6th International Forum on Advanced Technologies
March 9th-10 th, 2020,Tokushima, Japan

Switching Phenomena of Synchronization Phenomena
in Ladder-Coupled Chaotic Network Including Ring Structure
Katsuya Nakabai , Yoko Uwate and Yoshifumi Nishio
Tokushima University, Tokushima 770-8506, Japan
E-mail:{ nakabai, uwate, nishio}@ee.tokushima-u.ac.jp

1. Introduction
Synchronization in the network is one of the most interesting fields from the scientific points. Investigation ofthe synchronization is an important research for clarifying the nonlinear phenomenon in the natural world,which has been observed in various fields such as engineering, biology, and sociology. The network also hascharacteristics with different topologies. Therefore, it is important to investigate the dynamics due to thedifference in network structure, and research to analyze each topology is underway [1].
On the other hand, synchronization in coupled chaotic systems is a suitable model for describing various
high-dimensional nonlinear phenomena. Especially, in recent years, many studies of chaotic phenomena using
coupled chaotic circuits have been conducted. Circuit experiments and computer simulations of chaotic circuitswith simple configurations are considered to be suitable due to their high reproducibility. Investigation of
nonlinear phenomena related to chaotic phenomena will be an important issue in future engineering. Networks
using chaotic circuits are expected to be applied to modeling of the natural world and social networks [2].
Previously, our research group investigated the synchronization phenomena observed in ladder coupledsystems including ring structures. We confirmed perfect synchronization and chaotic synchronization indifferent network topologies [3]. In this study, we investigate the synchronization phenomena of a larger scalethan the previous network model. We particularly focus on the bridge positions that connect the ladder andring structures. Each coupled chaotic circuit sets parameters for generating a chaotic solution or a periodicsolution. By computer simulation, we investigate the synchronization phenomenon in each part by changing
the bifurcation parameter in the chaotic network. We also observe the switching effect of synchronous stability
in each part of the network.
2. Experimental setup
In this study, we use chaotic circuits. The model of chaotic circuit is shown in Fig. 1. This circuit consists of anegative resistor, two inductors, a capacitor and dual-directional diodes. This chaotic circuit is called
Nishio-Inaba circuit [4]. The normalized circuit equations are given as follows:(1)

Fig. 1. Chaotic circuit.

6th International Forum on Advanced Technologies
March 9th-10 th, 2020,Tokushima, Japan

In this study, we propose a network model withladder-coupled chaotic network including ring structure. Eachchaotic circuit is coupled by resistor. Figure 2 shows theproposed network model.
We set the parameters that the ladder position composed of
CC1 to CC5 generates a chaotic solution, and the five ring
positions composed of CC6 to CC20 generate periodicsolutions.
The normalized circuit equations of the systems are given as
follows:
Fig. 2. Network model.

(2)

In Eq. (2), N means the number of coupling circuits, γ is the coupling strength between circuits, and α isparameter indicating nonlinear degree. In this study, we set the parameters of the system as β = 3.0 and δ= 470.0. For the parameter α , the ladder part is set as α = 0.430 and the ring part is set as α = 0.412.
The coupling strength is defined as a bifurcation parameter. The coupling strength at the ladder and ring
positions is set as γ = 0.2. In this research, to investigation based on the combination of two topologicaltypes of ladder structure and ring structure, we simulate the synchronization by changing the positions ofbridge that couple between the two structures (for example,
the resistance that couples CC1 and CC6).
3. Results and discussion
We investigate the synchronization state by the computer
simulation. In this study, we simulate the synchronization of
chaotic network by dividing the coupling strength of thebridge part of γ = 0.01.
Voltage difference waveform between each circuit is used
as method for confirming the synchronization state. Whenthe voltage difference is small, it is considered that eachcircuit is performing the same output and synchronization isconsidered o occur. When the voltage difference is large, it
is considered that each circuit outputs a different output andthe state is regarded as asynchronous.

(a) Time advanced τ=50000-100000 in rings.

6th International Forum on Advanced Technologies
March 9th-10 th, 2020,Tokushima, Japan

Figure 3 shows the results when the value of couplingstrength at the bridge is γ = 0.01, which is much weaker thanother coupling strength. In this case, as shown in Fig. 3(a),we obtain the difference of each ring. As time advanced, theposition of synchronization or asynchronous state changes toanother place. On another front, Fig. 3(b) shows that the bridge
position is constantly asynchronous. The state is initiallyasynchronous in Ring3, but we confirm that the asynchronous
position moved to Ring1 and Ring5 over time. Subsequently,
the effect of switching the synchronization state on the rings
is repeated.
4. Conclusion(b) Time advanced τ=150000-200000 in rings.

In this study, we investigated the synchronization
phenomena in a network composed of coupled chaotic
circuits. The network topology is a ladder coupled system
including a ring structure.
As a result of investigation by computer simulation, we
observed synchronization states such as chaotic
synchronization and perfect synchronization by changing
the coupling strength of the bridge part. In the case of γ =0.01, the ladder position and ring position are asynchronous.
In our future works, we analyze the dynamicssynchronization in more intricate networksof(c) Time advanced τ=50000-100000 in rings.
Fig. 3. Voltage difference waveform.

References[1] D. Malagarriga, A. E. P. Villa, J. Garca-Ojalvo and A. J. Pons, “Consistency of heterogeneoussynchronization patterns in complex weighted networks” Chaos 27, 031102, 2017.
[2] T. Nishiumi, Y. Uwate and Y. Nishio, “Synchronization Phenomena of Chaotic Circuits with
Stochastically-Changed Network Topology”, Proceedings of International Symposium on Nonlinear
Theory and its Applications (NOLTA’14), pp. 811-814, Sep. 2014.
[3] K. Nakabai, Y. Uwate and Y. Nishio, “Synchronization in Ladder-Coupled Chaotic Circuits Including RingStructure” Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’19),
DOI:10.1109/ISCAS.2019.8702350, May 2019.
[4] Y. Nishio, N. Inaba, S. Mori and T. Saito, “Rigorous Analyses of Windows in a Symmetric Circuits” IEEETransactions on Circuits and Systems“, vol. 37, no. 4, pp. 473-487, Apr. 1990.