IEICE TRANS. FUNDAMENTALS, VOL.E91–A, NO.9 SEPTEMBER 2008
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Special Section on Nonlinear Theory and its Applications
Pulse Wave Propagation in a Large Number of Coupled Bistable Oscillators Kuniyasu SHIMIZU†a) , Student Member, Tetsuro ENDO†b) , Fellow, and Daishin UEYAMA†† , Nonmember
SUMMARY A simple model of inductor-coupled bistable oscillators is shown to exhibit pulse wave propagation. We demonstrate numerically that there exists a pulse wave which propagates with a constant speed in comparatively wide parameter region. In particular, the propagating pulse wave can be observed in non-uniform lattice with noise. The propagating pulse wave can be observed for comparatively strong coupling case, and for weak coupling case no propagating pulse wave can be observed (propagation failure). We also demonstrate various interaction phenomena between two pulses. key words: bistable oscillators, inductor coupling, pulse propagation/interaction phenomena, non-uniform lattice, propagation failure
1.
Introduction
Coupled oscillator systems have been investigated for many decades in various areas of engineering, physics, and mathematics [1]. Examples include parallel operation of microwave oscillators, Josephson junction arrays, model of information processing in the brain and beam-scanning control system. Rich behaviors of these systems, such as mutual entrainment, self-synchronization, chaotic itinerancy and so on, are observed [2]–[4]. In this paper, we investigate pulse wave propagation phenomenon in a large number of inductor-coupled bistable oscillators. The dynamics for weak nonlinear cases have been almost elucidated via averaging method [5]. In contrast, its dynamics for strong nonlinear case seems to be unexplored. Namely, for weak nonlinear case, the behavior of this oscillator array obeys averaging theory, and only standing wave patterns are possible. However, when nonlinearity becomes comparatively strong, there exist various kinds of propagating pulses. Although propagating pulses have been investigated extensively in reaction-diffusion systems [6], [7], there seems few studies on pulse wave propagation phenomenon observed in coupled oscillator arrays. Yamauchi et al. investigated wave propagation phenomenon in an inductor-coupled van der Pol oscillator array with soft nonlinearity [8], [9]. They demonstrated various propagating “phase” waves. In contrast to their research, the pulse waves in our case Manuscript received November 21, 2007. Manuscript revised February 28, 2008. † The authors are with the Dept. of Electronics and Bioinformatics, Meiji University, Kawasaki-shi, 214-8571 Japan. †† The author is with the Dept. of Mathematics, Meiji University, Kawasaki-shi, 214-8571 Japan. a) E-mail:
[email protected] b) E-mail:
[email protected] DOI: 10.1093/ietfec/e91–a.9.2540
are based on “oscillation amplitude,” namely several adjacent oscillators forming a pulse oscillate with large amplitudes and others show no oscillation. We demonstrate that there exists a propagating pulse wave consisting of several adjacent oscillators oscillating with large amplitude, and the part of large amplitude oscillation in the array propagates with a constant speed. In particular, the propagating pulse wave can be observed even in such a practical condition that each oscillator has a slightly different intrinsic oscillation frequency in addition to noise (non-uniform lattice with noise). The propagating pulse wave can be observed more frequently when the parameter α (= the coupling strength) becomes comparatively large. On the other hand, when α is small, there is no propagating pulse wave not only in nonuniform lattice with noise but also in uniform lattice without noise. It is known as “propagation failure” in lattice reaction-diffusion systems [10]–[12]. Therefore, it is important to investigate the propagation failure in terms of coupling coefficient in inductor-coupled bistable oscillator lattices. At last, we present the interaction phenomena between two pulses and confirm pulse unification, pulse vanishing, pulse repulsion and pulse passing through phenomenon. 2.
