IEEE Workshop on Nonlinear Circuit Networks
December 15-16, 2017

Propagation of Chaotic and Periodic Solutionsin Ten Coupled Chaotic Circuitswith Different Parameters
Katsuya Nakabai, Katsuki Nakashima, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email:{nakabai,nakashima,uwate,nishio}@ee.tokushima-u.ac.jp

Abstract—In this study, we investigate the state of propagationof chaotic solution and periodic solution. The state of propagation
is diversified with the types of topology. We propose the networkmodel that ten chaotic circuits are coupled on the ladderstructure. We use chaotic circuits which are coupled by resistors.

I. I NTRODUCTION
In reality network, various things are connected such as:
cities, traffic and human beings are connected each other.
These connects are important factor in the modern. Incidentally, these things have little different characters in each even if
they are the same type. In this situation, such a differentiationof characters are sometimes propagated to other things. There
are some cases that this propagation affects a great deal in thenetwork. For example, the traffic jam of the transportation network, the pandemic outbreak of viral infection, deteriorationof human relations and so on are the bad influences. Therefore,
the investigation of the propagation of these characters isimportant to block the effects. Then, in the circuit network,
we investigate the state of similar phenomenon by propagationof chaotic solution and periodic solution. The behavior of
the network affects various phenomena under some different
situations.
In this study, we investigate the propagation of chaotic
and periodic solution in coupled chaotic circuits. We propose the network that we couple ten chaotic circuits on theladder structure. These chaotic circuits are little differenceof each other by the bifurcation parameter. In this model,
five circuits generate three periodic solutions and other five
circuits generate chaotic solutions. This structure is coupled byresistor, and we change the value of resistors, in other word,changing the coupling strength. We observe how chaotic andperiodic solution propagate in this ladder structure by changing
coupling strength.

Fig. 1.

Chaotic circuit.

The circuit equations of this circuit are described as follows:di
L1 = v + ridt
diL2 = v − vddtdv C dt = −i1 − i2 .

(1)

The characteristic of nonlinear resistance is described asfollows:()rdV
Vvd =(2)i2 +− i2 −.
2rdrd

II. S YSTEM MODEL
A. Circuit model
The chaotic circuit is shown in Fig. 1. This circuit consistsof a negative resistor, two inductors, a capacitor and dualdirectional diodes. This chaotic circuit is called Nishio-Inabacircuit.

- 108 -By changing the variables and parameters,√√
CL1 CVx,
i=V yn , v = V z ni=n21LL
12√
√CL1L1 C
α=r, β=, δ = rd,
L1L2L2√ γ = 1 , t = L C τ.
1 2R
R is the resistor that couples each chaotic circuit.
The normalized circuit equations are given as follows:(3)III. S IMULATION RESULTSdxi
= αxi + zi dτ dyi= zi − f (yi )dτdzi
= −xi − βyi − γ(zi − zi+1 ) dτ(i = 1, 2, · · ·, N -1).

(4)

In Eq. (4), N is the number of coupled chaotic circuits andγ is the coupling strength. f (yi ) is described as follows:
1f (yi ) =2(11yi +− yi −δδ).

We investigate the propagation of chaotic solutions andthree periodic solutions in ladder coupled chaotic circuitsby changing the coupling strength. All coupling strength
is changed evenly. We define that when chaotic attractors
change three periodic attractors, three periodic attractors arepropagated. When three periodic attractors change chaoticattractors, chaotic attractors are propagated. First, we couple
the chaotic circuits from CC1 to CC10. We set the coupllingstrength as γ = 0.001. In this case, because coupling strengthis weak, propagation have not occured.

(5)

z1

In Eq. (4), N is the number of coupled chaotic circuits andγ is the coupling strength.

z2x1z

In our system, each chaotic circuit is coupled by one resistor
R. Figure 2 shows the system model. We couple ten chaotic
circuits in ladder structure. In this study, we set the parametersof the system as β = 3.0 and δ = 470.0. The parameters α
are set from 0.411 to 0.420 with step size 0.001 from CC1to CC10. Figure 3 shows the attractors from CC1 to CC5.
We define these attractors as three periodic solutions. AndFig. 4 shows the attractor from CC6 to CC10. We define theseattractors as chaotic solution.

x2z

6B. System modelz3x3z

7xx4x5z

9x7z5z8x6z410x8x910Fig. 5. Attractors (γ = 0.001).

