Research Article | Open Access
Nonnegative Periodic Solutions of a Three-Term Recurrence Relation Depending on Two Real Parameters
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  and a slightly modified model of which is described as follows. Let neurons be placed on the vertices of a regular -gon and let the time dependent state values of the neurons be denoted by , for Suppose the rate of change is determined by the resultant effect of a constant multiple of , plus a magnified on-off (or bang bang) state dependent control mechanism as well as its two near neighbors , in the form Then assuming “uniformity” among the neuron interactions, we havewhere is identified with while with for compatibility reasons. Here is the on-off Heaviside step function defined by
When the real parameters and are switched among different real values, it is expected (and verified by simulations) that complex dynamic behaviors will be more than abundant. Some of these behaviors can be explained (see [1–3]) but some not (at least to the best of our knowledge). Similar models of piecewise constant dynamic systems which exhibit similar behaviors with parameters can be found in many recent investigations; see for examples [4–10] and the references therein.
In this note, we discuss the steady state solutions of the above dynamic system (i.e., those that satisfy for all and for each , or ). Then we will face the existence problem of periodic solutions of the following three-term recurrence relation:In [1–3], we are lucky to obtain complete information about the periodic solutions of (4) when and , , or by breaking the solutions into two sequences, a companion and an error (more specifically, for a solution of (4) with and , the companion sequence is which is an even integral sequence so that , where is the greatest even integer that is less than or equal to , and the error sequence is defined by ).
Yet, for other values of and , simulations show complex periodic behaviors beyond our present comprehension, except when the periodic solutions are also “nonnegative” (or “negative”). Such exceptional results, when examined more closely, can be explained.
In this paper, we will devote ourselves to explaining the behaviors of these nonnegative (or negative) and periodic solutions of (7).
First, a real sequence is a solution of (4) if it renders (4) into an identity after substitution. It is nonnegative (or negative) if all its terms are nonnegative (respectively negative). If is a nonnegative solution of (4), then clearly satisfieswhile if is a negative solution of (4), thenHence it seems that we are back to the usual nonhomogeneous linear second-order difference equations studied in elementary theory of difference equations. Since in (5) and (6), we can further restrict ourselves in the sequel to the following difference equation:with arbitrary By means of the techniques of general solutions plus the method of variation of parameters, it may be argued that the existence of nonnegative (respectively negative) periodic solutions can be handled completely. Such an assertion may be true in theory, but the general solution here is - as well as -dependent and hence actual attempts lead to many complications.
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 of a solution as a point in the plane and invent a geometric method to track the movements of these points. First, the totality of such pairs is called the orbit of this solution. More precisely, let be a solution of (4). We define the orbit of byand the “positive” orbit byAs examples, let and let and be solutions of (4) with or , respectively. The orbits with and with are depicted in Figures 1 and 2, respectively. These figures clearly suggest that is “-periodic,” that is “-periodic,” and that there are accompanying “distinctive” features which can be exploited further.
Several sets will be encountered in the ensuing discussions and we denote them as follows:(i) is the entire set of real numbers.(ii) the set of integers(iii) the set of positive integers.(iv) the set of nonnegative integers.(v) the set of rational numbers.
Before we enter into discussions on the necessary and sufficient conditions for a solution of (7) to be nonnegative and periodic, we take a note that any solution of our three-term equation (7) is uniquely determined by two of its consecutive terms. Furthermore, in view of the fact that (7) can be written asas well asit is both “symmetric” (or “reflection invariant”) and “translation invariant.” More precisely, let be the solution of (7) that satisfies and Then the solution of (7) determined by and , called the reflection of and denoted by , will satisfywhile the solution of (7) determined by and for any , called the -translation of and denoted by , will satisfy
In view of these invariances, to study (7), we may simply concentrate our attention on the positive orbits of its solutions! Indeed, let be the solution of (7) with We investigate by observing the positive orbit defined by On the other hand, we analyze by studying the positive orbit Note that for all , where , , where In the sequel, if no doubt arises, we frequently let be a solution of (7) which is defined by and we denote so that
Proposition 1. Let be a solution of (7) defined by and be the “positive” orbit of Then is a period of if and only if Furthermore, if , then is aperiodic and if , then is neither a nonnegative nor a negative solution.
2. Some Basic Results of (7)
Periods and the prime periods of sequences are defined as usual. Furthermore, let be a nonnegative solution of (7). Note that for some distinct , if , then is a period of and the least one among the periods is the least or prime period of (we also say that is -periodic). A constant solution of (7) is -periodic. We first find the necessary and sufficient conditions for to be a constant solution.
