Abstract
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.
1. Introduction
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 [1] 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.
The statements of Proposition 1 can be verified directly from the definitions, as well as the invariant properties of (7), and, thus, the proof is omitted here.
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.
Proposition 7. is a solution of (5) if and only if is a solution of (6).
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., [11]) 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 If , where , then and when ; if , where , then and when
The argument of Corollary 11 can be verified directly from Theorem 10 based on the facts that the slope of is characterized by and the slope of is characterized by , where and , respectively, and, thus, the proof is omitted.
In Theorem 10, we showed the asymptotic behaviors of generated by the Tracking Procedure. We illustrate the movements of when in Figures 13, 14, and 15 where the arrows indicate the moving directions of as
Next, we consider the asymptotic behaviors of with or
Theorem 12. Suppose and let such that Let be generated by the Tracking Procedure with input Then and is a degenerate parabola made up of and . If , then and for all ; otherwise, as well as are not on the same line. Furthermore, if , then and when
Proof. For the sake of convenience, we let and First of all, we see that which implies Furthermore,which leads to being two parallel lines. Consequently, if , then ; otherwise, and are two distinct parallel lines. Suppose Then In view of the Tracking Procedure, it is found that if , then which implies for all ; otherwise, which leads to for all Next, suppose By the Tracking Procedure, we can directly observe (see Figure 10) that for an arbitrary , if , then and vice versa. Now, we discuss the asymptotic behaviors of By the Tracking Procedure, we first observe thatand clearly, the converses are also true. Note that for any Furthermore, by the Tracking Procedure again, if , then for all otherwise, we see that for some ,In view of (44), since , , and is a positive constant, there is such that as well as and, thus, by the previous discussions, Accordingly, by the previous discussions again, when and since the slopes of and are , as The proof is complete.
Next, we consider the case where
Theorem 13. Suppose and let such that Let be generated by the Tracking Procedure with input Then we have the following.(i)Suppose Then and is a degenerate parabola made up of two parallel lines and If , then and such that for all ; otherwise, and are on the same line and when (ii)Suppose Then is a parabola and when
Proof. Suppose By (20), and, thus, Accordingly, is a degenerate parabola which is made up of two parallel lines and as defined in (24) and (25), respectively. More precisely, if , then which leads to ; if , then which implies and are two distinct parallel lines. By the Tracking Procedure, if , then for all ; if , then and are on the same line. Furthermore, if , then we have and by the Tracking Procedure again, , where Accordingly, when Next, suppose Then is parabola. By (7), for , we have such that and thus which leads toHence, when Note that, if , then is concave upward (see Figure 15), while if , then is concave downward. Accordingly, (ii) is true and this completes the proof.
Corollary 14. Suppose and let such that Let be generated by the Tracking Procedure with input Then we have the following.(i)Suppose Then and is a degenerate parabola with two parallel lines, and If , then such that and for all ; otherwise, such that as well as are two distinct parallel lines and(ii)Suppose Then is a parabola. If , then is concave upward and when ; otherwise, is concave downward and when
The results stated in Corollary 14 can be derived directly from Theorem 13 by observing the asymptotic behavior of the points
Let be generated by the Tracking Procedure with inputs and In Theorems 10, 12, and 13, we have shown the asymptotic behaviors of with Next, we consider the case where and Suppose Then the conic section is an ellipse which implies is both bounded above and below. Moreover, by the Principal Axes Theorem, the nondegenerate is an ellipse which is rotated by and translated by from a standard ellipse with major axis as well as minor axis Thus, where can be expressed parametrically bywhere the polar angle For the sake of convenience, we letwhere By the previous discussions, if , then where is a point ellipse and in view of (49), we may arrive at the following result.
Theorem 15. Suppose and such that Let be generated by the Tracking Procedure with input For any ,
Proof. In view of (49), we first consider the case where By (49), it follows thatas well asNote that By (7) and (52), we also have In view of (52), with respect to , it follows that which leads to The case where can be handled similarly. Hence, (50) is true and this completes the proof.
By Theorem 15, we can find the necessary and sufficient condition for a solution of (7) with to be a periodic sequence.
Corollary 16. Suppose and such that Let be generated by the Tracking Procedure with input Then is periodic if and only ifFurthermore, suppose such that is irreducible and If is even, then is -periodic; otherwise, is -periodic.
