2012 International Symposium on Nonlinear Theory and its Applications
NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Analysis of Fifteen Coupled Chaotic Oscillators Composed of RC Circuits
Ayana Shimada† , Yoko Uwate† , Yoshifumi Nishio† and Jingmin Xin‡
† Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Phone: +81-88-656-7470
Email: {ayana,uwate,nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we analyze fifteen coupledchaotic oscillators composed of RC circuits. In particular,
we pay our attention on the cross correlation characteristics between neighboring oscillators. In addition, we investigate the cross correlation characteristics between thefirst oscillator and the others. By computer calculations,
we investigate how the cross correlation changes as a circuit parameter increases. Further, the oscillators in the laterstage are confirmed to have stronger correlation than thosein the early stage.
1. Introduction
Recently, many researchers have shown their interestsin chaotic systems. Chaos has been investigated not only
in engineering but also in various fields such as medicine,sociology and economics. In the field of electrical andelectronic engineering, researchers have proposed various
applications using chaos as chaos communications, chaos
cryptosystem, chaos neural networks and so on. In orderto realize chaotic engineering systems, it is important toinvestigate simple chaos-generating circuits.
In our previous study, a simple chaotic oscillator using
two RC circuits were proposed [1]. We carried out computer calculations and investigated chaotic behavior when
the number of RC circuits increased [2].
In this study, we analyze fifteen coupled chaotic oscillators composed of RC circuits. In particular, we pay ourattention on the cross correlation characteristics between
neighboring oscillators. In addition, we investigate thecross correlation characteristics between the first oscillator
and the others. By computer calculations, we investigate
how the cross correlation changes as a circuit parameterincreases. Further, the oscillators in the later stage are confirmed to have stronger correlation than those in the earlystage.
2. Circuit model
Figure 1 shows the circuit model. In this figure 15chaotic oscillators, which was proposed in [1], are coupled
via one-way coupling through simple comparators. An independent rectangular voltage source is connected only tothe first oscillator and the voltage across one of the two capacitors in the previous stage of oscillators is inputted to the‡ Xi’an Jiaotong University
28 Xianning Xilu, Xi’an, Shaanxi 710049, China
Phone: +86-29-8266-8802
Email: jxin@mail.xjtu.edu.cn
comparators in the next oscillator. Hence, the whole circuit
consists of one rectangular voltage source, 30 comparators,
30 resistors, and 30 capacitors.
Figure 2(a) shows the rectangular voltage waveform
VS (t) inputted to the first oscillator. Eα is the amplitudeof the rectangular voltage and T is the period of the waveform. The circuit equations are described as follows:{dv1−v1 + E (v2 > VS )RC=
−v1 − E (v2 < VS )
dt


dv2 
 −v2 − E (v1 > VS )RC=
 −v2 + E (v1 < VS )dtdvn 
 −vn + E (vn+1 > vn−1 )RC= −vn − E (vn+1 < vn−1 )dtdvn+1  −vn+1 − E (vn > vn−1 )
RC(1)
=
 −vn+1 + E (vn < vn−1 )dt(n = 3, 5, 7, 9, · · · , 29)
where E is the output voltage of the comparators, namely
the DC supply voltage of the operational amplifiers. By
using the following variables and the parameters,
vk = Exk , t = RCτ, T = RCβ,(2)(k = 1, 2, 3, 4, · · · , 30)
we obtain the normalized circuit equations. Because thecircuit equations are linear in each region, the rigorous solution of the circuit equations can be derived as follows:−τ (x10 − 1)e + 1 (x2 > Vα )x1 =  (x10 + 1)e−τ − 1 (x2 < Vα )−τ (x20 + 1)e − 1 (x1 > Vα )x2 =  (x20 − 1)e−τ + 1 (x1 < Vα )−τ (xn0 − 1)e + 1 (xn+1 > xn−1 )xn =  (xn0 + 1)e−τ − 1 (xn+1 < xn−1 ){(x(n+1)0 + 1)e−τ − 1 (xn > xn−1 )xn+1 =(x(n+1)0 − 1)e−τ + 1 (xn < xn−1 )(3)(n = 3, 5, 7, 9, · · · , 29)
where Vα corresponds to VS and shown in Fig. 2(b), andx10 , x20 and xn0 are initial values (n =3, 5, 7, 9, · · · , 29).

- 785 -

(a)(b)

Figure 2: Rectangular voltage waveform.

(1)

(2)

(3)

(a)
Figure 1: Circuit model.

(b)

(c)

3. Simulation results

Figure 3: Chaotic attractors obtained from computer calculations for β = 4.0. (a) x1 −x3 , (b) x13 −x15 , and (c) x29 −x30 .
(1) α = 0.058, (2) α = 0.080, and (3) α = 0.100.

