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The collective behavior of a ring of coupled identical van der Pol oscillators is numerically studied in this work. Constant, gaussian and random distributions of the coupling parameter along the ring are considered. Three values of the oscillators constant are assumed in order to cover from quasilinear to nonlinear dynamic performance. Single and multiple coupled frequencies are obtained using power spectra of the long term time series. Phase portraits are obtained from numerical simulations, and the coupled behavior is analyzed, compared and discussed.

It is known that the van der Pol oscillator can represent many oscillating systems in a wide variety of applications: biological rhythms [

Some works on the collective behavior of large rings of coupled van der Pol oscillators have been reported. In [

In [

The van der Pol oscillator is a well known oscillator which is mathematically represented as

where x is the oscillator position and a is the uncoupled oscillator constant. In this work the case of van der Pol oscillators in the form of a ring with each oscillator coupled to its two nearest neighbors is considered. For a ring of N oscillators the following expression holds

Ring of N coupled oscillators

where _{i} is the coupling parameter corresponding to the i^{th} oscillator, which depends on the angular position θ_{i} of the i^{th} oscillator on the ring. Besides,

Given that the coupling parameter depends on the oscillator position in the ring, a coupling vector B can be defined as

For a gaussian distribution of B along the ring, b_{i} is determined from

where σ is the standard deviation and μ is the mean or expectation of the gaussian distribution.

For a random distribution of the coupling parameter,

where λ, S_{1} and S_{2} are constants, and rand (N, 1) is a matrix of random numbers.

For a constant distribution of the coupling parameter,

B = c (6)

In order to allow a quantitative comparison among the above distributions, all of them must satisfy

where d is a constant. Equation (7) implies that if d = 2, then for a constant distribution in Equation (6) c = d/2 and in this case

Equation (2) corresponding to the ring of coupled oscillators was numerically solved through the fourth-order Runge-Kutta method [^{−5}, which is small enough to guarantee numerical stability and convergence. As in [

where

Equation (9) represents a wave of amplitude A, wave number k and frequency ω traveling in the positive θ direction. Then, the initial conditions of position and velocity are, respectively, given by

In the computer simulations, it was assumed that A = 1, k = 2 and_{1} = 0, S_{2} = 2; 3) for the constant distribution c = 1. All these values satisfy the condition given by Equation (7).

Histogram of the coupling parameter under gaussian distribution

Histogram of the coupling parameter under random distribution

row of

Phase portraits for the gaussian distribution case are depicted in Figures 6(a)-(c). As the oscillator constant is increased, the coupled behavior goes from a single frequency with constant phase shift for a = 0.1 to multiple frequencies with variable phase shift for a = 0.5 and a = 1. Main frequencies of 0.1603, 0.1580 and 0.1525 Hz are identical to those corresponding to the constant distribution case. However, the secondary frequencies are quantitatively different, as can be appreciated when the second and third rows of

. Ring frequencies (Hz) obtained from power spectra. Main frequency in bold

COUPLING | a = 0.1 | a = 0.5 | a = 1 |
---|---|---|---|

Null | 0.1591 | 0.1567 | 0.1500 |

Constant | 0.1603 | 0.1580, 0.1563, 0.1624 | 0.1526, 0.1450, 0.1603 |

Gaussian | 0.1603 | 0.1580, 0.1565, 0.1532 | 0.1525, 0.1571, 0.1619 |

Random | 0.1595, 0.1580 | 0.1580, 0.1592, 0.1608 | 0.1557, 0.1587, 0.1693 |

Phase portraits for a constant distribution of the coupling parameter. (a) a = 0.1; (b) a = 0.5; (c) a = 1

Long time series for a constant distribution of the coupling parameter. (a) a = 0.1; (b) a = 0.5; (c) a = 1. Oscillator 1 (solid), oscillator 50 (dashed)

Phase portraits for a gaussian distribution of the coupling parameter. (a) a = 0.1; (b) a = 0.5; (c) a = 1

The phase portraits for random distribution are shown in Figures 7(a)-(c). In accordance to the fourth row of

The collective behavior of the ring is mainly determined by the nature of the coupling distribution. However, the oscillators’ constant plays also a significant role in this behavior. As the oscillator constant is increased the ring nonlinearity is increased too. For a = 0.1 the coupled performance is little influenced by the coupling distribution, as is observed in the second column of

The collective behavior of a ring of coupled identical van der Pol oscillators with different coupling schemes was numerically studied. The nature of this behavior strongly depends on the kind of distribution of the coupling parameter and on the value of the oscillators constant.

1) For a small value of the oscillators constant and a constant distribution of the coupling parameter, the ring behavior goes from quasilinear with a single frequency to multiple frequencies and variable phase shifts.

Phase portraits for a random distribution of the coupling parameter. (a) a = 0.1; (b) a = 0.5; (c) a = 1

Long time series for a random distribution of the coupling parameter. (a) a = 0.1; (b) a = 0.5; (c) a = 1. Oscillator 1 (solid), oscillator 50 (dashed)

Power spectra of the first (a) and 50^{th} (b) oscillators for a = 1 under a random coupling distribution, corresponding to the time series of Figure 8(c)

2) A random distribution of the coupling constant, combined with a high value of the oscillator constant, causes a coupled behavior which ranges from anti-phase to chaotic.

3) The coupled behavior becomes more complex as the oscillator constant is increased, irrespective of the distribution of the coupling parameter.

4) In the all cases considered, the coupled main frequency is greater that the uncoupled frequency, irrespective of the value of the oscillators constant or the nature of the coupling distribution.

M. A. Barron appreciates the generous help and guidance of Prof. Mihir Sen, from Notre Dame University, Notre Dame, IN, during the conception and writing of this paper.

The author declares that there is no conflict of interests regarding the publication of this article.