Synchronization in Two Rings of Three Coupled van der Pol Oscillators Daiki Nariai,Minh Hai Tran, Yoko Uwate and Yoshifumi Nishio Department. of Electrical and Electronic Engineering, Tokushima University 2-1 Minami-Josanjima, Tokushima 770-8506, Japan E-mail: {nariai, minhhai, uwate, nishio}@ee.tokushima-u.ac.jp Abstract In this study, we investigate the synchronization phenomena in two rings of van der Pol oscillators coupled by resistors. We propose a novel coupled oscillatory system comprising two rings of van der Pol oscillators with different coupling schema. We focus on the coupling strengths of the coupled van der Pol oscillators. By computer simulation, we investi- gate how the synchronization phenomena change by chang- ing the coupling strengths. In these results, we observe vari- ous synchronization phenomena. 1. Introduction The synchronization phenomena of coupled oscillators are familiar. Synchronization phenomena have been studied in various fields for many years, such as in electrical systems, mechanical systems and biological systems. Among them, the synchronization phenomena of van der Pol oscillators are similar to natural phenomena when the frequency is changed. A coupled system of van der Pol oscillators is simple and easy to handle. Many researchers have proposed various coupled oscillatory networks of van der Pol oscillators [1]- [3]. We fo- cus on the coupling strengths of coupled oscillatory networks consisting of two van der Pol oscillators. The van der Pol oscillator is a simple circuit. It consists of a resistor, inductor, capacitor and nonlinear resistor. It was invented by the electrical engineer Balthasar van der Pol. The equation of a van der Pol oscillator is a second-order differ- ential equation. In this study, we propose a novel coupled oscillatory sys- tem comprising two rings of van der Pol oscillators coupled by resistors. The first ring consists of three van der Pol os- cillators connected by resistors. The second ring consists of three van der Pol oscillators connected by inductors and re- sistors. By computer simulation, we investigate the synchro- nization phenomena observed in the proposed circuit system by changing the coupling strengths. 2. System Model C L i gn v Cn i Ln VDP Figure 1: Circuit of van der Pol oscillator circuit 2 circuit 3 circuit 6 circuit 4 circuit 1 VDP VDP VDP R R2 R3 2L 2L 2L 2L 2L 2L i1a i1b i3a i3b i2a i2b i4a i4b i5b i6a i6b i5a ir4 ir5 ir3 ir1 ir2 ir6 R R R1 circuit 5 NC NC NC R' R' R' i gn C v Cn NC Figure 2: Circuit model Figure 1 shows the circuit of a van der Pol oscillator. We call this circuit VDP. We use six van der Pol oscillators in this study. Figure 2 shows a model of the system with six van der Pol oscillators. We use two ring circuits of van der Pol oscil- lators. The three VDP of the first ring are connected by resis- tors. The three NC of the second ring are connected by induc- tors and resistors. When the two rings are not connected, the oscillators in the first ring exhibit in-phase synchronization and oscillators of the second ring exhibit three-phase synchro- nization. The first and second rings are connected by resistors (R 1 , R 2 , R 3 ). We observe the synchronization phenomena of adjacent oscillators. We investigate how the synchronization phenomena change upon changing the values of R 1 , R 2 and R 3 . The circuit equations of the first ring are given as follows: Journal of Signal Processing, Vol.21, No.4, pp.121-124, July 2017 SELECTED PAPER AT NCSP'17 Journal of Signal Processing, Vol. 21, No. 4, July 2017 121