Proof. Note that if and , then by the Tracking Procedure and Proposition 1, which implies for all Suppose and By (49), we have By Theorem 15, for any , is constant such that Hence If is periodic with period , then we see that there is some such that Accordingly, which impliesOn the other hand, if (56) holds, then, by the previous discussions and (58), is periodic with period Note that by Propositions 2 and 4, if is -periodic with , then Next, suppose , where Since where are periods of , we can first restrict ourselves to , where Also, if , then and this allows us to further concentrate on Note that is bijective on Hence, for each such that , where , there is a unique pair such that , where and are coprime. In view of (58), it is easy to see that if is even, then is a period of ; if is odd, then is periodic with period In the former case, we claim that is -periodic. To this end, if is a period of , then, by the previous discussions, there is such that as well are coprime and Hence, which is not possible. The latter case can be handled similarly and this completes the proof.
In view of Corollary 16, we have found the necessary and sufficient condition for , which is generated by the Tracking Procedure with input such that and , to be periodic. Let , , or ; for instance, by (58), we take , , or , correspondingly, so that is -, -, or -periodic, respectively. For another example, by (58) again, if or , then is -periodic where or , respectively.
For the analysis of nonnegative or negative solutions of (7) with , we need one more result.
Theorem 17. Let Suppose mod where Then is dense in
In view of Theorem 17, let mod , where Then we see that is dense in (see the details in [12] in pages 280 and 281). Suppose Since , there exists a subsequence of such that , when , from which it follows that Accordingly, is dense in as desired.
Recall that by the Principal Axes Theorem, and are the principal axes of , where and , For and such that , one property of which is generated by the Tracking Procedure with input is revealed as follows.
Corollary 18. Suppose and such that Let be generated by the Tracking Procedure with input The set cannot lie entirely in one of the half planes divided by
Proof. Since and , is an ellipse. By Corollary 16 and Propositions 2 and 4, if , then is -periodic, where ; otherwise, is aperiodic. Suppose Then is -periodic, where such that First of all, by the Tracking Procedure, we see that moves on in a clockwise manner. Secondly, is symmetric with respect to Hence, without loss of generality, we suppose Since move on in a clockwise manner; by the Tracking Procedure, we can be sure that for some , Note that if , then we may investigate the orbit or so that or , respectively. Suppose Without loss of generality, we suppose and such that Then by Theorem 17, there is a subsequence such that as required and this completes the proof.
The previous discussions about the asymptotic behaviors of the solution of (7) help us to understand the periodicity of To further analyze the positivity of solutions of (7), we require one specific solution with , , and We note that Also, some properties of are useful for the ensuing analysis and they are as follows.
Proposition 19. For , the lines and intercept with at and , respectively. Furthermore, if and , then is a nondegenerate conic section and the lines and are tangents of at the points and , respectively.
Proof. First of all, we show that meets the line at To this end, note thatand, thus, is the unique solution to and Accordingly, the line intercept with at By the previous discussions, if , then is a degenerate parabola made up of the line; if and , then is a degenerate parabola made up of the line. Note that the is a parabola with Accordingly, if and , then is not a degenerate conic section which implies is tangent to and at as well as , respectively. The proof is complete.
We illustrate Proposition 19 by several plots of , namely, ellipses with plotted in Figure 16, ellipses with plotted in Figure 17, and hyperbola with plotted in Figure 18.
In the following section, we establish the relation between the nonnegative (or negative) solution of (7) and a special solution in the geometric setting.
4. A Special Positive Orbit
Let be the solution of (7) with By (12) and (13), we see that which implies for all Hence, it is found that and by (15), we haveNote that (61) allows us to obtain from plotting only. In the sequel, we focus on the the “positive” orbit of when the orbit is discussed. By the Tracking Procedure with , we can generate the positive orbit Next, we establish the connection between the nonnegative solutions and the specific solution
Let Let of (7) with for some Then by the definitions of and , we can be sure that(1) is a nonnegative solution if and only if ;(2) is a negative solution if and only if
Now, it is our objective to find and and in several special cases where and , we will show that and can be expressed as the intersections of certain half planes in
Proposition 20. Let and be solutions of (7) with and . Let and be solutions of (7) with , with , and with . Then we have
Proof. For the sake of convenience, we suppose First of all, if and , then which implies (63) is true when Next, we show that (63) holds for To this end, by (7), it follows thatas required. Note that in view of (66), as well as (67), we haveNext, we consider the case where Let such that Thus, we may reformulate (63) for , where asFurthermore, in view of (68), and which implies (63) is true and this completes the proof.
