1. Introduction

Oscillatory behavior is ubiquitous in all physical systems in different disciplines ranging from biology and chemistry to engineering and physics, especially in electronic and optical systems. In radio frequency and lightwave communication systems, oscillators are used for transformation signals and channel selection. Oscillators are also present in all digital electronic systems, which require a time reference. Coupled oscillators are oscillators connected in such a way that energy can be transferred between them. Since about 1960, mathematical biologists have been studying simplified models of coupled oscillators that retain the essence of their biological prototypes: pacemaker cells in the heart, insulin-secreting cells in the pancreas; and neural networks in the brain and spinal cord that control such rhythmic behavior as breathing, running and chewing. Ring of coupled oscillators is cascaded combination of delay stages, connected in a close loop chain and it has a number of applications in communication systems, especially in anatomic – organs such as the heart, intestine and ureter consist of many cellular oscillators coupled together. An ideal oscillator would provide a perfect time reference, i.e., a periodic signal. However all physical oscillators are corrupted by undesired perturbation noise. Hence signals, generated by physical oscillators, are not perfectly periodic.

Mathematical basis of coupled oscillators has been established in [1], [2] and mutual synchronization of a large number of oscillators has been investigated in [3]. Very interesting applications can be found in [4] and [5]. In [6] the authors have investigated a ring array of van der Pol oscillators and clarified each mode structure. Their results are based on the method of equivalent linearization of the nonlinear terms using Krylov-Bogoliubov method. An active element is characterized by a simple symmetric cubic nonlinearity, that is, with V-I characteristic

.

A ring oscillator is described by the following integro-differential system:

(1)

The usually accepted approach (cf. [6]) is to exclude the current functions after differentiation and to obtain a second order system of van der Pol differential equations

Here we consider the original integro-differential system (1) and exclude current functions without differentiation. So we reach the following first order (instead of second order) integro-differential system:

or

.

In view of the boundary conditions for the ring connection

.

the above system yields the following system of N equations for N unknown functions.

(2)

satisfying the initial conditions

.

We define an operator acting on suitable function space and its fixed point is a periodic solution of the above system. The advantages of our method is its simpler technique and obtaining of successive approximations beginning with simple initial functions.

We formulate a periodic problem: to find a periodic solution on of the system (2). To solve the periodic problem we use the method from [7].

By we mean the space of all differentiable -periodic functions with continuous derivatives.

First we introduce the sets (assuming ):

The set turns out into a complete metric space with respect to the metric:

where

It is easy to see .

Introduce the operator B as a n-tuple

defined on every interval by the expressions ;

, ,

where

2. Preliminary Results

Now we formulate some useful preliminary assertions for our investigation.

Lemma 1. If then are -periodic functions.

Proof: Indeed, we have

Lemma 1 is thus proved.

Lemma 2. If then are -periodic functions.

The proof is straightforward based on the previous Lemma 1.

Lemma 3. For every

it follows

Proof:

We notice that in view of Lemma 2 the functions are -periodic. Let us define the functions

and

and rewrite them in the form

,.

Changing the variable we obtain

Consequently

Lemma 3 is thus proved.

Lemma 3 shows that operator function

is - periodic one.

Lemma 4. The initial value problem (2) has a solution iff the operator B has a fixed point , that is,

Proof: Let be a periodic solution of (2). Then integrating (2) we have:

But

.

Therefore is equivalent to

and then the solution of (2) is a fixed point of B.

Conversely, let be a fixed point B, that is, .

Therefore by definition of B we obtain

,

or

We show

which implies

.

Indeed, put and then we have

Obviously as that implies .

Therefore the operator equation

becomes

Differentiating the last equalities we obtain (2).

Lemma 4 is thus proved.

3. Main Result

Theorem 1. Let the following assumptions be valid:

1) ;

2) For sufficiently large

.

Then there exists a unique -periodic solution of (2).

Proof: First we show that

Indeed

We have

.

Using the estimates from Lemma 3 we obtain

and

Therefore for sufficiently large we have

It remains to show that is a contractive operator on .

Indeed,

Then

and

It follows

.

Thus

For the derivative we obtain

.

We have

Thus

The above inequalities imply

and then

where .

Theorem 1 is thus proved.

4. Numerical Example

The inequalities guaranteeing an existence-uniqueness of a periodic solution are

.

Let us consider the case when . We assume the active elements have characteristics . Then

;

If for instance , and . Then we have to choose and the second inequality can be disregarded. Then that implies and hence

.

5. Conclusions

Successive approximations to the solution can be obtained beginning with the following initial functions . It is easy to calculate the next approximation from the right-hand side of the above system:

Principal Remark. Once we have found the voltages in view of (1) we solve the equations with respect to :

,

where are already known functions.

The currents are periodic functions too:

.