Circuit Model
Figure 1 presents a lattice of inductor-coupled bistable oscillators connected in a ring structure. We assume the hard-type nonlinearity for NC, i.e., iNC = g1 V − g3 V 3 + g5 V 5 , g1 , g3 , g5 > 0. Namely, each isolated oscillator has two steady-states: no oscillation and periodic oscillation depending on the initial condition. Assuming also a certain amount of time-varying noise, the circuit equation can be written by the following system after normalization [5]:
Fig. 1 A lattice of inductor-coupled bistable oscillators in a ring structure.
c 2008 The Institute of Electronics, Information and Communication Engineers Copyright
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x˙i = yi + A · ni (t) y˙ i = −εωi (1− βx2i + x4i )yi −ω2i (1− α)xi + ω2i α(xi−1 − 2xi + xi+1 ),
(1)
i = 1, 2, · · · , N, x0 = xN , xN+1 = x1 , (· = d/dt) where N is the number of oscillators† . The xi denotes the normalized output voltage of the i-th oscillator, yi denotes its derivative and ωi denotes the intrinsic angular frequency of the i-th oscillator. Parameter ε (> 0) shows the degree of nonlinearity. Parameter α (0 α 1) is a coupling factor; namely α = 1 means maximum coupling, and α = 0 means no coupling. Parameter β controls amplitude of oscillation. The ni (t) is a time-varying uniform random number distributing from −1 to +1, which is introduced at each iteration of numerical integration (hence, this term can be regarded as a noise to this system). Although we take N = 100 throughout this paper, pulse wave propagation phenomenon can be observed in an arbitrary number of coupled oscillators†† . We adopt the ring structure as a manner of coupling, because the effect of both ends can be neglected. However, for a large number of coupled oscillators, the dynamics of inside arrays are less influenced by the boundary condition. Therefore, we can observe almost qualitatively the same phenomena in other coupled lattices including the linear coupled lattice. 3.
Pulse Wave Propagation Phenomenon
3.1 A Single Propagating Pulse Wave in the Uniform Lattice without Noise At first, we will show a typical wave propagation phenomenon observed in coupled identical oscillator case without noise. Figure 2 presents a typical propagating pulse wave for ε = 0.36, α = 0.3 and β = 3.2††† . Each figure shows the snapshot at time t. It is observed that a single pulse wave propagates from left to right with a constant speed. Which direction a traveling pulse propagates depends mainly on the initial condition. The propagating pulse wave consists of several adjacent oscillators oscillating with large amplitude. Such a propagating pulse exists in a wide regime in the parameter space. Figure 3 presents the existence regions of the propagating pulse and other phenomena for ε = 0.36†††† . It is noted that if we set β to an appropriate value (nearly from 3.15 to 3.30), there exists the propagating pulse in a wide range of α (the “P” regime in Fig. 3). In the left-hand side of the propagating pulse regime (the “S” regime in Fig. 3), in which α is smaller than 0.1 approximately, there exists a standing pulse††††† instead of the propagating pulse. This corresponds to the “propagation failure” regime. It should be noted that propagation failure phenomenon occurs not only in reaction-diffusion equations but also in other type of equation as Eq.(1). Around the region above the propagating pulse regime (the “W” regime in Fig. 3), whole oscillation such as all oscillators oscillate
Fig. 2 Snapshots of typical pulse wave propagation for α = 0.3, β = 3.2 and ε = 0.36. Initial condition is x22 = −0.9, x23 = −0.7, x24 = 1.6, x25 = 1.4, y22 = 0.9, y23 = −1.6, y24 = −1.4, y25 = 1.2 and all other variables are zero. P denotes the propagating pulse wave at time t. Oscillator number has a ring structure, i.e., number 100 is connected to number 1.
with large amplitude can be observed. In the region below †
This is not the reaction-diffusion equation. If we adopt a lattice of resistance-coupled bistable oscillators, the equation becomes a reaction-diffusion type. †† We observe this phenomenon even for the N = 6 case. ††† All numerical integrations are carried out by the fourth-order Runge-Kutta method with step size = 0.01. †††† This result can be derived by computer simulation. we gradually vary α and β with step size = 0.01, and then we check whether propagating pulse exists or not at each point. ††††† We call the pulses (oscillations) which are stationary in space as “standing pulse.” They can be periodic, quasi-periodic, and chaotic.
