Investigation of Synchronization Phenomena in
Coupled Two-degrees-of-Freedom Chaotic Circuits
Naoto Yonemoto, Katsuya Nakabai, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
E-mail: {yonex34, nakabai, uwate, nishio}@ee.tokushima-u.ac.jp

II. SYSTEM MODEL
The circuit model is shown in Fig. 1. Two two-degrees-offreedom chaotic circuits are coupled by a resistor R. The chaotic
circuit consists of negative resistor and two Inaba’s circuits. Inorder to change the frequencies of chaotic circuits, we change
the value of L and C.

Abstract—This paper considers synchronization phenomena incoupled two-degrees-of-freedom chaotic circuits by a resister. It isconsidered that studying various cases of synchronization
phenomena when using chaotic circuits showing asynchronous
simultaneous oscillation will be useful in clarifying non-linear
phenomena that exist around us. By means of the circuit
experiments and computer simulations, chaotic attractors andLissajous figures are shown. From the results, synchronization
phenomenon was confirmed between the circuits farthest from theconnection part.
Keywords; Synchronization; Asynchronous
oscillation; Two-degrees-of-freedom chaotic circuitI.

simultaneous

INTRODUCTION

There are many synchronization phenomena in nature.
Representatives include frog chorus and firefly issuance. As forengineering applications, communication encryption anddecryption, and synchronization phenomena in neurons of thehuman brain have been confirmed, so application to brain
computers is expected. Therefore, research on synchronization
phenomena in the field of chaos has attracted much attention.
Observation and analysis of these phenomena using electric
circuits is one of the useful means because it is able tocorrespond to the actual physical system. Studies have beenconducted on the variation of synchronization phenomena andthe complexity of generated chaos for each parameter when two
three-dimensional chaotic circuits are connected [1].

Figure 1: Circuit model.

Asynchronous simultaneous oscillation is one of the typical
non-linear phenomena and is considered to exist in various highdimensional systems in nature. Therefore, several synchronous
phenomena have been investigated when connecting an
asynchronous simultaneous oscillator with resistors [2] orinductors [3]. High-dimensional chaotic circuits are actively
studied, and a two-degrees-of-freedom chaotic circuit has been
proposed [4]. The circuit shows asynchronous simultaneous
oscillation when connecting two sets of chaotic circuits withdifferent natural frequencies in series. A set of circuits in a twodegrees-of-freedom chaotic circuit is called a subcircuit. In thisstudy, we investigate the synchronization phenomenon using
two chaotic circuits with asynchronous simultaneous oscillation
connected by resistors. In the circuit experiment and computer
simulation, chaotic attractors and Lissajous figures are shownfor three natural frequency cases.

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The parameters are described as follows:
𝐿𝐿𝐿𝐿𝐶𝑖1𝐿12𝐿13𝐿14