With defined in Proposition 20, we define the -variable functionand the setThen we construct a set which is an intersection of half planes:We claim thatLet be a solution of (7). Note that by Corollaries 3 and 5 and the definition of (72), we see that is nonempty which implies If , for some , then, by the definition, is nonnegative which implies for all . Since each is characterized by , in view of (72), we see that On the other hand, if , then is not nonnegative which implies there is some such that Accordingly, we have and by (72), it follows that which is not possible.
Theoretically, we can calculate for all to determine (73) explicitly. Suppose is an -periodic solution of (7) with given and . Then we only have to calculate , where , to determine (or ). Even though this is feasible for us, it is still difficult to illustrate explicitly on when is sufficiently large. In the sequel, we find , where and , by one specific Note that the cases where , with or , can be handled separately. By means of investigating the specific and where and , can be found in an explicit way. Recall that if and , then the -axis and the -axis are tangent to at as well as , respectively. Let be a solution of (7) with The tangent line of at is given by , whereIn addition, we also let be the line so that and are symmetric with respect to the line. Now, we establish the connection between and the set of intersection of half planes divided by
Theorem 21. Given , where and , then we have
Proof. First of all, we consider the case where and we show that , where To this end, we show the following results,are true for all Let be with Since and , by (70) and (74), it follows that (75) holds for Now, we suppose that for some such that (75) holds for and Then we have seen that for For , by (68), (76), and (7), it follows that as required. Next we consider and By (68), (77), and (7) again, in the former case, as desired; the latter case can be handled similarly. By induction, we see that for all , is true. Now, we consider the case where By the reflection and translation invariances, we transform in Proposition 20 and and by similarly applied arguments for , we see that , for all Note that which implies , where Accordingly, if , then ; if , then Finally, since , we see that (75) holds and this completes the proof.
Corollary 22. Let be a solution of (7) with , for some If , then there is some such that
Proof. By the definitions of in (75), if , then is on the tangent line at , for some By (75) again, without loss of generality, we may further suppose that , for some For the sake of convenience, we let and such that Since , by (74), we have Next, we show that To this end, we first haveWe apply and to (81) and by simplifying, it is found that which implies as desired. Since and are the and lines, respectively, by the previous discussions, there is some such that and this completes the proof.
In this section, we introduced the definitions of (or ) so that the necessary and sufficient conditions for a solution of (7) to be nonnegative (or negative) can be obtained, respectively. Also, in Theorem 21, we discussed the specific solution of (7) with and the intersection of the closed half planes generated by the tangent lines at , for all For the case where and , we will show that some can be expressed explicitly in terms of in the sequel. Note that by Proposition 7 and Theorem 8, some and with and can be derived from the results of the cases where and Hence, with all the results derived in the previous discussions together, we are ready to discuss the necessary and sufficient conditions for to be nonnegative (or negative).
5. Nonnegative and Negative Solutions when
Let be a solution of (7) defined by In the last section, we have completed the discussions about the periodicity of and the asymptotic behaviors of which is generated by the Tracking Procedure with input Note that , where is generated by the Tracking Procedure with input To facilitate the ensuing discussions, we define two sets and on whereNote that is defined by four closed half planes and is defined by two open as well as two closed half planes. Also note that by the definitions in (24) and (25), and are both independent of Now, we are able to apply these results to derive necessary and sufficient conditions for to be nonnegative (or negative).
Theorem 23. Let be a solution of (7) with Suppose , for some . Then is neither a nonnegative solution nor a negative solution.
Proof. Without loss of generality, we let Let such that Let and be generated by the Tracking Procedure with inputs and , respectively. Then is a hyperbola (or a degenerate one made up of two lines and ) such that For the sake of convenience, we let If , then, by Proposition 2, , for all Suppose By the previous discussions, we first consider the case where If , then and by Corollary 11, it is found that when ; if , then we take and we see that so that when as required. Next, suppose is a hyperbola. Then by Corollary 11 again, when Accordingly, if , then is neither a nonnegative nor a negative solution and this completes the proof.
