4th International Forum on Advanced Technologies
March 8th-9 th, 2018,Tokushima, Japan

Synchronization Phenomena of Coupled Chaotic Circuitswith Symmetrical and Asymmetric Structures
Katsuki Nakashima , Kazuki Ueta , Yoko Uwate and Yoshifumi Nishio
Tokushima University, Tokushima 770-8501, Japan* E-mail: nakashima@ee.tokushima-u.ac.jp
1. Introduction
Nonlinear phenomena of coupled chaotic circuits are drawing attention from many researchers [1],[2]. In thisstudy, we investigate the synchronization phenomena of coupled two symmetrical structures and the influenceof the network topology when we use chaotic circuits. One structure generates chaotic attractors and the otherstructure generates three-periodic attractors. Moreover, we observe the synchronization phenomena ofnetworks with asymmetrical structures and we compare networks with symmetrical structures to networks withasymmetrical structures.
2. Circuit model
The chaotic circuit is shown in Fig. 1 and the system model is shown in Fig. 2. This chaotic circuit consistsof two inductors L1 and L2, one capacitor C, negative resistor −r and two diodes.

Fig.1 Chaotic circuit.
The normalized equations of chaotic circuits are given as follows:(i, j = 1, 2, · · ·, 6).

where γ is the coupling strength.
Figure 2 shows the proposed system models. We propose the system models that the two symmetricalstructures are coupled by a resistor in Fig. 2(a). In addition, we propose the system models that the two
asymmetric structures are coupled by a resistor in Fig. 2(a’). In Fig. 2(a) and (a’), we set CC1 to CC3 as
chaotic solution and CC4 to CC6 as three periodic solution. In this study, we set the coupling strength γ as
0.2, the coupling strength γ1 between the topologies as 0.1 and γ2 as 0.01 in all models.

(a) 3-3Coupled model.
(a') 1-3Coupled model.
Fig.2 System model.
3. Simulation results
We investigate the synchronization phenomena in each model and the influence of topology. Figure 3 shows
attractor of each chaotic circuit, Fig. 4 shows lissajous figures and Fig. 5 shows the volage of different
waveform. In both model(a) and model(a’), CC4, CC5 and CC6 are generated chaotic attractor. Also, between

4th International Forum on Advanced Technologies
March 8th-9 th, 2018,Tokushima, Japan
CC3 and CC4 are synchronized. CC4-CC5 and CC4-CC6 are asynchronous. CC5 and CC6 are chaotic
synchronization in the model(a). Therefore, in CC5 and CC6, synchronous and asynchronous states changes
irregularly in the simulation time.

(a) 3-3Coupled model.

(a') 1-3Coupled model.

Fig.3 Attractors.

(a) 3-3Coupled model.

(a') 1-3Coupled model.

Fig.4 Lissajous figures.

(a) 3-3Coupled model.

(a') 1-3Coupled model.

Fig.5 Different waveform.
4. Conclusion
In this study, we have proposed system models using two symmetrical structures and asymmetrical structuresthat are coupled by a resistor. We have investigated the synchronization phenomena by influence of topologyof the system. By the computer simulations, we have observed synchronization phenomena in each model. As
a result, network with coupled symmetrical structures (model(a)) has a strong influence on one circuit that arecoupled between the structures. However, network with coupled asymmetric structures (model(a’)) has theinfluence on all of the circuits in topology.
In the future works, we will investigate the synchronization phenomena by changing the coupling strength.
Moreover, we observe the influence for the topology of the network.
Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for Scientific Research 16K06357
References[1]Y. Nishio and A. Ushida, “On Synchronization Phenomena in Coupled Chaotic Circuits Networks,”
Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’96), vol. 3, pp. 92-95,May. 1996.
[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 itsApplications (NOLTA’14), pp. 811-814, Sep. 2014.