It is known that the van der Pol oscillator can represent many oscillating systems in a wide variety of applications: biological rhythms [1] , heartbeat [2] [3] , chemical oscillations [4] , circadian rhythms [5] , and so on. The coupled behavior of van der Pol oscillators has been studied in the past considering different conditions and applying diverse analysis techniques, for example: diffusive displacement coupling [6] ; weak displacement coupling using harmonic balance [7] ; weak displacement and velocity couplings using perturbation methods [8] ; strong coupling with detuning using perturbation methods [9] ; strong diffusive coupling using matched asymptotic expansions [10] ; weak and moderate amplitude coupling with numerical solution [11] ; weak and strong bath coupling using Floquet theory and numerical solution [5] ; Whittaker method to determine the synchronized states in a ring of four mutually coupled non-identical van der Pol oscillators [12] .

Some works on the collective behavior of large rings of coupled van der Pol oscillators have been reported. In [13] the authors consider the synchronized states in a ring of mutually coupled self-sustained electrical oscillators described by coupled Van der Pol equations. They analyze the stability of the synchronization of the ring using the Floquet theory, the Whittaker method and numerical investigations. In [14] the dynamic behavior and synchronization of a very large ring of coupled van der Pol oscillators is studied by linearization and subsequent conversion of the original equations into a canonical Hill’s equation; the influence of the coupling parameter value on the ring stability is investigated by determining the value and sign of the non-periodic term of the Hill's equation. In a recent work [15] one of the present authors consider a ring of van der Pol oscillators, and the focus of his work is on the emergent behavior of a large number of oscillators. He determines the conditions under which time-independent solutions are obtained, and the linear stability of these solutions is investigated. Besides, the effect of the singularity of the coupling matrix on the ring dynamics is explored: when this becomes singular, an infinite number of steady states are present, and the phenomenon of oscillation death arises.

In [13] -[15] , the value of the coupling parameter remains constant along the ring. No variations or variable distributions of the coupling parameter along the ring are studied or even considered. In this work a ring of N identical van der Pol oscillators, as is sketched in Figure 1, is considered. Three variable distributions: constant, gaussian and random of the coupling parameter along the ring are proposed. The collective behavior of the ring under the variable distributions is numerically studied and compared with that of a constant distribution. The influence of the oscillator constant on the ring behavior is also considered. Three values of the oscillator constant are assumed, namely 0.1, 0.5 and 1; the dynamic performance of the ring of coupled van der Pol oscillators goes from quasilinear to strongly nonlinear, depending on the kind of coupling distribution.

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

where and b_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, and.

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, S₁ and S₂ 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 [16] . The time step used for integration was 1 × 10⁻⁵, which is small enough to guarantee numerical stability and convergence. As in [15] , it was considered that N = 100 is a sufficiently large amount of oscillators for analysis, and that any increase in the amount of oscillators does not change the global behavior of the ring. The Initial conditions for position and velocity of oscillators were determined from [15]

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. Identical oscillators were considered in the numerical simulations. Three values of the oscillator constant, namely 0.1, 0.5 and 1, were considered in order to change the dynamic performance of the ring from quasilinear to strongly nonlinear. For N = 100 and d = 2, the following parameter values were defined: 1) for the gaussian distribution σ = 4, μ = 1; 2) for the random distribution λ = 2, S₁ = 0, S₂ = 2; 3) for the constant distribution c = 1. All these values satisfy the condition given by Equation (7).

Figure 2 and Figure 3 show the histograms corresponding to the gaussian and random distributions of the coupling parameter, respectively. The typical bell-shaped form of a gaussian data distribution is appreciated in Figure 2, whereas an irregular coupling parameter of a random distribution is observed in Figure 3. Both distributions satisfy the condition given by Equation (7). Phase portraits for the three coupling parameter distributions and the three considered values of the oscillators constant were constructed using long term time series of the first and the fifty oscillators. This pair of oscillators was selected because is one of the pairs most apart in the ring.

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.

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.