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Fig. 3 Existence regions of propagating pulse and other phenomena in uniform lattice without noise for ε = 0.36. The “P,” “S,” “W” and “Z” are existence regions of propagating pulse wave, standing pulse, whole oscillation and no oscillation, respectively. This figure is obtained from the same initial condition as Fig. 2.
the propagating pulse regime (the “Z” regime in Fig. 3), no oscillation exists. These three regions seem to overlap to some extent with the propagating pulse regime. 3.2 A Single Propagating Pulse Wave in the Non-uniform Lattice with Noise Next, we assume that each oscillator has a slightly different intrinsic oscillation frequency in addition to noise. We distribute the intrinsic angular frequencies according to Gaussian distribution with average (≡ a) equal to 1.0, and with various standard deviation (≡ σ). We also give time-varying noise to this system by adding a uniform random number A · ni (t) distributing from −A to +A to the first equation of Eq.(1) at each iteration of numerical integration. Figure 4(a) presents a bird’s eye view plot of a single propagating pulse wave for σ = 0.02 and A = 0.2 with the same initial condition and parameters in Fig. 2. An example of actual distribution of the intrinsic angular frequencies can be seen in Fig. 4(b). It is clear that the propagating pulse wave with almost constant speed exists in spite of both intrinsic oscillation frequency fluctuation (hereafter, abbreviated as “fluctuation”) and noise. In such a practical condition, other phenomena, i.e. standing pulse, no oscillation and whole oscillation, etc., may be observed with the same parameters and the same initial condition, because the observed phenomenon is probabilistic. However, as we increase the coupling factor α, the propagating pulse is observed more frequently than other phenomena. Here, we make 30 trials for fixed α in order to check whether the propagating pulse exists or not. In each trial, the intrinsic angular frequencies are changed according to Gaussian distribution. The noise amplitude A is set to 0.2 which can be regarded as sufficiently large noise. Then, we calculate the probability: P’ = p / q, where p is the number of emergence of the propagating pulse and q is the trial
Fig. 4 A single propagating pulse under both frequency fluctuation and noise for σ = 0.02, A = 0.2, α = 0.3, β = 3.2 and ε = 0.36. Initial condition is same as that of Fig. 2.
number. We plot the α versus P’ characteristic in Fig. 5 for β = 3.2 and ε = 0.36 by changing σ. Comparing the results for each σ, qualitatively the same characteristics can be observed, though the critical point (the point at which the probability becomes positive) of each curves are different. Namely, it is noted that when α is small, no propagating pulse exists. In particular, when α is approximately smaller than 0.1, the standing pulse exists for all values of σ; namely no propagating pulse exists. When α becomes larger, the propagating pulse exists in several trials out of 30 trials. The critical point of α increases with the increase of σ; namely, for larger fluctuation cases, the propagation failure regime of α becomes larger. Near the critical point of each curves, interesting random walk patterns can be observed in addition to usual propagating pulse waves as in Fig. 6. The random walk pulse moves both right and left directions randomly due to noise. This is in contrast to the pulse wave moving in one direction shown in Fig. 4(a) un-
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Fig. 5 Probability of emergence of the propagating pulse in terms of coupling strength α for various values of σ (A = 0.2, β = 3.2, ε = 0.36).
Fig. 7 An example of the same-phase and the reverse-phase initial condition for pulse interaction. (a) The same-phase initial condition. Namely, the x22 and x67 , x23 and x66 , x24 and x65 , etc. are given with the same-phase initial condition. (b) The reverse-phase initial condition. Namely, the x22 and x67 , x23 and x66 , x24 and x65 , etc. are given with the reverse-phase initial condition.
Fig. 6 An example of the random walk pattern observed near the critical point (α = 0.13) of the curve of σ = 0.02 in Fig. 5 (A = 0.2, β = 3.2 and ε = 0.36).
