PAPER Special Section on Nonlinear Theory and its Applications
Masayuki YAMAUCHI îª ) , Yoshifumi NISHIO tt , Members , Akio USHIDA ttt , Fellow , and Mamoru TANAKA t , Member SUMMARY In this study, nonlinear wave phenomena related to trans- missions and reflections of the phase-inversion waves around a discontinu- ity of a coupled system consisting of two kinds of arrays of van der Pol os- cillators are investigated. By computer simulations, behavior of the phase- inversion waves around the discontinuity in the coupled system is classified into eight types. Further, the mechanisms of the transmission and the reflec- tion of a phase-inversion wave at the discontinuity are explained. Circuit experiments confirm the simulated results. key words:  coupled oscillators, synchronization, wave propagation, phase difference, phase-inversion wave
Introduction
1.
Large number of coupled limit-cycle oscillators are useful as models for a wide variety of systems in natural fields, for example, diverse physiological organs including gastroin- testinal tracts and axial fiber of nervous systems, convect- ing fluids, arrays of Josephson junctions and so on. Hence, it is very important to analyze synchronization and the re- lated phenomena observed in coupled oscillators in order to clarify mechanisms of generations or in order to control the generating-conditions of various phenomena in such natural systems [1]–[4].  In the field of the electrical engineering, a lot of studies on synchronization phenomena of coupled van der Pol oscillators have been carried out up to now [5]– [12].  Recently, we have discovered very interesting wave propagation phenomena of phase states between two adja- cent van der Pol oscillators coupled by inductors as a ladder [13]. In the study, we named the continuously existing wave of changing phase states between two adjacent oscillators from in-phase to anti-phase or from anti-phase to in-phase as "phase-inversion wave."  Further, we reported the detail on "phase-wave" in [14], which is propagation of the phase difference less than 180 degrees between two adjacent os- cillators. The phase-waves exist only in the transient states unlike the phase-inversion waves.  When the phase differ- ence of a phase-wave is smaller than angle drawn into the anti-phase, the phase-wave disappears. Phase difference de- creases as the phase-wave propagates. When the phase dif- ference of a phase-wave is larger than angle drawn into the anti-phase, the phase-wave changes to the phase-inversion wave.  Phase difference increases as the phase-wave prop- agates [5], [14].  Thus, the phase-waves change to phase- inversion waves or disappear. In this study, by computer simulations, we observe be- havior of the phase-inversion waves around a discontinuity in a coupled system consisting of two kinds of arrays of van der Pol oscillators and classify the phenomena.  We explain the mechanisms of the transmission and the reflection of the phase-inversion waves at the discontinuity by using the re- lationship between phase difference and instantaneous fre- quency. We also carry out circuit experiments.
Circuit Model
2.
Circuit model is shown in Fig. 1. In the circuit, N 2 identical van der Pol oscillators are coupled as an array by inductors. From the 1st to the N 1 th oscillators they are coupled by in- ductors L 00 , while from the N 1 th to the N 2 th oscillators they are coupled by L 01 . Consequently, there is a discontinuity at the N 1 th oscillator. At first, the v – i characteristics of the nonlinear neg- ative resistors in the circuit are assumed by the following functions. (1) The circuit equations governing the circuit in Fig. 1 are expressed as [1st oscillator]
[2nd ~ ( N 1 − 1)th oscillators]
[ N 1 th oscillator]
[N2th oscillator] where and
(3)
The above variables and parameters are shown in Fig. 1. It should be noted that α 0 and α 1 correspond to the couplings and that ε corresponds to the nonlinearity. Equa- tions (2)–(8) are calculated by using the fourth-order Runge- Kutta method with the stepsize h = 0.001.
Phase-Inversion Wave
3.
In this section, we explain the phase-inversion waves ob- served from an array of van der Pol oscillators without dis- continuity, which was investigated in [13]. Figure 2 shows an example of the phase-inversion waves. In the computer calculations, we produced the phase- inversion waves as follows.  Almost same initial conditions are given for all oscillators to produce complete in-phase synchronization, which is one of the steady states in the sys- tem. Next, invert the voltage and the current of one oscilla- tor. In Fig. 2, a pair of phase-inversion waves is generated at the first oscillator. In the diagrams, vertical axes are sums of voltages of adjacent oscillators and horizontal axis is time. Hence, black regions show phase states near in-phase synchronization, and white regions show phase states near anti-phase syn- chronization.  We can observe the generation of a pair of phase-inversion waves, which is continuously existing wave propagation to change the phase states from in-phase to anti- phase and from anti-phase to in-phase.
Classification
4.
