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Simple dynamic systems representing time varying states of interconnected neurons may exhibit extremely complex behaviors when bifurcation parameters are switched from one set of values to another. In this paper, motivated by simulation results, we examine the steady states of one such system with bang-bang control and two real parameters. We found that nonnegative and negative periodic states are of special interests since these states are solutions of linear nonhomogeneous three-term recurrence relations. Although the standard approach to analyse such recurrence relations is the method of finding the general solutions by means of variation of parameters, we find novel alternate geometric methods that offer the tracking of solution trajectories in the plane. By means of this geometric approach, we are then able, without much tedious computation, to completely characterize the nonnegative and negative periodic solutions in terms of the bifurcation parameters.

Simple dynamic systems representing time varying states of interconnected compartments or “neurons” may exhibit extremely complex behaviors when bifurcation parameters are switched from one set of values to another. An example has been given in several of our previous studies [

When the real parameters

In this note, we discuss the steady state solutions of the above dynamic system (i.e., those that satisfy

Yet, for other values of

In this paper, we will devote ourselves to explaining the behaviors of these nonnegative (or negative) and periodic solutions of (

First, a real sequence

That said, in this paper, we will handle our equation from a novel approach and the crux of which is based on representing each pair of two consecutive terms

The orbit of

The orbit of

Several sets will be encountered in the ensuing discussions and we denote them as follows:

Before we enter into discussions on the necessary and sufficient conditions for a solution of (

In view of these invariances, to study (

Let

The statements of Proposition

Periods and the prime periods of sequences are defined as usual. Furthermore, let

Let

Suppose

Suppose

Suppose

The proof is complete.

Let

The results can be obtained from Proposition

In view of Proposition

Let

Suppose

Let

Suppose

In this paper, since we are interested in the periodicity of the solutions of (

Let

Suppose

Suppose

Suppose that

Next, we discuss nonconstant solutions of (

We will need the following quadratic function:

If

If

If

The hyperbolae with

The hyperbolae with

The ellipses with

The parabolae with

Unless indicated by the adjective “degenerate,” a conic section is meant to be nondegenerate. The conic sections

To discuss the principal axes further, we define the two variable functions:

Let

The conic sections will be of great help in the analysis of solutions of (

Let

Without loss of generality, we let

Let

Plot

Take

Let

Draw a vertical line

The line

If

Note that by the Principal Axes Theorem, the level curve

The orbit on

The orbit on

The orbit on

The orbit on

The orbit on

The orbit on

As we have shown previously, the

We first consider the case where

Suppose

Suppose

Suppose

Note that

In view of Theorem

Suppose

Suppose

and if

If

For the sake of convenience, we let

Suppose

Suppose

The proof is complete.

Suppose

The argument of Corollary

In Theorem

The asymptotic behavior of

The asymptotic behavior of

The asymptotic behavior of

Next, we consider the asymptotic behaviors of

Suppose

For the sake of convenience, we let

Next, we consider the case where

Suppose

Suppose

Suppose

Suppose

Suppose

Suppose

Suppose

The results stated in Corollary

Let

Suppose

In view of (

By Theorem

Suppose

Note that if

In view of Corollary

For the analysis of nonnegative or negative solutions of (

Let

In view of Theorem

Recall that by the Principal Axes Theorem,

Suppose

Since

The previous discussions about the asymptotic behaviors of the solution of (

For

First of all, we show that

We illustrate Proposition

The

The

The plot of

In the following section, we establish the relation between the nonnegative (or negative) solution of (

Let

Let

Now, it is our objective to find

Let

For the sake of convenience, we suppose

With

Theoretically, we can calculate

Given

First of all, we consider the case where

Let

By the definitions of

In this section, we introduced the definitions of

Let

Let

Without loss of generality, we let

Next, we consider the case where

Let

Note that in view of (

Next, we discuss the case where

Let

Suppose

Suppose

Without loss of generality, we let

Let

From Proposition

This section shows that both

Let

Let

Suppose

Suppose

From the previous discussions,

Let

Suppose

Suppose

Suppose

where

Recall that

The set

In view of Theorem

In the previous discussions, we have analyzed the cases where

Let

Let

Next, we consider the corresponding results when

Let

Suppose

which is the interior of the ellipse

Suppose

Without loss of generality, we let

Now, we show

The proof is complete.

Theorems

Let

Suppose

Suppose

where

Without loss of generality, we let

Suppose

Let

In view of Theorem

Corollary

The set

The set

In the previous sections, we have analyzed the nonnegative (or negative) solution

Now, we summarize the necessary and sufficient conditions for

Suppose

Suppose

(where

Suppose

Suppose

Suppose

(where

Suppose

With the help of the

In [

The nice thing here is that by means of our geometric method, not only can we track the orbit of a solution

Finally, in this paper we have only touched upon one class of steady state solutions of our original neural network. Much work has to be done before the dynamic behaviors of its solutions can be fully understood. Our contributions here, however, are that elementary geometric methods can be used to explain some of the complex phenomena obtained from simulations (see, e.g., Figures

The authors declare that there are no conflicts of interest whatsoever regarding the publication of this paper.

The third author is partially supported by the Ministry of Science and Technology, R.O.C. under Grant MOST 104-2221-E-007-061-MY3 which also complies with the declaration of conflicts of interest stated above.