2004 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'04)
Hawaii, USA, Mar. 5-7, 2004.

Response of Coupled Chaotic Circuits to Various Type Signals
Tomoya Hayashi and Yoshifumi Nishio
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, FAX:+81-88-656-7471
Email: {hayatan or nishio}@ee.tokushima-u.ac.jp

Abstract

2. Circuit model

In this study, we investigate synchronization phenomena of
coupled chaotic circuits to input signals in time domain. Synchronization state of two chaotic signals changes dependingon the input signals. By using phase difference of two chaotic
signals, it can express the states of quasi-synchronization. We
confirmed its behavior to parameter of input signals in time
domain.

Figure 1 shows the circuit model. The circuit in Fig. 1(a)is a three-dimensional autonomous circuit generating chaos.
and was proposed by Inaba and Mori [4]. In the circuit in Fig.
1(b), two Inaba’s circuits are coupled by one coupling resistor
R and an input signal is added to the coupling resistor as acurrent I3 .
The circuit equations governing the circuit in Fig. 1(b)are given as Eq. (1). We assume the i − v characteristics ofthe diodes in the circuit by the two-segment piecewise linear
function as Eq. (2).
3dIk
= rik + rIk − vk − RIj
L1
dt
j=1
dik
= rik + rIk − vk − vd (ik )
L2
(1)
dt
dvk
= ik + IkCdt(k = 1, 2)(2)
vd (ik ) = 0.5(rd ik + E − |rd ik − E|)

1. Introduction
Studies on engineering application of chaos, such as chaos
communication systems and chaos cryptosystems, attract
many researchers attentions. Because chaotic signals process
infinite information, it could be possible to create novel engineering systems with great advantages. However, at thismoment, there are only a few systems with some kinds of advantages over conventional engineering systems. Hence basic
researches on chaotic phenomena with future engineering applications in mind are still quite important.
On the other hand, the authors have researched about many
oscillatory circuits connected by a resistor [1]-[3]. Using
this connection of circuits, chaotic signals generated fromchaotic circuits change the state of synchronization to minimize the total of currents flowing in coupled resistor. Moreover, we confirm the phenomena of the coupled chaotic circuits when sinusoidal input is added [4]-[5]. The coupledsystem changes the state of synchronization against the input
signal with very limited range of angular frequency. Therefore this coupled system is useful for signal processing of input signals.
In this study, we investigate synchronization phenomena of
coupled chaotic circuits to input signals in time domain. Synchronization state of two chaotic signals changes dependingon the input signals. By using phase difference of two chaoticsignals, it can express the states of quasi-synchronization. We
confirmed its behavior to parameter of input signals in time
domain.

By using the variables and the parameters in Eq. (3), thecircuit equations in Eq. (1) are normalized as Eq. (4).
CC
Exk , ik =Eyk ,Ik =L1
L1
L1,(3)
vk = Ezk , t = L1 Cτ, α =L2
CCCβ=r, γ=R, δ = rd
L1L1L1- 273 -3dxk= β(xk + yk ) − zk − γxjdτj=1dyk= α {β(xk + yk ) − zk − fd (yk )}dτdzk= xk + ykdτ(k = 1, 2)

(4)

I1

I1

i1

I3
L1

L1

4. Quasi-synchronization of chaotic signals

I2
i2

i1
C

v1

L2-r

vd

L1

C-r

v1

C

L2-r

vd

v2

L2vd

R
(a)

(b)

Figure 1: Chaotic circuit and the coupled chaotic circuits.
α = 7.0, β = 0.14 and δ = 100.
3. Input signals
It is possible to consider various kinds of signals as input signal x3 in future, for example, simple signals represented by basic mathematical functions, chaotic signals obtained from other chaotic circuits, audio/visual signals in realengineering systems and so on.
In this study, as the first step toward the total understanding of the response of the coupled circuits to various input
signals, we consider the case that a simple sinusoidal, rectangular and chaotic signal is input.
Namely, we use the following sinusoidal functions with thenormalized amplitude A m and the normalized angular frequency ω as the input x 3 .
x3 (τ ) = Am sin ωτIn this study, we use the quasi-synchronization state of twochaotic signals. In this state, a chaotic signal S i (t) from a sub
circuit of coupled system and the other chaotic signal from theother sub circuit S j (t) have the relation such as |S i (t)±Sj (t−T )| < ε where a constant T represents phase shift between
two signals and a constant ε is much smaller than the average
amplitude of each chaotic signal. This is not mathematicaldefinition and is used only for qualitative explanation of theobserved phenomena.
In previous research[4], we have investigated the synchronization of coupled chaotic circuits to input signal which time
passed enough. In order to investigate the synchronization,
we define the phase difference of the two chaotic signals.
By using the time when the chaotic signals take extrema as
shown in Fig. 2, the phase difference θ[deg] is defined asfollows.
τ20 − τ10× 360.
(9)θ=τ11 − τ10(5)x1x2