In this section, we observe transmission and reflection of the phase-inversion waves for the case of ( N 1 , N 2 ) = (11, 20). We fix the parameters ε = 0.20 and α 0 = 0.050 and α 1 is varied as a control parameter. A pair of phase-inversion waves are generated at the first oscillator. Figures 3 and 4 show the observed results.  Figure 4 shows the phase differences between adjacent oscillators along time.  Vertical axes are the phase differences between adjacent oscillators.  Horizontal axes are time.  Φ x H is a phase difference between x th oscillator and y th oscillator. According to the parameter value of α 1 , the observed phenomena are classified into eight types as follows, (a) ~around 0.002: A pair of phase-inversion waves is re- flected at the discontinuity almost completely. (b)  around 0.003 ~ 0.025:  A pair of phase-inversion waves is reflected and is transmitted.  Complex phenomena are observed after the reflection and the transmission. (c) around 0.026 ~ 0.044: A pair of phase-inversion waves is reflected and is transmitted.  The phase-inversion waves reflected at the discontinuity change to a pair of phase-waves. (d)  around 0.045 ~ 0.054:  A pair of phase-inversion waves is transmitted almost completely. (e)  around 0.055 ~ 0.123:  A pair of phase-inversion waves is transmitted. A new pair of phase-waves is generated after the phase-inversion waves are transmitted. (f) around 0.124 ~ 0.174: A pair of phase-inversion waves YAMAUCHI et al.: PHASE-INVERSION WAVES IN OSCILLATORS COUPLED BY TWO KINDS OF INDUCTORS AS A LADDER (b) α₁ = 0.020. is reflected and is transmitted.  The phase-inversion waves change to phase-waves after the reflection and the transmission. (g)  around 0.175 ~ 0.232:  A pair of phase-inversion waves is reflected and is transmitted.  The phase-inversion (a) αι = 0.001. waves transmitted through the discontinuity change to a pair of phase-waves. (h) around 0.233 ~: A pair of phase-inversion wave is re- flected and is transmitted. The phase-inversion waves change to phase-waves after the transmission.  It is difficult to observe in the figure, but the first phase- inversion wave is reflected at the discontinuity, collides with the second phase-inversion wave and changes to a pair of phase-waves. It should be noted that we can observe different trans- mission phenomena for the cases of L 01 ≫ L 00 and L 01 < L 00 .  Namely, for L 01  ≫  L 00 (α 1  ≪ α 0 ), phase-inversion wave is not transmitted beyond the discontinuity. While, for L 01 < L 00 (α 1 > α 0 ), phase-inversion wave is transmitted as phase-wave beyond the discontinuity.  This interesting dif- ference is explained in the Sect. 5. Figure 5 summarizes the parameter regions corre- sponding to the above-mentioned eight types of the observed phenomena.
Mechanisms of Transmission and Reflection
5.
In this section, the transmission mechanism and the reflec- tion mechanism of the phase-inversion waves at the discon- tinuity are explained according to the computer calculated results in the previous section. It has been already known that oscillation frequency of in-phase synchronization of oscillators coupled by induc- tors is different from that of anti-phase synchronization [5]. If we define the oscillation frequency of in-phase synchro- nization as f in and the oscillation frequency of anti-phase synchronization as fanti , fin is smaller than fanti [8]. Further, the difference between f in and f anti increases as coupling in- ductance increases.  For example, when even van der Pol oscillators are coupled by inductors as a ring and ε is very small, theoretical values are obtained by Eqs. (9) and (10) [10]. [Frequency of in-phase synchronization]
Computer calculated f in and f anti are slightly different from these values and are obtained by computer calculations as follows: When α = 0.0500 and ε = 0.2000, fin = 0.159[1/τ] and fanti = 0.174 [1/τ]. We use this characteristic to explain the mechanisms of the transmission and the reflection of the phase-inversion waves. f in and f anti in Figs. 6, 7 and 8 show these values.
Transmission Mechanism
5.1
In order to explain the transmission mechanism, we con- sider the case of α 0  ≈ α 1 .  Because the transmission is a typical phenomenon observed for that case. We can say the transmission mechanism is similar to the propagation mech- anism [13] of a phase-inversion wave, if α 0 ≈ α 1 is satisfied. Hence, the transmission mechanism is explained as follows: 1. Let us assume that OSC11 ~ OSC20 are in-phase syn- chronization and that the wave changing from in-phase into anti-phase is going to reach OSC11 from the di- rection of OSC1 (see (a) in Fig. 6). 2. As phase difference between OSC10 and OSC11 Φ 10,11 approaches −180° (see (b) in Fig. 6), instantaneous fre- quency of OSC11 f 11 changes from f in to f anti (see (c) in Fig. 6). 3. The change of f 11 causes decrease of Φ 11,12 (see (d) in Fig. 6).  Speed of the decrease is decided by the differ- ence between f 11 and f 12 . This means that propagation speed of the wave is decided by the difference between f in and f anti 4. When Φ 11,12 reaches almost − 180°, f 6 is equal to fanti (see (e) in Fig. 6). As a result, the phase-inversion wave is transmitted through the discontinuity.