𝛼 = 𝑔√ 𝑖1 , 𝛽11 = 11 , 𝛽12 = 11 , 𝛽13 = 11 ,𝐿𝐿𝐿𝐿

𝐿21𝐿22𝐿23𝐿24

𝛽21 = 11 , 𝛽22 = 11 , 𝛽23 = 11 , 𝛽24 = 11𝐶

𝐶

𝐶1𝐿1𝐿

𝛾1 = 11 , 𝛾21 = 11 , 𝛾22 = 11 , 𝛿 = √ 𝑖1 , 𝜀 = √ 𝑖1 .
𝐶𝐶𝐶𝑅 𝐶𝑟𝐶12185

2122𝑖1𝑑𝑖1ISOCC 2020

The normalized circuit equations are described as follows:𝑥̇ 11 = −(𝑦11 + 𝑧11 ) + 𝛼(𝑥11 + 𝑥12 ) − δ(𝑥11 + 𝑥12 − 𝑥21 − 𝑥22 )𝑦̇ 11 = 𝑥11
𝑧̇11 = 𝛽11 (𝑥11 − 𝑓(𝑧11 ))𝑥̇ 12 = −𝛾1 (𝑦12 + 𝑧12 ) + 𝛼𝛾1 (𝑥11 + 𝑥12 ) − 𝛾1 𝛿(𝑥11 + 𝑥12 − 𝑥21 − 𝑥22 )𝑦12 = 𝛽12 𝑥12
𝑧̇12 = 𝛽13 (𝑥11 − 𝑓(𝑧12 ))𝑥̇ 21 = −𝛾21 (𝑦22 + 𝑧21 ) + 𝛼𝛾21 (𝑥21 + 𝑥22 ) + 𝛾21 𝛿(𝑥11 + 𝑥12 − 𝑥21 − 𝑥22 )𝑦̇ 21 = 𝛽21 𝑥21
𝑧̇21 = 𝛽22 (𝑥21 − 𝑓(𝑧21 ))𝑥̇ 22 = −𝛾22 (𝑦22 + 𝑧22 ) + 𝛼𝛾22 (𝑥21 + 𝑥22 ) + 𝛾22 𝛿(𝑥11 + 𝑥12 − 𝑥21 − 𝑥22 )𝑦̇ 22 = 𝛽23 𝑥22
𝑧̇22 = 𝛽24 (𝑥21 − 𝑓(𝑧22 )){.

In the case Fig. 2(c), although the subcircuits closest to theconnection is asynchronous, it is confirmed that the lower
subcircuits far from the connection are in-phase synchronized.
The circuit parameters are changed from case Fig. 2(a) to β21 =2.0, β22 = 12.0, γ21 = 2.0
B. Circuit experiment results
Figure 3 shows the circuit experiment results. As shown inFig. 3, in-phase synchronization is able to be observed between
the two underlining subcircuits, when natural frequencies are setto ω1 ≠ ω2, ω3 ＝ ω4.
Element values are 𝐶11 = 100.0[nF], 𝐶12 = 33.0[nF],
𝐿11 = 180.0[mH], 𝐿12 = 30.0[mH], 𝐿13 = 60.0[mH],𝐿14 = 10.0[mH], 𝐶21 = 68.0[nF], 𝐶22 = 33.0[nF],
𝐿21 = 120.0[mH], 𝐿22 = 20.0[mH], 𝐿23 = 60.0[mH]III.

𝐿24 = 10.0[mH], 𝑅 = 12[k𝛺].

RESULTSIn this study, various natural frequency cases are considered.
The frequencies of x11, x21, x12, and x22 are corresponding to ω1,ω2, ω3, and ω4, respectively.
A. Computer simulation results
Figure 2 shows the computer simulation results.
In the case Fig. 2(a), in-phase synchronization phenomena are
confirmed between the upper subcircuits and between the lowersubcircuits. The circuit parameters are chosen as α = 0.2, β11 =6.0, β12 = 3.0, β13 = 18.0, β21 = 1.0, β22 = 6.0, β23 = 3.0, β24 = 18.0,
γ1 = 3.0, γ21 = 1.0, γ22 = 3.0, δ=0.1, and ε = 0.01.

Figure 3: Attractors and synchronization phenomenain the circuit experiment.
IV. CONCLUSION
In this study, synchronization phenomena using two twodegrees-of-freedom chaotic circuits are investigated. We
confirmed that directly connecting two previously separated
circuits via a resistor induced synchronization between the twounderlining subcircuits farthest from the newly createdconnection, while the two closest subcircuits stayedasynchronous. As our future works, we will investigate
synchronization phenomena in the circuit experiment and wewill observe synchronization phenomena when connecting
circuits with an inductor.

In the case Fig. 2(b), in-phase synchronization phenomena are
confirmed only between the upper subcircuits. The circuit
parameters are changed from case Fig. 2(a) to β12 = 2.0, β13 =12.0, γ1 = 2.0.

(a) ω1 = ω2, ω3 = ω4.

REFERENCES(b) ω1 = ω2, ω3 ≠ ω4.

[1]

[2]

(c) ω1 ≠ ω2, ω3 = ω4.
[3]

Figure 2: Attractors and synchronization phenomenain the computer simulation.

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[4]

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