τ 10 τ 1 1

τ 20τ

Figure 2: Definition of the phase difference θ.
Function of rectangular input signal is defined as follows.
It is defined with Eq. 5, using the parameter angular frequency
ω and Am .
5. Response to input signal in time domain
−Am sin(ωτ ) < 0
In this section, response of coupled chaotic circuits is inx3 =(6)Am sin(ωτ ) > 0
vestigated, to various type signals in time domain. Calculated phase difference θ which shows the synchronization
And for chaotic input signals, we use the chaotic input sig- state of chaotic signals in coupled chaotic circuits, are shown
nal which is generated from same chaotic signal used cou- in Fig. 3- 10.
pled chaotic circuits. The differential equations of chaotic
θ is calculated on the basis of one chaotic signal of two
circuits generating chaotic input signal are shown as Eq. (7). chaotic signals. From this reason, with the signal made in to a
We wrote it subscript in Eq. (7) for convenience.
standard, phase difference has two displays in one state. Then
ε is the parameter for the frequency of oscillation of chaotic in order to display simply, we make the state where there iscircuits. Amc controls amplitude of input chaotic signal. no input signal as a standard, θ is shown by 180[deg] using
Chaotic signal x4 is amplified by Amc and inputted to cou- |θ − 180|. And in every results in Fig. 3- 10, input of signal
pled chaotic circuits as chaotic input signal x 3 .
is begin after 50,000 iterations(τ ).
In there results, phase difference θ of two chaotic sigdx42nals
varied depending on the input signals. Obviously,= ε4 β(x4 + y4 ) − ε4 z4 − ε4 γx4dτθ changes only to input signal with limited frequency.
(Fig. 3, 4, 6, 8, 10) Then value of θ is denepnding on thedy4(7) amplitude of input signals. (Fig. 5, 7, 9) And between two= α {ε4 β(x4 + y4 ) − ε4 z4 − fd (y4 )}dτregion, without a response of coupled chaotic circuits (θ isdz4
round 0[deg]) and with a response (θ is around 120[deg]), θ= x4 + y4dτis changes according to the value of parameter which a input
signal have. θ varies not suddenly but gradually and changes
x4 (τ ) = Amc × x3 (τ )(8) like a sinusoidal signal.

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6. Conclusions
We investigated that the quasi-synchronization phenomena in coupled chaotic circuits to various type signals. By
using phase difference θ of chaotic signals, states of quasisynchronization of chaotic signals are shown in time domain
to varied parameters of input signals. It is confirmed thatphase difference θ varied like a sinusoidal signal between region with coupled chaotic circuits response to input signal
and not response.

iteration:tau ->Phase difference : theta ->[deg]

References[1] Yoshifumi Nishio and Akio Ushida, “Spatio-Temporal
Chaos in Simple Coupled Chaotic Circuits,” IEEE Transactions on Circuits and Systems I, vol. 42, no. 10, pp.
678-686, Oct. 1995 .

parameter: omega ->Phase difference : theta ->[deg][2] Yoshifumi Nishio, Martin Hasler and Akio Ushida,“Markov Chain Modeling of Chaotic Wandering in Simple Coupled Chaotic Circuits,” Proc. of ECCTD’01, vol.
2, pp. 105-108, Aug. 2001.
[3] Yoshifumi Nishio and Akio Ushida, “Chaotic Wanderingand Its Analysys in Simple Coupled Chaotic Circuits,”
IEICE Transactions on Fundamentals, vol. E85-A no. 1,pp. 248-255, Jan. 2002.
[4] Tomoya Hayashi, Yoshifumi Nishio, Martin Hasler andAkio Ushida, “Response of Coupled Chaotic Circuits to
Sinusoidal Input Signal,” Proc. of ISCAS’02, vol. 3, pp.
461-464, May 2002.

<-iteratio

auetarama ->meger: on:tp

Figure 3: Sinusoidal signal is input: Phase difference θ toτ (iteration) and ω. α = 7.0, β = 0.14, γ = 0.03, δ = 100,
Am = 1.17.

[5] Tomoya Hayashi, Yoshifumi Nishio, Martin Hasler andAkio Ushida, “Information Transmission Using Signal Processing by Coupled Chaotic Circuits,” Proc. ofNOLTA’02, vol. 2, pp. 821-824, Oct. 2002.
[6] Naohiko Inaba and Shinsaku Mori, “Chaotic Phenomenain Circuits with a Linear Negative Resistance and an Ideal
Diode,” Proc. of MWSCAS’88, pp. 211-214, Aug. 1988.
iteration:tau ->Phase difference : theta ->[deg]parameter: omega ->Figure 4: Sinusoidal signal is input: θ to τ (iteration) and ω.
α = 7.0, β = 0.14, γ = 0.03, δ = 100, Am = 1.17.

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[deg]

Phase difference : theta ->iteration:tau ->Phase difference : theta ->[deg]<-iteratio

n:t

au: Ametermpara->
parameter: epsilon ->Figure 8: Chaotic signal is input : θ to τ (iteration) and ε.
Figure 5: Sinusoidal signal is input: θ to τ (iteration) and A m .
α = 7.0, β = 0.14, γ = 0.03, δ = 100, Amc = 1.0, β(input
α = 7.0, β = 0.14, γ = 0.03, δ = 100, ω = 1.037.
signal)= 1.0.

[deg]

iteration:tau ->Phase difference : theta ->Phase difference : theta ->[deg]<-iteratio

parameter: omega ->auc ->: Ametermpara

Figure 9: Chaotic signal is input : θ to τ (iteration) and A mc .
α = 7.0, β = 0.14, γ = 0.03, δ = 100, ε = 1.037, β(inputsignal)= 1.0.

[deg]

Phase difference : theta ->[deg]iteration:tau ->Phase difference : theta ->Figure 6: Rectangular signal is input: θ to τ (iteration) and ω.
α = 7.0, β = 0.14, γ = 0.03, δ = 100, Am = 1.17.

n:t<tioraiteuta
n:: Ametermpara->

parameter: beta ->Figure 7: Rectangular signal is input: θ to τ (iteration) andAm . α = 7.0, β = 0.14, γ = 0.03, δ = 100, ω = 1.037.

Figure 10: Chaotic signal is input : θ to τ (iteration) and β.
α = 7.0, β = 0.14, γ = 0.03, δ = 100, Amc = 1.0, ε = 1.0.

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