2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

Investigation of Coupled Chaotic Circuits with Ladder and Ring Structures
Katsuya Nakabai, Katsuki Nakashima, Yoko Uwate and Nishio Yoshifumi†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
In this study, we investigate synchronization phenomena inladder-coupled chaotic circuits including ring structures modeling sapling of plant. We focus on the effect of the positionof the ring structures in the network to the synchronization of
the whole network. In our proposed network model, chaotic
circuits are coupled by resistors. We set the parameters of
the circuits to generate periodic solutions or chaotic solutions.
Computer simulations confirm that the various synchronization phenomena occur in different situations.

circuit.
In this study, we investigate the stability in our proposedmodel by ladder and rings modeling sapling of the plant bysynchronization phenomena. In our proposed network model,
chaotic circuits are coupled by resistors. These chaotic circuits are little difference of each other by the bifurcation parameter. We set the parameters of the circuits to generate
periodic solutions or chaotic solutions. The number of rings
changes to each model. And we observe the synchronization
phenomena in different network models.
2. System model

1. Introduction
Synchronization in a network is one of the interesting topics in scientific disciplines. Investigation of this is importantwork in order to clarify nonlinear phenomena in the natural
world. It is observed over the various fields, for example, engineering, biology, sociology and so on [1]. Incidentally, thenetworks have different topology and characters in each even
if they are the similar type [2], [3]. Therefore, the investigation of dynamics in different of the network is significant andresearch has proceeded to analysis the dynamics of synchronization on each topology.
On the other hand, synchronization of coupled chaotic
system is interesting models to describe various higherdimensional nonlinear phenomena [4], [5]. Recently, studying chaotic phenomena by using coupled chaotic circuits increases. On the chaotic circuit with simple elements, circuit
experiment and computer simulation are suitable for studying
in terms of that experiment time is short and repeatability ofexperiments is high. Networks by using coupled chaotic circuits can be modeled natural world or social networks. Moreover, it is important for future engineering to investigate nonlinear phenomena like chaotic synchronization.
In modern society, agenda of vegetation exist about environment or food. The study of plant bioelectric potential
has attracted rising attention in recently from the viewpointof plant growth [6]. Bioelectric potential is known that itchanges depending on surrounding environment. Therefore,
we deem the bioelectric potential to the voltage phase difference of chaotic circuit and we model the plants by chaotic

In this study, we use chaotic circuits. The model of chaotic
circuit is shown in Fig. 1. This circuit consists of a negative resistor, two inductors, a capacitor and dual-directional
diodes. This chaotic circuit is called Nishio-Inaba circuit [7].

Figure 1: Chaotic circuit.

We propose ladder-coupled chaotic circuits including ringstructures. Figure 2 shows examples of the network models.
In this case of Fig. 2(a), we set five chaotic circuits in ladderstructure and one ring structure coupling to CC1 by resisters.
Likewise, we propose three network models that coupled ring
structure to CC1, CC2 or CC3. We set the parameters of thesystem as β = 3.0 and δ = 470.0. Next, we change thecondition in terms of the parameter α. α denotes degree ofchaos. The network models are divided into two types by thevalue of α. One pattern is to unify the value of all α (unifiedsystem) like shown in Fig. 2(a), and the other is to change the550

2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

value of all α (ununified system) like shown in Fig. 2(b) . In
case of Fig. 2(a), the value of α is unified 0.412. And in caseof Fig. 2(b), the value of α is not unified. The parameters α
are set from 0.411 to 0.415 with step size 0.001 from CC1 toCC5. The α of CC6 and CC7 are added 0.001 from the α ofcoupled circuit.
Subsequently, we augment the number of rings in the caseof one ring and three rings. Likewise in ununified systems,
the α of CC6, CC7, CC8, CC9, CC10 and CC11 are added
0.001 from the α of coupled circuit. We show the models
from Fig. 2(c) and Fig. 2(d). Figure 2(c) is unified systems
with three rings and Fig. 2(d) is ununified systems with threerings.
By changing the variables and parameters,√√C
L1 C
V x n , i2 =V yn , v = V zni1 =
L1L2
√
√
CL1
L1 C
α=r, β=, δ = rd,LLL
122√ γ = 1 , t = L C τ.
1 2R(b) Ununified system with one ring.

(c) Unified system with three rings.

(1)(d) Ununified system with three rings.

R is the resistor that couples each chaotic circuit.
The normalized circuit equations of the systems are given
as follows: dxi

= αxi + zi


dτ



dyi


= zi − f (y)
dτN∑
dzi


=−x−βy−γij (zi − zj )ii dτi,j=1
(i, j = 1, 2, · · ·, N ).