Figure 3 shows some examples of chaotic attractors obtained from computer calculations. We compare Fig. 3(a2)with (c2), then we can see that the degree of chaos becomeslarger for the later stage. When the value of α becomeslarger, only periodic attractors are observed in the earlystages as shown in Fig. 3(a3).
Figures 4(a) and (b) show the time waveform, when thevalues of α are 0.058 and 0.080, respectively. There exist
regular and irregular parts in the waveforms. We pay ourattentions on the relation between the time waveforms ofneighboring oscillators.
We show the cross correlations of neighboring oscillators in Figs. 5(a) and (b). Vertical axis is the correlation coeﬃcient R x,y between xm and xm+2 (m = 1, 3, 5, 7, 17, 27).
Horizontal axis shows the delay time ∆τ. We find thatneighboring oscillators correlate from Figs. 5(a) and (b).
In Fig. 5(a), when the value of ∆τ is 0.18, the cross correlation coeﬃcient of x1 − x3 takes the largest value 0.53. Thecross correlation coeﬃcient of x27 − x29 takes the largest
value 0.67 also for ∆τ = 0.18. Therefore, we can say thatthe cross correlation becomes larger for the later stages. Wesummarize the values of ∆τ for which the cross correlationcoeﬃcient takes the maximum value max(R xy ) in Table 1.
We find that the neighboring oscillators have strong positive correlations.
Figures 6 and 7 show the attractors reconstructed by considering the time delay ∆τ shown in Table 1. From theshapes of these attractors, we can also see that the neighboring oscillators are correlated with some amount of timedelay.
Next, Fig. 8 shows the cross correlations of the first oscillator and others. Vertical axis is the correlation coeﬃcient R x,y between x1 and xl (l = 3, 5, 11, 17, 23, 29). Horizontal axis shows the delay time ∆τ. We can see thatthe correlation coeﬃcients between x1 and others are verysmall except x3 and x5 . This means that the first oscillator
does not have strong correlations with the oscillators on thelater stages.
Figure 9 shows the attractors reconstructed by considering the appropriate time delay obtained from Fig. 8. We
cannot see any strong correlations.

- 786 -

(a)

(a)

(b)
Figure 5: Graph of the cross correlation value of neighboring oscillators for β = 4.0. (a) α = 0.058 and (b) α = 0.080.
4. Conclusion(b)
Figure 4: Time waveforms for β = 4.0. From top: x1 ,x3 , x5 , x7 , x9 , x11 , x13 , x15 , x17 , x19 , x21 , x23 , x25 , x27 , x29 .
(a) α = 0.058 and (b) α = 0.080.
Table 1: Time delay ∆τ with the maximum values of correlation coeﬃcient max(R xy ).

x1 − x3
x3 − x5
x5 − x7
x7 − x9
x17 − x19x27 − x29

α = 0.058∆τmax(R x,y )
0.18
0.53
0.22
0.64
0.21
0.61
0.18
0.54
0.21
0.70
0.18
0.67

α = 0.080∆τmax(R x,y )
0.20
0.63
0.20
0.56
0.22
0.61
0.26
0.72
0.20
0.66
0.21
0.73

In this study, we have analyzed fifteen coupled chaotic
oscillators composed of RC circuits. In particular, we paid
our attention on the cross correlation characteristics between neighboring oscillators. In addition, we investigatedthe cross correlation characteristic between the first oscillator and others. By computer calculations, we investigated the cross correlation changes as a circuit parameterincreases. We could observe the cross correlation characteristic between neighboring oscillators. However, wefound that non-adjacent oscillator does not have correlation. The chaotic oscillators in the later stage were confirmed to have stronger correlation than those in the earlystage.
In our future work, we will subject detailed research of
the cross correlation of neighboring oscillators and analyze
with Lyapunov exponent.

Acknowledgments
This work was partly supported by JSPS Grant-in-Aidfor for Scientific Research 22500203.

- 787 -(a)

(d)

(b)

(e)

(c)
(a)

(b)

(c)

(d)

(e)

(f)

(f)

Figure 6: Attractors reconstructed with time delay for α =0.058 and β = 4.0. (a) x1 (τ) − x3 (τ + 0.18), (b) x3 (τ) −x5 (τ+0.22), (c) x5 (τ)− x7 (τ+0.21), (d) x7 (τ)− x9 (τ+0.18),(e) x17 (τ) − x19 (τ + 0.21), and (f) x27 (τ) − x29 (τ + 0.18).
References

Figure 7: Attractors reconstructed with time delay for α =0.080 and β = 4.0. (a) x1 (τ) − x3 (τ + 0.20), (b) x3 (τ) −x5 (τ+0.20), (c) x5 (τ)− x7 (τ+0.22), (d) x7 (τ)− x9 (τ+0.26),(e) x17 (τ) − x19 (τ + 0.20), and (f) x27 (τ) − x29 (τ + 0.21).

[1] S. Masuda, Y. Uchitani and Y. Nishio, “Simple ChaoticOscillator Using Two RC Circuits,” Proc. of NCSP’09,pp. 89-92, 2009.
[2] A. Shimada, H. Kumeno, Y. Uwate, Y. Nishio andJ. Xin, “Coupled Chaotic Oscillators Composed of RC
Circuits,” Proc. of NCSP’12, pp. 17-20, 2012.

Figure 8: Cross correlation values of the first oscillator andothers for α = 0.080 and β = 4.0.

(a)

(b)

(c)

(d)

Figure 9: Attractors reconstructed with time delay for α =0.080 and β = 4.0. (a) x1 (τ)− x5 (τ+0.30), (b) x1 (τ)− x9 (τ+1.33), (c) x1 (τ) − x17 (τ + 0.17), and (d) x1 (τ) − x29 (τ + 1.35).

- 788 -