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

Abstract

2. System model

In this study, we investigate the propagation of chaotic soThe chaotic circuit is shown in Fig. 1. This circuit consists
lutions and periodic solutions in ladder coupled chaotic cir- of a negative resistor, two inductors, a capacitor and dualcuits. The state of propagation is different for each type of directional diodes. This chaotic circuit is called Nishio-Inaba
circuit networks. We change the parameters of chaotic cir- circuit.
cuits by regularity in the ladder structure. We use chaotic
circuits which are coupled by resistors.

1. Introduction
In reality network, various things are connected such ascities, traffic and human beings. These connects are important factor in the modern. Incidentally, these things have littledifferent characters in each even if they are the same type.
In this situation, such a differentiation of characters is someFigure 1: Chaotic circuit
times propagated to other things. There are some cases that
this propagation affects a great deal in the network. For example, the traffic jam of the transportation network, the pandemic outbreak of viral infection, deterioration of human reThe circuit equations of this circuit are described as follations and so on are the bad influences. Therefore, the inveslows:
tigation of the propagation of these characters is important toblock the effects. Then, in the circuit network, we investigatedi
L1 = v + ri
the state of similar phenomenon by propagation of chaotic
dt
solution and periodic solution. The behavior of the network
diaffects various phenomena under some different situations.
L2 = v − vd(1)dtIn this study, we investigate the propagation of chaotic and
dv
periodic solution in coupled chaotic circuits. These chaotic C dt = −i1 − i2 .
circuits have little difference of each other by the regularlychanged of parameter. In this model, five circuits generate three periodic solutions and other five circuits generatechaotic solutions. Ten chaotic circuits are coupled by reThe characteristic of nonlinear resistance is described assistor, and we change the value of resistors, in other word, follows:changing the coupling strength. We observe how chaotic and()periodic solution propagate in this ladder structure by changrdVV
ing the coupling strength. Moreover, we change the number
vd =i2 +− i2 −.
(2)
2rdrdof chaotic circuits, and we confirm the state of propagation.

- 549 -dxn
= αxn + zn

dτ

dyn

 dτ = zn − f (yn )CCn
 dzn = −xn − βyn

dτ


− γ(zn − zn−1 ) − γ(zn − zn+1 )(n = 2, 3, · · ·, 9).
dx10


= αx10 + z10

dτ

 dy
10
= z10 − f (y10 )
CC10
dτ


 dz10= −x10 − βy10 − γ(z9 − z10 ).
dτ

By changing the variables and parameters:√√
CL1 C
i1 =V x n , i2 =V yn , v = V zn


LL12
√√
CL1L1 Cα=r,
β=,
δ=r,d
LLL
122
√t = L1 C2 τ.

(3)The normalized circuit equations are given as follows:

dx1


= αx1 + z1

dτ

 dy
1= z1 − f (y1 )
dτ



dz1


= −x1 − βy1 .
dτ(4)

(8)

(9)

In this study, we set the parameters of the system as β =3.0 and δ = 470.0. The parameters α are set from 0.411 to0.420 with step size 0.001 from CC1 to CC10. Figure 3 shows
the attractors from CC1 to CC5. We define these attractors as
three periodic solutions. Figure 4 shows the attractor fromCC6 to CC10. We define these attractors as chaotic solution.

f (yi ) is described as follows:
1f (yi ) =2(11yi +− yi −δδ).

(5)Figure 2: System modelIn 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.
By changing the variables and parameters:√√
CL1 C
V x n , i2 =V yn , v = V zni1 =LL

12
√
√
C
L1L1 C
α=r,
β=,
δ
=r,d

LL
L
122

√
1

γ = , t = L1 C2 τ.
Rz1

z2x1x2z4x3

z5x4x5Figure 3: Three periodic attractots

(6)

zz

6In Eq. (6), γ is the coupling strength. R is the resistor thatcouples each chaotic circuit.
The normalized circuit equations are given as follows:dx1

= αx1 + z1

dτ

 dy
1= z1 − f (y1 )
CC1
dτ

dz1


= −x1 − βy1 − γ(z1 − z2 ).
dτz3

z

7x6z

8x

7x8

Figure 4: Chaotic attractors(7)- 550 -z

910x

9x10

3. Simulation results

z1

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 chaotic
attractors, chaotic attractors are propagated. First, we couple
the chaotic circuits from CC1 to CC10. The result is shownin Fig. 5. We set the coupling strength as γ = 0.001. In thiscase, because coupling strength is weak, propagation has notoccured.

