in both ends ofPoincare section of attractor is set between the first andthe fourth quadrant. The number of dots to describe attractor is counted from Poincare section. First 10,000 laps ofattractor are ignored because attractor is in an unstable state
at first. Total dots from 10,001 laps to 30,000 laps are measured for obtaining average dots per one lap. In addition,
average dots are replaced the cycle of the chaotic attractor. We can obtain exact value of oscillation frequency byassigning the cycle T to Eq. (9).

- 214 -

3.3. Definition of Synchronous Statez
θ = arctan .
x

Oscillation frequency : F[Hz]Synchronous condition is necessary to investigate relationship between synchronous state and transition of oscillation frequency. In order to derive synchronous state numerically, the phase diﬀerence θ is used. Equation of thephase diﬀerence is shown as(10)

The phase diﬀerence is obtained from chaotic attractor
as with oscillation frequency. For example, we measure thephase diﬀerence between circuits 1 and 2. The phase diﬀerence is obtained by assigning coordinate of dot drawn thechaotic attractor of circuit 2 to Eq. (10) when the chaotic
attractor of circuit 1 runs through the poincare section. We
define that two circuits 1 and 2 synchronize when the phase
diﬀerence θ of each circuit is below 30 degrees.

The coupling strength :Figure 7: Transition of oscillation frequency when the value of
α are set nearer α5 (α2 = 0.45, α3 = 0.46, α4 = 0.47).

4. Simulation Results
Oscillation frequency : F[Hz]We set parameter α1 = 0.40 and α5 = 0.48. Figures 5∼8
show transition of oscillation frequency when the couplingstrength is changed.

n(1-2)n(1-3)n(1-4)n(1-5)

The coupling strength :Figure 8: Transition of oscillation frequency when the value of
α are set nearer medium value(α2 = 0.43, α3 = 0.44, α4 = 0.45).

Figure 5: Transition of oscillation frequency when the value ofOscillation frequency : F[Hz]α are set equability(α2 = 0.41, α3 = 0.435, α4 = 0.46).

The coupling strength :Figure 6: Transition of oscillation frequency when the value of
α are set nearer α1 (α2 = 0.41, α3 = 0.43, α4 = 0.44).

There are four patterns to set parameter α2 , α3 and α4 .
We set these parameters : approximately at equal interval while avoiding the values that aﬀect periodic solution,nearer α1 , nearer α5 and nearer medium value.
From the Figs. 5∼8, n(a − b) is indicated circuit state between circuit number a and b. In addition, we surround thecoupling strength with circle when the phase diﬀerence isbelow 30 degrees. It is clear that relationship is uncommon
between synchronous state and oscillation frequency in any
patterns of α2 ∼ α4 . Each circuit does not synchronize
when oscillation frequency converges on a steady value.
And, oscillation frequencies are obtained a stable value before entire circuit becomes synchronization.
Next, we pay attention to timing of synchronization ofamong the coupled chaotic circuits. Timing of synchronization forced on oscillation frequency and the phase difference when the values of α2 ∼ α4 are set equability isshown in Tables 1 and 2. Table 1 is visualized Fig. 9 tobe easily understandable timing of synchronization. There- 215 -are four symbols ○, ◎, △, ◇. These symbols show thatcircuits have some condition. For example, when the coupling strength is set 0.002, chaotic circuit of circuit number
n=1,2 becomes synchronization as shown in Table 1. With
respect to Tab. 2, when coupling strength is set 0.003, oscillation frequency is obtained a stable value in circuit number
n=1,3 and n=2,3. It will be easy to understand this methodby compare Table 1 and 2 with Fig. 5. By the expectation,
we consider that it has similarities for two investigations.
As compared to two table, there is no similarity in eachtiming.

5. Conclusions
In this study, we have investigated relationship between
synchronous state and oscillation frequency in the coupledchaotic circuit. From simulation results, we could obtaina steady value of oscillation frequency before entire circuit
becomes synchronization. These results are not determinedby the value of α. That is, synchronous state in the coupledchaotic circuit is lowly dependent to oscillation frequency.
References[1] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer,“Synchronization and graph topology,” Int. J. Bifurcation and Chaos, vol.15, no.11, pp.3423-3433, Nov.
2005.
[2] R. M. May, “Simple mathematical models with very
complicated dynamics,” Nature 261, 459 467 (10June 1976).
[3] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronization of nonlinear electronic oscillators for neural computation,” IEEE Trans. Neural Networks, vol.15, no.5, pp.1315-1327, Sep. 2004.

Figure 9: Timing of synchronization based on phase diﬀerence.

σTable 1: Timing of synchronization based on phase diﬀerence.
Circuit number : n
1
2
3
4
50.002 ○ ○0.006○ ○, ◎ ◎0.007 ○○
0.021○○
0.024 ○
○
0.033
○ ○
0.045
○
○
0.050 ○
○
0.051
○
○

[4] Y. Nishio, N. Inaba, S. Mori and T. Saito, “Rigorousanalyses of windows in a symmetric circuit,” IEEETrans. Circuits Syst., Vol. 37, No. 4, pp. 473-487,
1990.
[5] Y. Takamaru, H. Kataoka, Y. Uwate and Y. Nishio,“Clustering Phenomena in Complex Networks ofChaotic Circuits”, Proc. ISCAS’ 12 Mar. 2012.
[6] R. Stoop, P. Benner and Y. Uwate, “Real-world existence and origins of the spiral organization of shrimpshaped domains,” Phys. Rev. Lett., Vol. 105, 074102,
2010.
[7] C. Bonatto and J. A. C. Gallas, “Periodicity hub andnested spirals in the phase diagram of a simple resistive circuit,” Phys. Rev. Lett., Vol. 101, 054101, 2008.

σTable 2: Timing of synchronization based on oscillation
frequency.

0.002
0.003
0.005
0.021

1○
○
○
○

2○
◎◎
◎Circuit number : n34○, ◎△△○, ◎, △◇5○, ◎, △, ◇- 216 -