Next, we consider the case where and This case is a little more complicated than the one described in Theorem 23, and we give the result of first.
Theorem 24. Let be a solution of (7) with and Suppose , for some such that Then is a nonnegative solution if and only if ; and is a negative solution if and only if
Proof. Note that in view of (82), is symmetric with respect to the line and and are symmetric with respect to the line. Let and be generated by the Tracking Procedure with inputs and , respectively. Here, , where , is a hyperbola (or a degenerate one made up of two lines and ). Note that is a cone with two rays and such that , where First of all, we show that To this end, we investigate and directly, and by Theorem 10, it is found that for all if and only if Accordingly, we have and the argument about can be handled similarly. Note that both and are symmetric with respect to the line. Hence, the cases where can be handled by the invariances of translation and reflection. The proof is complete.
Next, we discuss the case where and
Theorem 25. Let be a solution of (7) with and Suppose for some such that (i)Suppose Then is a nonnegative solution if and only if which is the intersection of the closed half planes defined in (75). Furthermore, is a negative solution if and only if (ii)Suppose . is a nonnegative solution if and only if ; and is a negative solution if and only if , where is the intersection of the closed half planes defined in (75).
Proof. Without loss of generality, we let and Since , is a hyperbola (or a degenerate one, two lines and ). Let and be generated by the Tracking Procedure with inputs and , respectively. Suppose Then By Theorem 21, it follows that as desired. Next, by Corollary 11, if , then , for all Conversely, if for all , then, by Theorem 10 and Corollary 11, Now, we consider the case where From the previous discussions, Proposition 7 and Theorem 8, we can be sure that Next, we claim that , where is given by (75). From the previous discussions and Corollary 22, we can be sure that is a positive solution if and only if Thus, from Proposition 7 and Theorem 8, it follows that and this completes the proof.
Corollary 26. Let be a solution of (7) with . is a nonnegative and periodic solution if and only if and for some ; is a negative and periodic solution if and only if and for some
From Proposition 2, Theorem 10, Corollary 11, and Theorems 23, 24, and 25, we can easily see that the statement of Corollary 26 holds.
This section shows that both and are nonempty and if is not a constant solution of (7) with , then is not periodic. Next, we will discuss and where
6. Nonnegative and Negative Solutions when
Let be a solution of (7) with and Let Recall that is on the conic section which is a parabola or degenerate parabola that is made up of either one line or two distinct parallel lines. In this section, we now discuss the nonnegative and negative solutions of (7) by investigating and
Theorem 27. Let be a solution of (7) with Suppose for some (i)Suppose Then cannot be a negative solution; furthermore, is a nonnegative solution if and only if(ii)Suppose Then cannot be a nonnegative solution; furthermore, is a negative solution if and only if
Proof. From the previous discussions, , where , is a degenerate parabola made up of two lines, and defined by (24) and (25), respectively. Moreover, if , then ; otherwise, and are two distinct parallel lines. We first consider the case where By Theorem 12 and the Tracking Procedure, if such that , then is periodic with period and which implies is a nonnegative solution. Conversely, if is a nonnegative solution, then and by Theorem 12, which implies Accordingly, we see that On the other hand, by Theorem 12 again, we can be sure that cannot be a nonpositive solution which implies is empty. Next, we suppose From the previous discussions and Proposition 7, if is a nonnegative solution then is a solution of (7) with which is not possible. Also, and are symmetric with respect to line and from Proposition 7 and Theorem 8, it follows that and this completes the proof.
Theorem 28. Let be a solution of (7) with Suppose for some .(i)Suppose Then is a nonnegative solution if and only if and which is the line; furthermore, is negative solution if and only if and (ii)Suppose Then cannot be a negative solution; furthermore, is a nonnegative solution if and only if which is the intersection of the closed half planes defined by (75).(iii)Suppose Then cannot be a nonnegative solution; furthermore, is a negative solution if and only if where is the intersection of the closed half planes defined by (75).
