Synchronization Phenomenaof Two Coupled Chaotic Circuits
Using Stochastic Coupling
Takahiro Hattori, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
E-mail: {hattori, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— In this research, synchronization phenomena instochastic coupling are investigated using two chaotic circuits. As
a result, new synchronization patterns are observed by switching
resistors with reference to the positive value and the negativevalue of the chaotic circuit.
Keywords; Synchronization; Stochastic; Chaotic circuit

I.

INTRODUCTION(a) Chaotic circuit.

Synchronization phenomena are phenomena in which two
or more oscillating bodies behave in the same way by
influencing each other and can be observed in many situations
in everyday life [1]. The observation and analysis ofphenomena using electrical circuits are also one of the mosteffective tools because they can be applied to real physicalsystems. Investigating synchronization phenomena in chaotic
circuits is very important, therefore, for future engineering
applications. Synchronization phenomena in chaotic circuits
are mostly investigated in statically coupled chaotic circuit
networks, and fewer in dynamically varied coupling chaotic
circuit networks [2]. For these reasons, in this research,
synchronization phenomena of chaotic circuits in dynamicallyvarying resistors coupled chaotic circuits (stochastic coupling)
are investigated [3].
II.

(b) Attractor(γ = 0.5).

(c) Attractor(γ = 0.65).

Figure 1. Circuit model.

SYSTEM MODEL

The chaotic circuit used in this research is shown in Fig.
1(a). The attractor of this chaotic circuit is shown in Fig. 1(b)and (c). The chaotic circuit is called as the Mori-Shinriki circuitand consists of a negative resistor, an inductor, two capacitorsand dual-directional diodes. In this research, two Mori-Shinriki
chaotic circuits are used to connect two resistors Ra and Rb. Raand Rb are then varied stochastically to construct a systemmodel. The system model is shown in Fig. 2. The circuit
equation is described as Eq. (1).

Figure 2. System model.

Normalized resistor R is set as the coupled strength δ. The
normalized equation of this circuit is given as follows:

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The nonlinear function f() corresponds to the i-v
characteristics of the nonlinear resistor consisting of thediodes and are described as follows:III.

(a) Coupling step time rates of δa and δb for switching.

SETTING THE COUPLED STRENGTH

The normalized coupling strength of Ra and Rb is set to δaand δb. In this research, the coupling strength is switchedbetween the in-phase state for δa and the asynchronous state forδb.

(b) The overall δa and δb coupling time rate for switching.

(a) In-phase state (δa = 0.22). (b) Asynchronous state (δb = 0.01).
Figure 3. The coupling strength.

The positive value and the negative value of the waveformof y in the Mori-Shinriki circuit (γ = 0.65) are also used as
standards for switching the coupling strength and are shown in
Fig. 4. Switching is performed with δa for time the positivevalue and δb for time the negative value.

Figure 4. The standard waveforms for switching the coupling strength.

IV.

(c) Phase differences.
Figure 5. Simulation results.

V. CONCLUSIONSThis research investigated synchronization phenomena ofchaotic circuits in coupled chaotic circuits with dynamicallyvarying resistors (stochastic coupling). The results ofdynamically switching the coupling strength with reference tothe positive value and the negative value of the chaotic circuit
showed new synchronization phenomena. As a future work, wewould like to carry out circuit experiments and compare theresults with numerical simulations.

RESULTSIn this research, the time interval is set to (τ = 400,000, step
= 0.005τ) using numerical simulation. The synchronization rateis investigated by considering phase differences within 20
degrees to be in-phase states. Results are shown in Fig. 5. It isshowed coupling step time rates of δa and δb for switching,classified by 5,000 in Fig. 5(a). It is also showed the overall δaand δb coupling time rate for switching in Fig. 5(b), and phasedifferences in Fig. 5(c). These show that in-phase andasynchronous states are repeated. The synchronization rate is68.9 [%]. It is also considered that the coupling step times of δamore than 20,000 are relatively in-phase state.

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[1]

[2]

[3]

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