Proposition 2. Let be a solution of (7) with (i)Suppose Then is a constant solution if and only if (ii)Suppose Then is a constant solution if and only if and
Proof. Suppose If is a constant solution of (7), then and so that Accordingly, The converse can be verified by direct iteration. Next, suppose If is a constant solution of (7), then and by (7)which leads to Conversely, if and , then by iteration is clearly a constant solution.
The proof is complete.
Corollary 3. Let be a solution of (7) with Let where Then is -periodic. is a nonnegative solution if and only if ; and is a negative solution if and only if
The results can be obtained from Proposition 2 directly and, hence, we omit the proof.
In view of Proposition 2, it is easy to see that the orbit of a constant solution is just the set containing the only point
Proposition 4. Let be a solution of (5) with Then is -periodic if and only if , , and
Proof. Suppose is -periodic. Then as well as and by (5), and which lead us to and , respectively. Accordingly, we have and Note that since is -periodic, we can be sure that Hence, if , then which is a contradiction; if , then and The converse may be checked by direct substitution into (5) and this completes the proof.
Corollary 5. Let be a solution of (7) with and Suppose Then is periodic with period . is nonnegative if and only if ; and is negative if and only if
Proof. Suppose First of all, if , then, by Proposition 2, is -periodic; otherwise, by Proposition 4, is -periodic. Hence, is periodic with period If is a nonnegative solution, then which implies Furthermore, since is periodic with period , we can see that for all If , then is satisfied and is a nonnegative solution. The case where is a negative solution can be handled similarly and this completes the proof.
In this paper, since we are interested in the periodicity of the solutions of (4) which are also nonnegative (or negative), the following results will be useful.
Proposition 6. Let be a nonnull solution of (7). If and , then cannot be a nonpositive solution; if and , then cannot be a nonnegative solution.
Proof. Suppose and If is nonpositive, then and Hence , which implies is null, contrary to our assumption.
Suppose and If is nonnegative, then and so that A contradiction is arrived and this completes the proof.
Proof. Suppose that is a solution of (5) with By (5), we have Let such that Accordingly, and we see thatwhich satisfies (6). By induction, it follows that is a solution of (6) and this completes the proof.
Next, we discuss nonconstant solutions of (5) (although -periodic solutions are discussed in Proposition 4, we may include them in the following discussions). The behavior of solution of (7) with least period (or ) was quite easy to analyze directly, yet it is difficult to conduct similar analyses for with larger least period (or is aperiodic) by similar manner. Hence, we will investigate the behavior of from a new perspective.
3. The Orbits of Solutions of (7)
We will need the following quadratic function:and its -level curve is defined bySince is a quadratic function, its level curves are plane conic sections. The properties of these conic sections are well known. First, we see that the line is one of the principal axes of and if ; then is the center of and the other principal axis is the line. In particular (see, e.g., ) can be classified by the value of its discriminant and the values as well as :(1)If , then is a hyperbola (see the green and red curves in Figures 3 and 4) when or a degenerate hyperbola consisting of two intersecting lines when (see the blue lines in Figures 3 and 4).(2)If , then is an ellipse (see Figure 5) when or a degenerate point ellipse when .(3)If , then is a parabola when or a degenerate parabola consisting of a single or two distinct parallel lines when (see the green and blue lines in Figure 6).
Unless indicated by the adjective “degenerate,” a conic section is meant to be nondegenerate. The conic sections are symmetric with respect to the line. This property is a simple consequence of the fact that is a symmetric function in and Hence, the line is one of the principal axes of the conic sections.
To discuss the principal axes further, we define the two variable functions:and the associated line Also, for , we letand Furthermore, let , where and for We define the following two variable functions:as well aswith and Note that in view of (24) and (25), as well as are independent from when
Let Define Then and are symmetric with respect to the line, while and are symmetric with respect to the line. The boundary of the plane set will be denoted by
The conic sections will be of great help in the analysis of solutions of (4). The most significant result is the following.
Theorem 8. Let be a solution of (5). If for some , then Furthermore, if , where , then
Proof. Without loss of generality, we let and Let Then By (21), we see that and by (7), it follows so that which leads to By induction, for any , which implies Next, we consider , where Let such that and Since is symmetric with respect to the line, and by the previous discussions, we can be sure that which implies , where Accordingly, and by (15), it follows that Furthermore, by the definitions, if , where , then and since itself is symmetric with respect to the line, it is clear as desired. The proof is complete.