Figure 2: Network models.

zizixixi(2)(a) Three periodic attractor.

(b) Chaotic attractor.

Figure 3: Attractors.

3. Simulation results
In Eq. (2), N is the number of coupled chaotic circuits andγ is the coupling strength as γ = 0.008 in this study. We set
the parameters of the circuits to generate periodic solutions or
chaotic solutions. When we set the parameter as α = 0.411
to α = 0.415, the circuits generate three-periodic solutions.
And when the parameter is more than α = 0.416, the circuits
generate chaotic solutions. The attractors are shown in Fig. 3.

We investigate synchronization phenomena in laddercoupled chaotic circuits including ring structures. In thistime, we use attractors and Lissajous figures to research synchronization. Lissajous figures show the phase difference
between each chaotic circuit. When the phase difference isclose to 0, the Lissajous figures become like Fig. 4(a). In
this case, the system causes phase synchronization (perfectsynchronization) and this condition is stable. And when thephase differential increases, the Lissajous figures approach
the shape of Fig. 4(b). This shows that the system is asynchronous and this condition is unstable. Moreover, the phasedifference is to some extent, the Lissajous figure become likeFig. 4(c). In this case, the system causes chaotic synchronization. We compare the synchronization with these threepatterns.

(a) Unified system with one ring.

551

2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019zizizjzizjzj(a) Synchronization. (b) Asynchronous. (c) Chaotic synchronization.

Figure 4: Lissajous figures.
Figure 6: The number of chaotic attractors in case of three rings.

First, we simulate the circuit model with one ring, and theresult is shown in Fig. 5 and Tables 1 and 2. In this case, despite of that the circuits are set three periodic attractors, theboth unified system and ununified system cause chaotic synchronization and observe chaotic attractors by coupling ringto CC1 or CC2. On the other hand, when we make the ring
structure with CC3, all chaotic circuits are generated threeperiodic attractors and cause perfect synchronization in bothcases.
Tables 3 and 4 show the percentage of each synchronizedstate in these models. We obtain the result that network modelwith one ring in ununified system is no asynchronous point
and it indicates a stable state.

Table 1: Coupling one ring in unified system.

Table 2: Coupling one ring in ununified system.

Next, we simulate the model with three rings, and the resultis shown in Fig. 6 and Tables 5 and 6. In this case, on the bothsystems, the positions of bridge on the ladder and ring tend tobe asynchronous. However, in the case of coupling to CC1,CC3 and CC5, both systems are occured synchronization onthe bridge. Furthermore, in the case of coupling rings to CC2,CC3 and CC4, synchronization is difference in both systems.
Tables 7 and 8 show the percentage of each synchronization state in these models. In case of coupling to CC1, CC2
and CC3, and in case of coupling to CC2, CC3 and CC4,
we obtain the result that the percentage of asynchronous decreases. Though, in case of coupling to CC1, CC2 and CC4,
the percentage of asynchronous increases.

Table 3: Percentage of synchronization state in unified systemwith one ring.

From these results, we consider that in ununified system,
the percentage of asynchronous tend to decrease.

Table 4: Percentage of synchronization state in ununified systemwith one ring.

Figure 5: The number of chaotic attractors in case of one ring.

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2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

Table 5: Coupling three rings in unified system.

Table 6: Coupling three rings in ununified system.

4. Conclusions
In this study, we have investigated synchronization phenomena in ladder-coupled chaotic circuits including ring
structures modeling sapling of the plant. The models are twotypes named unified system and ununified system. We have
set the parameters of the circuits to generate periodic solutions or chaotic solutions. And we have proposed that wechange the number of ring structure. We set the parametersof the circuits to generate periodic solutions or chaotic solutions. By the computer simulations, we have confirmed thatwhen we changed the number of rings, we can obtain various
synchronization.
In case of one ring, the large difference of dynamics does
not exist, however the positions of synchronization are different. In case of three rings, synchronization phenomenon iseasier to occur in ununified system than unified system. Furthermore, we have considered that in ununified system, thepercentage of asynchronous tend to decrease.
In our future works, we investigate synchronization phenomena by using α as the random value in ununified system.
Furthermore, we carry out circuit experiment and confirm thereproducibility.
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Table 7: Percentage of synchronization state in unified systemwith three rings.

[3] D. Malagarriga, A. E. P. Villa, J. Garca-Ojalvo and A.
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Table 8: Percentage of synchronization state in ununified systemwith three rings.

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