Proof. Recall that is the line which is one principal axis of , where Suppose By Theorem 13, if , then is neither a nonnegative nor a nonpositive solution. If , then is -periodic and we see that (i) holds by direct verification. Suppose Then by Theorem 21, we can be sure that as required and is depicted in Figure 19. For the case of (iii), from Proposition 7 and Theorem 8, it follows that (87) is true and this completes the proof.
In view of Theorem 28. we have shown that which is the intersection of the half planes defined by the tangent lines at all the points of on the parabola Figure 19 illustrates where the red-dashed curve is the parabola The discussions of the cases where are done and move forward to analyze the last cases where
7. Nonnegative and Negative Solutions when
In the previous discussions, we have analyzed the cases where and derived the necessary and sufficient conditions for a solution to be nonnegative (or negative). Let be a solution of (7). From our analyses of the case where , the periodicity of has no significant role in the analysis (albeit may be periodic with least period ); yet the periodicity of is of great importance when discussing the case where In Corollary 16, we illustrated the necessary and sufficient conditions for to be a periodic solution of (7) with and we will discuss the case where , , and in the sequel.
Theorem 29. Let be a solution of (7) with and Then cannot be a negative solution; furthermore, is a nonnegative solution if and only if is the null solution.
Proof. Let such that Let be generated by the Tracking Procedure with input Suppose From the previous discussions, which implies is a point ellipse. On the other hand, by Proposition 2, which leads to , for all and, thus, is a nonnegative solution with least period Now, we suppose Then is an ellipse and we show that is neither a nonnegative nor a negative solution. To this end, we consider and Suppose By Proposition 6, cannot be a nonpositive solution which implies is not negative. If is a nonnegative solution, then by Proposition 7, is a nonpositive solution which is a contradiction. The case where can be handled similarly and this completes the proof.
Next, we consider the corresponding results when and
Theorem 30. Let be a solution of (7) with and Suppose and let for some (i)Suppose Then cannot be a negative solution; furthermore, is a nonnegative solution if and only if which is the interior of the ellipse (ii)Suppose Then cannot be a nonnegative solution; furthermore, is a negative solution if and only if , where
Proof. Without loss of generality, we let Also, for the sake of convenience, let and Also, let be generated by the Tracking Procedure with input Then we consider two cases where and Here, if , then, by the previous discussions, is a point ellipse so that , for all Note that is tangent to lines and at and , respectively. Suppose We first show that cannot be a nonpositive solution. If , then, by Proposition 6, is not a nonpositive solution. If is nonpositive with , then, by the previous discussions, and by Corollary 18, there is some such that or Furthermore, if , then, by (7), Accordingly, is not a nonpositive solution when
Now, we showwhere Note that , where If (88) holds, then, by Theorem 8, we can be sure that , where Accordingly, for all . On the other hand, suppose (88) does not hold. Since , by Theorem 17, there is some such that (or some such that ) which implies is not nonnegative. Hence, (90) is true. Next, suppose First of all, by the previous discussions and Corollary 22, we see that is a positive solution if and only if Thus, from Proposition 7 and Theorem 8, it follows that as required.
The proof is complete.
Theorems 30 and 17 and Proposition 7 help us to analyze the necessary and sufficient conditions for an aperiodic solution of (7) to be nonnegative (or negative). For the case when is periodic, we have to resort to the help of the specific solution of (7).
Theorem 31. Let be a solution of (7) with and Suppose and , for some . Then is periodic and we have the following.(i)Suppose Then cannot be a negative solution; furthermore, is a nonnegative solution if and only if which is the intersection of the closed half planes defined by (75).(ii)Suppose Then cannot be a nonnegative solution; furthermore, is a negative solution if and only if where is the intersection of the closed half planes defined by (75).
Proof. Without loss of generality, we let Also, let be generated by the Tracking Procedure with input Suppose From the previous discussions, if , then is a nonnegative solution with least period If , then, by Corollary 16, is periodic and is an ellipse, where Also, by Theorem 21, we can be sure that as desired. Next, we show that is not a nonpositive solution. First of all, by Proposition 6, is not nonpositive when Secondly, suppose is a nonpositive solution of (7) with Then by Corollary 18, there is some such that or If , then, by (7), which is a contradiction. Accordingly, is not a nonpositive solution.
