Suppression of Chaos Propagation in Ladder Chaotic
Circuits by Local Switching of Coupling Strength
Naoto Yonemoto, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
E-mail: {yonex34, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— Various phenomena have been observed in coupledchaotic circuits. One of the phenomena in chaotic synchronizationis the chaotic propagation. It has been investigated what kind offrustrations affect the chaotic propagation. These have been
investigated for both simple and complex networks. In this study,
the chaos propagation in the ladder-coupled chaotic circuits isstudied by computer simulation, using the voltage difference
between the adjacent circuits as the threshold for switching thecoupling strength.

Figure 1: Circuit model.

Keywords; Chaotic circuit; Chaos propagationI.
I NTRODUCTION
Various phenomena have been observed in coupled chaoticcircuits. The collapse of the chaotic synchronization [1] and theclustering phenomena [2] have attracted attention and havebeen studied. However, it is believed that there are still many
unexplained phenomena in chaotic synchronization. Therefore,
it is important to discover, model, and investigate suchphenomena in order to understand and utilize them.

Figure 2: System model.
The parameters are described as follows:
𝑖1 = √

One of the synchronous phenomena observed in coupledchaotic circuits is chaos propagation, which occurs when one
chaotic circuit is set to generate chaos and the others are set togenerate a three-period attractor. The propagation time and thepropagation rate have been considered by the coupling strength[3], changing the network structure, and the location where thechaos is generated [4]. In the past, a method to change thenetwork topology by probability [5] has been proposed as oneof the frustrations to the chaos synchronization.

𝑖2 =√𝐿1𝐶𝑉𝑦𝑛 ,
𝐿2𝐶,
𝐿1𝐿1,
𝐿2

𝛿 = 𝑟𝑑𝛼 = 𝑟√
1{ 𝛾 = 𝑅,𝛽=𝑣 = 𝑉𝑧𝑛

√𝐿1 𝐶,
𝐿2𝑡 = √𝐿1𝐶𝜏.

The normalized circuit equations are described as follows:𝑑𝑥𝑛

In this study, we utilize this method to investigate the effect ofswitching the coupling strength on the propagation of chaos in
ladder chaotic circuits. The chaos propagation is investigated inthree different ways of changing the coupling strength by meansof computer simulation.

𝑑𝜏𝑑𝑦 𝑛𝑑𝜏𝑑𝑦 𝑛= 𝛼𝑥𝑛 + 𝑧𝑛
= 𝑧𝑛 − 𝑓(𝑦𝑛 )= −𝑥𝑛 − 𝛽𝑦𝑛 − 𝛾𝑛,𝑛−1 (𝑧𝑛 − 𝑧𝑛−1)−𝛾𝑛,𝑛+1 (𝑧𝑛 − 𝑧𝑛+1 ){ (𝑛 = 1,2,∙∙∙, 𝑁 )

(1)

𝑑𝜏

II. SYSTEM M ODELThe chaotic circuit is shown in Fig. 1. This circuit consists ofa negative resistor, two inductors, a capacitor and dualdirectional diodes. The system model is shown in Fig. 2. In thissystem, the circuit in the one end of the system generates
chaotic attractor and the other circuits generate three-periodic
attractors. Ten chaotic circuits next to each other are coupled
together with resistors.

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𝐶𝑉𝑥 ,𝐿1 𝑛183

where
𝛾1,0 = 𝛾𝑁,𝑁+1 = 0𝛿11()𝑓 𝑦𝑛 = (|𝑦𝑛 + | − |𝑦𝑛 − |).
2𝛿𝛿For the computer simulations, we calculate Eq. (1) using thefourth-order Runge-Kutta method with step size ℎ = 0.005.
We set the parameters of this circuit model as follows; 𝛼𝑐 =0.460, 𝛼𝑝 = 0.413, 𝛽 = 3.0 and 𝛿 = 470.0.

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III.

RESULTS(2) Switching each coupling
When the coupling strength is switched for each coupling,propagation can be classified into four patterns as shown in theFig. 6.