Then, we set the coupling strength as γ = 0.0055, and threeperiodic attractor is propagated CC6 in Fig. 6 .

z1z2x1z6

z3x2x3z8z7x6z4x4x5z10z9x8x7z5x9x10Fig. 6. Attractors (γ = 0.0055).
System model.

Fig. 2.

z1z2x1

z3x2

z4x3

z5x4x5Furthermore, when we increase the coupling strength, threeperiodic attractors are propagated the chaotic attractors gradually one by one. Figure 7 shows that three periodic attracters
are propagated to CC7. Figure 8 shows that three periocidattractors are propagated to CC8. Figure 9 shows that threeperiodic attractors are propagated to CC9. Finally, Fig. 10shows three periodic attractors are propagated to all coupledchaotic circuits.

Three periodic attractots.

Fig. 3.

z1zz

6z

7z8z

910z2x1

6xxFig. 4.

Chaotic attractors.

78x

9x2z7z6xz3z4x3z8z5x4x5z10z9x

10x6x7x8x9Fig. 7. Attractors (γ = 0.0064).

- 109 -x10

z1z2x1z3x2z7z6x3z8x7x6z4z5x4z10z9x8x5x9x10

Fig. 8. Attractors (γ = 0.0068).

z1z2x1z6z3x2x3z8z7x6z4x4Fig. 12. Relationship between the number ofthree periodic attractors and coupling strength .

x5z10z9x8x7z5x9

IV. C ONCLUSIONSx10Fig. 9. Attractors (γ = 0.007).

z1z2x1zz3x2z6x3z7xx4x5z9x7z5z8x6z410x8x910Fig. 10. Attractors (γ = 0.01).

However, when we set the coupling strength as γ = 0.011,
chaotic attractors are propagated all three periodic attractors.
Differently from the propagation of three periodic attractors,
chaotic attractors propagate all three periodic attractors atonce. Then, even though we increase coupling strength more
over, these attractors do not change. Therefore, we obtain theresult that when we set the coupling strength as more thanγ = 0.011, chaotic attractors are propagated at all times.

z1z2x1zz3x2z

66x3z

7xz47x4z

8xz5z

9x8x5
10x9x10Fig. 11. Attractors (γ = 0.011).

We investigate how the relationship between propagationof three periodic attractors and coupling strength. Figure 12shows the graph of this result. The vertical line shows thenumber of three periodic attractors. The horizontal line shows
the coupling strength.

In this study, we have proposed the system model thatten chaotic circuits with different parameters are coupled byresisters in ladder structure. The parameters α are set from0.411 to 0.420 with step size 0.001 from CC1 to CC10. We
have investigated the propagation of chaotic solutions andperiodic solutions by changing coupling strength γ. By thecomputer simulations, we confilm that the state of propagationof these attractors is altered when we change the couplingstrength. As a result, in ladder ten coupled chaotic circuits,
when we increase the coupling strength from γ = 0.0055 to0.01, three periodic attractors are propagated chaotic attractors
gradually one by one. Also, when we set the coupling strength
as γ = 0.011, chaotic attractors are propagated all threeperiodic attractors at once.
In the future works, we will develop this model to couple some ladders and investigate the propagation of chaotic
attractors and three periodic attractors.

R EFERENCES[1] Y. Nishio, N. Inaba, S. Mori and T. Saito, “Rigorous Analyses ofWindows in a Symmetric Circui“ IEEE Transactions on Circuits andSystems, vol. 37, no. 4, pp. 473-487, Apr. 1990.
[2] R. L. V. Taylor,“Attractors: Nonstrange to Chaotic“ Society for Industrialand Applied Mathematics, Undergraduate Research Online, pp. 72-80,
21 6 2011.
[3] Y. Uwate and Y. Nishio,“Chaos Propagation in a Ring of Coupled Circuits Generating Chaotic and Three-Periodic Attractor“ Proceedings ofIEEE Asia Pacific Conference on Circuits and Systems (APCCAS’12),
pp. 643-646, Dec. 2012.
[4] K. Ago, Y. Uwate and Y. Nishio,“Influence of Local Bridge on a Complex Network of Coupled Chaotic Circuit“ Proceedings of International
Symposium on Nonlinear Theory and its Applications (NOLTA’14), pp.
731-734, Sep. 2014.
[5] T. Chakazawa, Y. Uwate and Y. Nishio,“Investigation of Spreading
Chaotic Behavior in Coupled Chaotic Circuit Networks with VariousFeature“ Proceedings of RISP International Workshop on Nonlinear
Circuits, Communications and Signal Processing (NCSP’17), pp. 337340, Feb. 2017.

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