z1z2x1zz3x2z

6z4x3z7z5x4z8z6x1

z7

z3x2z8x7x6z4x3

z9x8

z5x4

z10x9x5x10Figure 7: Attractors (γ = 0.0064)

z1z6x5z2x1z7x6z9z2z3x2z8x3z9x8x710z4z5x4z10x9x5x10

Figure 8: Attractors (γ = 0.0068)

xx6x7x8x910Figure 5: Attractors (γ = 0.001)

When we set the coupling strength as γ = 0.0055, threeperiodic attractor is propagated CC6 as shown in Fig. 6. Furthermore, when we increase the coupling strength, three periodic attractors are propagated the chaotic attractors gradually
one by one. Figure 7 shows that three periodic attractors arepropagated to CC7. Figure 8 shows that three periocid attractors are propagated to CC8. Figure 9 shows that three periodic attractors are propagated to CC9. Finally, Fig. 10 shows
three periodic attractors are propagated to all coupled chaotic
circuits.

z1

z2

z3

z4

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 the coupling strength
moreover, these attractors do not change. Therefore, we obtain the result that when we set the coupling strength as morethan γ = 0.011, chaotic attractors are propagated at all times.
Finally, we change the number of coupled chaotic circuits
from N = 7 to N = 10. The parameters α are set with step
size 0.001 from CC1 to CC7, from CC1 to CC8, from CC1to CC9 and from CC1 to CC8. Likewise, the parameter αof CC1 is 0.411. We focus on the number of three periodic
attractors. We investigate how the relationship between propagation of three periodic attractors and coupling strength ischanged depending on the number of coupled chaotic circuits.

z5z1z6x1z7x2z8x3z9x4z10x7x8x9z3z4z5x5
z6x6z2x1

z7x2

z8x3

z9x4

z10x5x10x6

Figure 6: Attractors (γ = 0.0055)

x7x8x9Figure 9: Attractors (γ = 0.007)- 551 -x10

z1z2x1zz3x2z6x3
z7xx4x5z9x710x84. Conclusions

z5z8x6z4x910

Figure 10: Attractors (γ = 0.01)

z1z2x1zz3x2z66x3z7xz47x4z8xz5x8x5z

910x9x10

Figure 11: Attractors (γ = 0.011)

Figure 12 shows the graph of this result. The vertical line
shows the number of three periodic attractors. The horizontal
line shows the coupling strength. We ascertain that when wereduce the number of coupled chaotic circuits, three periodic
attractors propagate by weaker coupling strength. Similarly,
chaotic attractors propagate all at once by weaker couplingstrength.

In this study, we have proposed the system model that ten
chaotic circuits with different parameters are coupled by resisters 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 and periodic solutions by changing the coupling strength γ. By thecomputer simulations, we confirm that the state of propagation of 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 couplingstrength as γ = 0.011, chaotic attractors are propagated all
three periodic attractors at once. Next, we change the number of coupled chaotic circuits, and we investigate the propagation likewise. When we decrease the number of coupledchaotic circuits, propagation of chaotic attractors and threeperiodic attractors is occurred by weaker coupling strength.
In the future works, we will develop this model to couple
some ladders and investigate the propagation of chaotic attractors and three periodic attractors.

References[1] 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[2] T. Chikazawa, Y. Uwate and Y. Nishio,“Investigation ofSpreading Chaotic Behavior in Coupled Chaotic Circuit
Networks with Various Feature” Proceedings of RISPInternational Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP’17), pp. 337340, Feb. 2017.

[3] Y. Nishio, N. Inaba, S. Mori and T. Saito, “RigorousAnalyses of Windows in a Symmetric Circui” IEEE
Transactions on Circuits and Systems, vol. 37, no. 4,
pp. 473-487, Apr. 1990.
Figure 12: Relationship between the number of three periodic
attractors and the coupling strength by the value of N .

[4] R. L. V. Taylor, “Attractors: Nonstrange to Chaotic” Society for Industrial and Applied Mathematics, Undergraduate Research Online, pp. 72-80, 21 6 2011.

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