der the same-fluctuation/noise level. As α becomes larger, the probability tends to 1.0, which means that the propagating pulse exists for all trials. Therefore, we can say that the propagating pulse wave becomes stronger against fluctuation and noise as α becomes larger. Furthermore, propagation speed becomes faster as α becomes larger. 3.3 Interaction of Two Propagating Pulses in the Nonuniform Lattice with Noise When we give large initial values to more than one places at the same time, multiple propagating pulses emerge. They collide at a certain time and interact with each other. In this section, we will investigate various pulse interaction phenomena under fluctuation and noise. For simplicity, we
show interaction (collision) among two pulses. At first, we will define the same-phase and the reversephase initial condition. Figure 7(a) presents an example of the same-phase initial condition. Namely, two pulses whose corresponding units present the same-phase synchronization are given. Similarly, Fig. 7(b) presents an example of the reverse-phase initial condition; namely, two pulses whose corresponding units present the reverse-phase synchronization are given. The actual values for the same- and the reverse-phase initial condition are given in Table 1 and Table 2, respectively. We employ the above two initial conditions. In uniform lattice without noise shown in Sect. 3.1, there exist the pulse repulsion and the pulse passing through phenomena depending mainly on initial condition [13]. Namely, if we give the reverse-phase initial condition to two places on the ring array, two pulse waves propagating in the opposite direction collide and repel. In contrast, when we give the same-phase initial condition, two pulse waves collide and pass through. However, in non-uniform lattice with noise, the above phenomena can’t be observed easily. For example, in 60 trials of the same-phase and the reversephase initial condition, in which intrinsic angular frequencies obey Gaussian distribution (σ = 0.02) in addition to noise (A = 0.2) as in Sect. 3.2, we observe a few samples of repulsive and passing through phenomena. Instead, we observe other phenomena such as pulse unification and pulse vanishing phenomena in most of trials. Figures 8(a) and (b) demonstrate these two typical phenomena under fluctuation and noise for the same-phase initial condition. Fig-
IEICE TRANS. FUNDAMENTALS, VOL.E91–A, NO.9 SEPTEMBER 2008
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The actual values of the same-phase initial condition in
x22
y22
x23
y23
x24
y24
x25
y25
−0.9
0.9
−0.7
−1.6
1.6
−1.4
1.4
1.2
x67
y67
x66
y66
x65
y65
x64
y64
−0.9
0.9
−0.7
−1.6
1.6
−1.4
1.4
1.2
4.
Other xi , yi = 0.0
Table 2 Fig. 7(b).
The actual values of the reverse-phase initial condition in y22
x23
y23
x24
y24
x25
−0.9
0.9
−0.7
−1.6
1.6
−1.4
1.4
1.2
x67
y67
x66
y66
x65
y65
x64
y64
0.9
−0.9
0.7
1.6
−1.6
1.4
−1.4
−1.2
x22
the reverse-phase initial condition, the typical phenomena we observe are those of Figs. 8(a) and (b); namely, pulse unification and pulse vanishing phenomenon. In particular, pulse unification phenomenon can be observed more frequently than pulse vanishing phenomenon for large α.
y25
Other xi , yi = 0.0
Conclusion
A simple model of inductor-coupled bistable oscillators is shown to exhibit various propagating pulse waves. It is confirmed numerically that there exists the propagating pulse wave even under both fluctuation and noise (non-uniform lattice with noise) for comparatively large coupling strength. In contrast, we observe mainly the standing pulses for small coupling strength. This corresponds to the propagation failure phenomenon. Moreover, we present the interaction phenomena among two pulses in the non-uniform lattice with noise. These interactions present the interesting behaviors such as pulse repulsion, pulse passing through, pulse unification and pulse vanishing phenomenon, etc. Based on the obtained results, it is possible to assume that the pulse wave may be observed in practical circuit. As a future problem, we will investigate this system theoretically and also implement this system in actual circuit. Further, we will investigate a lattice of resistance-coupled bistable oscillators which becomes the well-known reaction-diffusion equation. References
Fig. 8 A bird’s-eye view plots of interaction among two pulses for α = 0.3, β = 3.2 and ε = 0.36. In case (a), two propagating pulses pass through once before only one pulse survives. In case (b), two propagating pulses pass through two times before they vanish. Initial condition of case (a) and (b) is x22 (x67 ) = −0.9, x23 (x66 ) = −0.7, x24 (x65 ) = 1.6, x25 (x64 ) = 1.4, y22 (y67 ) = 0.9, y23 (y66 ) = −1.6, y24 (y65 ) = −1.4, y25 (y64 ) = 1.2 and all other variables are zero. Oscillator number 100 is connected to number 1 due to ring structure.
ure 8(a) shows that two propagating pulses pass through a few times before only one pulse survives (the other pulse may disappear or merge). Figure 8(b) shows that two propagating pulses pass through a few times before they vanish. It should be noted that as α is increased, the vanishing phenomenon is more easily observed for the same-phase initial condition. For example, the pulse vanishing phenomenon can be observed in 51 samples out of 60 trials ( 85%) for α = 0.9, though it can be observed only in 12 samples out of 60 trials ( 20%) for α = 0.3. Next we will explain the phenomena observed for the reverse-phase initial condition. In uniform lattice without noise, there exists “the standing wave slip” phenomenon for the reverse-phase initial condition [13]. However, this phenomenon can not be observed in the non-uniform lattice with noise. After all, in non-uniform lattice with noise for
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