Let be a solution of (5). Then is uniquely determined by two consecutive terms and Although we can calculate the next term directly, the conic sections allow us to easily “track” the movements of the points from to . Indeed, by Theorem 8, we see that , where Then by means of this conic section, we may plot the corresponding orbit as follows.
Tracking Procedure. Input (1)Plot where on the plane.(2)Take and start at the point (3)Let be the point on which is symmetric to with respect to the line.(4)Draw a vertical line through (5)The line and the conic section can intersect at and possibly at another point If , then let ; otherwise, take (6)If , then stop; otherwise, let and go to step (3).
Note that by the Principal Axes Theorem, the level curve is defined by setting , which is a polynomial of degree . The conic section has one principal on the line which implies any vertical line through meets at least once and at most twice through two points or If , then is on the line and this can be verified directly by (7). In view of the Tracking Procedure, we can generate each as by initializing and it is not difficult to see that the above algorithm can yield the “positive” orbit ; on the other hand, the “negative” orbit of can be plotted by the “positive” orbit which is generated by the Tracking Procedure with input In Figures 7 and 8, we illustrate the orbits of on ellipses; Figures 9 and 10 depict the orbits on a parabola and a degenerate parabola correspondingly; and in Figures 11 and 12, we show the orbits on a degenerate hyperbola and a hyperbola, respectively.
As we have shown previously, the -level curve of is a conic section and it can be classified into hyperbola, ellipse, or parabola (or the degenerate curves). Let be a solution of (7) defined by By means of the Tracking Procedure, we can easily analyze the asymptotic behavior of when In the sequel, we will discuss the sequence which is generated by the Tracking Procedure with input depending on , , and
We first consider the case where By the previous discussions, the points generated by the Tracking Procedure with input move on where and is a hyperbola (or a degenerate one made up of two lines). Since is a hyperbola, we first locate the asymptotes of Note that is the center of In view of (21), we rewrite the equation as and suppose are the oblique asymptotes of Then we have As for , since lines intersect at , we apply to the lines and obtain Accordingly, we see that and can be expressed by as well as , respectively. For , one relation between and as well as defined in (24) and (25), respectively, is revealed byas well asMoreover, with the oblique asymptotes and , we may illustrate the asymptotic behaviors of where
Theorem 9. Suppose and let such that Let be generated by the Tracking Procedure with input Then is a hyperbola (or a degenerate one made up of two lines and ) and we have the following.(i)Suppose is a degenerate hyperbola. If , then for all ; if , then for all .(ii)Suppose is a hyperbola. If , then and are sitting on different branches for all ; if , then and lie on the same branch for all .
Proof. Note that and By Theorem 8 and (7), we see that Suppose or In the former case, and by (31), as well as (32), which implies ; in the latter case, by similar arguments, we may see that Note that if , then and by the Tracking Procedure, which implies for all Suppose Then is a hyperbola and is on one branch of By the Tracking Procedure, we may see that if , then “jumps” to another branch (see Figure 3), while if , then will stay on the same branch (see Figure 4). By induction, we may see that (ii) holds. This completes our proof.
In view of Theorem 9, we may follow the movements of generated by the Tracking Procedure when This enables us to see the asymptotic behavior of when in Theorem 10. Before introducing Theorem 10, we denote the distance between two points and on by and the distance from a point to the line by .
Theorem 10. Suppose and let such that Let be generated by the Tracking Procedure with Then is a hyperbola (or a degenerate one made up of two lines and ) and we have the following.(i)Suppose If , then for all ; if and , then and if and , then(ii)If and is a hyperbola, then ; and if and is a hyperbola, then
Proof. For the sake of convenience, we let and so that and Let By Proposition 2, if , then is a constant solution of (7) which implies Now, we suppose Since , is a hyperbola (or a degenerate one). Hence, we discuss the two cases where is on a degenerate hyperbola or a hyperbola.
Suppose and Then by Theorem 9, and are on the same line. In view of (24) and (25), we let , , , and For the sake of convenience, we also letSince is symmetric with respect to the line, we see that if , then ; if , then Suppose If and , then and so that and , respectively. Also, by the symmetry of with respect to the line, we can be sure that so thatOn the other hand, we calculateand by (35) and (36), as well as (37), it follows that By induction, we have , where and Accordingly, (34) holds. The case where and can be handled similarly with , where and By similar arguments, (33) is also true.
Suppose We first consider the case where Then and by (24)By induction, it follows that , where Consequently, case (ii) is true when and by similar arguments, we also see that holds with , where and
The proof is complete.
Corollary 11. Suppose and let such that Let be generated by the Tracking Procedure with input Then is a hyperbola (or a degenerate one made up of two lines and ). Suppose