A. Fixed the couplings strength
Chaos propagation is researched in ladder chaotic circuits,
when the coupling strength is fixed at 𝛾 = 0.002, 0.004,0.006, and 0.008. The horizontal axis is the value of 𝑥𝑛
representing the current flowing in 𝐿1, and the vertical axis
shows the value of 𝑧𝑛 representing the voltage applied to C.

Figure 6: Four patterns of chaos propagation.
When the first threshold width is 0 ≤ |𝑧𝑛 − 𝑧𝑛+1| ≤ 0.2, thechaos propagation can be suppressed as shown in the Fig. 6(a)(c). As the second threshold width increases to 0.1, 0.2 and 0.3,
the number of circuits for chaotic propagation becomes larger
to 4th, 5th and 6th as well as when the first threshold widths are
0.1 and 0.3. However, when threshold widths are 0.1, 0.1 and
0.4 from 1st to 3rd step in the algorithm, the number of circuits
propagated chaos becomes the largest as shown in Fig. 6(d).

Figure 3: Chaos propagation under fixed coupling strength
The propagation state at almost the same time is shown inFig. 3. As the coupling strength increases (𝛾 = 0.004, 0.006,and 0.008), the number of circuits for chaotic propagation
becomes larger. In particular, no propagation was observed incase of a coupling strength 𝛾 = 0.002.
B. Switching coupling strength
The chaos propagation is changed by switching the couplingstrength with the difference of 𝑧𝑛 and 𝑧𝑛+1 as a threshold. The
first method is to switch all the couplings simultaneous, and thesecond method is to switch each coupling at a certain. How toswitch the coupling strength is shown in Fig. 4.

IV.
CONCLUSION
In this study, chaos propagation in a ladder-coupled chaotic
circuit was studied when the coupling strength was switched.
The propagation was regulated in three ways: fixed couplingstrength, simultaneous switching of all the couplings, andswitching at each coupling. The results show that thepropagation of chaos can be controlled locally, and even in thecase of switching at each coupling, the propagation can besuppressed as well as in the method of switching all thecouplings at the same time by considering the decision criteria.
In the future, we would like to focus on the propagation process
such as recovery from chaos and indirect propagation.
REFERENCES

Figure 4: The algorithm for controlling chaotic propagation bycoupling strength (The threshold widths are all 0.2).
(1) Switching all couplings simultaneous
When the voltage difference at one coupling exceeds thethreshold, all couplings switch. In Fig. 5, the maximum numberof circuits propagated chaos is fifth, when the third threshold
width is 0.1. For the other threshold widths, that is fourth. In afew seconds, the coupling strength decreased from 0.008 to0.002 and then remained at 0.002.

[1]

Masahiro WADA, Yoshifumi NISHIO and Akio USHIDA, "Analysis ofBifurcation Phenomena in Two Chaotic Circuits Coupled by an Inductor",
IEICE Transactions on Fundamentals, vol. E80-A, no. 5, pp. 869-875,May 1997.

[2]

Yuji TAKAMARU, Yoko UWATE, Thomas OTT and YoshifumiNISHIO, "Dependence of Clustering Patterns on Density of ChaoticCircuits in Networks", RISP Journal of Signal Processing, vol. 17, no. 4,
pp. 103-106, Jul. 2013.

[3]

Yoko UWATE and Yoshifumi NISHIO, "Chaos Propagation in a Ring of
Coupled Circuits Generating Chaotic and Three-Periodic Attractors",
Proceedings of IEEE Asia Pacific Conference on Circuits and Systems(APCCAS'12), pp. 643-646, Dec. 2012.

[4]

Takahiro CHIKAZAWA, Yoko UWATE and Yoshifumi NISHIO,"Spread of Chaotic Behavior in Scale-Free and Random Networks",
Proceedings of Asia Pacific Conference on Postgraduate Research in
Microelectronics and Electronics (PrimeAsia'17), pp. 21-24, Oct. 2017.
Toshiaki NISHIUMI, Yoko UWATE and Yoshifumi NISHIO,"Synchronization Phenomena of Chaotic Circuits with StochasticallyChanged Network Topology", Proceedings of International Symposiumon Nonlinear Theory and its Applications (NOLTA'14), pp. 811-814, Sep.
2014.

[5]Figure 5: Chaos propagation after a certain time.
(Same time as Fig. 3)

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