Suppose By Corollary 22, Proposition 7, and Theorem 8, it follows that (91) holds. Also, if is a nonnegative solution of (7) with , then, by Proposition 7, is a solution of (7) with which is not possible and this completes the proof.
Corollary 32. Let be a solution of (7) with and Suppose such that and is irreducible. Then is a closed polygon with sides which is circumscribed around the ellipse and is either or depending on whether is even or odd, respectively.
Proof. In view of Theorem 21, which is defined by the tangent lines of Furthermore, all the lines are tangent to at the points which are generated by a specific solution of (7) with Since , by Corollary 16, is -periodic where is dependent on and the parity of Note that Accordingly, we see that is defined by distinct lines. To see is a closed polygon with sides which is circumscribed around , it is sufficient to show that is closed. If , then is -periodic such that if , then, by (70), such that Accordingly, is closed, where Suppose To see is closed, it is sufficient to show that there is some such that Since , in view of Corollary 18, there is some such that as required and this completes the proof.
Corollary 32 gives an explicit description of , where , and two examples for the cases and are in Figures 20 and 21 respectively.
8. Conclusions
In the previous sections, we have analyzed the nonnegative (or negative) solution of (7) by investigating the pairs for all which are on a conic section (or degenerate one). Specifically, the orbit is the collection of for all which is a subset of a conic section and by the Tracking Procedure, is uniquely determined by giving By means of this geometric method, many qualitative properties of (7) can be obtained. In particular, we obtain the necessary and sufficient conditions for to be a nonnegative (or negative) solution.
Now, we summarize the necessary and sufficient conditions for to be a nonnegative (or negative) periodic solution of (7) as follows (recall that for a set , is the set on such that and are symmetric with respect to the line).(1)Suppose . is a nonnegative and periodic solution if and only if , and for some , Furthermore, such must satisfy for all (and hence is -periodic).(2)Suppose . is a nonnegative and periodic solution if and only if (where is irreducible and ) and for some , which, by Corollary 32, is a closed -side polygon where is either or depending on whether is even or odd, respectively. Furthermore, such must either satisfy for all (and hence is -periodic) or is -periodic.(3)Suppose . If , then is a nonnegative and periodic solution if and only if , and for some , and is on the line; if , then is a nonnegative and periodic solution if and only if , and for some , , and lie on the line. Furthermore, such is either -periodic if (and -periodic if ) when or -periodic when As for the negative periodic solutions, we have the following.(1)Suppose . is a negative and periodic solution if and only if and for some , Furthermore, such must satisfy for all (and hence is -periodic).(2)Suppose . is a negative and periodic solution if and only if (where is irreducible and ) and for some , , where by Corollary 32, is a closed -side polygon and is either or depending on whether is even or odd, respectively. Furthermore, such must either satisfy for all (i.e., is -periodic) or is -periodic.(3)Suppose If , then is a negative and periodic solution if and only if , and for some , and is on the line; if , then is a negative and periodic solution if and only if , and for some , and lie on the line. Furthermore, such is either -periodic if (and -periodic if ) when or -periodic when
With the help of the Tracking Procedure, the behaviors of the solution of (7) can be visualized (instead of investigating each term of by the method of general solutions). By means of our geometric method, we may save lots of calculation and most properties of which may be obtained by the general solutions are also available from our geometric discoveries.
In [1–3], we conducted complete analyses on the periodic solutions (7) with and By means of breaking a solution into two subsequences, the companion sequence and the error sequence, we successfully show that both the companion sequence and error sequence are periodic so that is periodic. Note that for , , and , we have , , and , respectively.
The nice thing here is that by means of our geometric method, not only can we track the orbit of a solution on easily but also we may obtain the conditions for to be nonnegative (or negative) quickly (especially for the case where and is periodic). By [1–3], we have observed the importance of nonnegative solutions of (7) with , , and , respectively. However, we also realize the difficulties of analyzing the solution of (7) with since the breaking of into companion and error sequences is not applicable for all the solutions anymore.
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 1 and 2) and these methods are also expected to be applied in other piecewise smooth dynamic systems.
Conflicts of Interest
The authors declare that there are no conflicts of interest whatsoever regarding the publication of this paper.
Acknowledgments
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.