5.2
Reflection Mechanism
In order to explain the reflection mechanism, we consider two typical cases for the reflection.  Namely the coupling parameter for one side of the discontinuity is much larger or much smaller than that of the other side.
5.2.1
When L 01 ≫ L 00 (i.e. α 1 ≪ α 0 )
1. Let us assume that OSC10~OSC20 are in-phase syn- chronization and that a phase-inversion wave from in- phase to anti-phase reaches at OSC10 from the direc- tion of OSC1. 2. Instantaneous frequency of OSC10 f 10 begins to change from f in toward f anti , because in-phase syn- chronization between OSC9 and OSC10 breaks by the phase-inversion wave (see (a) in Fig. 7). 3. As f 10 changes from f in to f anti , phase difference be- tween OSC10 and OSC11 Φ 10,11 approaches −180° (see (b) in Fig. 7). 4. As Φ 10,11 approaches − 180°, f 11 begins to change from fin to fanti (see (c) in Fig. 7). 5. Change of f 11 does not propagate to OSC12 immedi- ately, because L 01 is much larger than L 00 (see (d) in Fig. 7). 6. Hence, f 11 cannot reach f anti and begins returning to f in (see (e) in Fig. 7). 7. As f 11 returns to f in , Φ 10,11 continues to increase until reaching −360° (see (f) in Fig. 7). 8. By the effect of the decrease of Φ 10,11, f 10 begins to decreases again from fanti toward fin (see (g) in Fig. 7). 9. Φ 11,12 becomes –180°, because in spite of constant value which is f 12 , f 11 changes from f anti and returns to f in again (see (h) in Fig. 7). Remark: As L 01 approaches L 00 , it becomes easier that change of f 11 propagates to OSC12. Hence, we could ob- serve that the phase-inversion wave is transmitted beyond the discontinuity as shown in Figs.2(b), (c) and 3(b), (c). On the other hand, the reflection becomes smaller, because being hard to change f 11 is the main reason of the generation of the reflection. 5.2.2 When L 01 ≪ L 00 (i.e. α 1 ≫ α 0 ) 1. Let us assume that OSC10~OSC20 are in-phase syn- chronization and that a wave from in-phase to anti- phase reaches at OSC10 from the direction of OSC1. 2. Instantaneous frequency of OSC10 f 10 begins to change from f in toward f anti , because in-phase syn- chronization between OSC9 and OSC10 breaks by the phase-inversion wave (see (a) in Fig. 8). 3. As f 10 changes from f in to f anti , phase difference be- tween OSC10 and OSC11 Φ 10,11 approaches −180° (see (b) in Fig. 8). 4. As Φ 10.11 approaches −180°, f 11 changes from f in to fanti (see (c) in Fig. 8). 5. Change of f 11 propagates to OSC12 immediately, be- cause L 01 is much smaller than L 00 (see (d) in Fig. 7). 6. Hence, in-phase synchronization between OSC11 and OSC12 is hard to break (see (e) in Fig. 8). And f 11 and f 12 return to f in together (see (f) in Fig. 8). 7. As f 11 returns to f in , Φ 10,11 continues to increase until reaching −360° (see (g) in Fig. 8). Hence, when L 01 is much smaller than L 00 , the trans- mission always exists with the reflection. Remark: As L 01 approaches L 00 , it becomes more difficult that phase states of OSC11 and OSC12 hold in-phase syn- chronization.  Hence, the reflection of the phase-inversion wave disappears as shown in Figs. 2(d), (e) and 3(d), (e).
6.
Circuit Experiments
In this section, we confirm the generation of the transmis- sion and the reflection of the phase-inversion wave by cir- cuit experiments.  The typical result for ( N 1 , N 2 ) = (7, 10) is shown in Fig. 9. The corresponding computer simulated result is shown in Fig. 10. We observe that the circuit exper- imental result and the computer simulated result agree well. The small amount of the difference between the two results may come from the resistance in real inductors or difficulty of setting initial phase difference in real circuit experiments.
7.
Conclusions
In this study, we observed behavior of a pair of phase- inversion waves around the discontinuity of the system con- sisting of two parts of coupled van der Pol oscillators. Ob- served phenomena by computer calculations were classified into eight types. We explained the mechanism of the trans- mission and the reflection of the phase-inversion wave at the discontinuity and made clear that different mechanisms re- flect the waves for the cases of L 01 > L 00 and L 01 < L 00 . Further, we observed a pair of phase-inversion waves by cir- cuit experiments.  The circuit experimental results agreed with the numerical calculated results very well.