- freely available
Mathematics 2018, 6(4), 58; https://doi.org/10.3390/math6040058
2. Studying the Sigmoid Function
2.1. Introducing the Considered Sigmoid
- a continuous increasing function,
- the limit of as x approaches negative infinity is 0,
- the limit of as x approaches infinity is 1,
- and ,
2.2. About and Parameters
2.3. Fixed Point of
- If , then on , and we have and , which leads to the variations depicted in Figure 5.Consequently, if , then has one unique solution, i.e., has one unique fixed point.
- If , then there exist two real numbers and such that the variation table of Figure 6 is satisfied for H.
- . As , we deduce that . Additionally, and , then . As a conclusion, .As , H is increasing on , H decreases on , and the limit of H of x as x approaches equals , we can deduce that has an unique solution on , on a point . In particular, has one unique fixed point in this interval.
- We show that , and so has no fixed point on ., then . So:Furthermore, , and is increasing on (as on this interval). As , we can deduce that . Consequently, . Let . Since , thus is negative if and only if . We now investigate the sign of j.As , it is sufficient to study j on the interval ., so j is strictly decreasing on I. The discriminant of the quadratic equation being , the latter has two solutions , which are equal when (i.e., when ). Note that only may belong to I, and that the latter is equal to = = . We successively haveIf belongs to , then is positive and is negative.Let us show that . As , we then have , and so . Finally, . As stated at the beginning of the proof, since is a root of H, is a fixed point for and thanks to (7)., and so , as j is decreasing with . To put it in a nutshell, .
- The unique fixed-point can be found as follows: start with an arbitrary element in and define a sequence by , then .
- For , as , we can deduce that is Lipschitz continuous, with a Lipschitz constant equal to a. As a well-known consequence, the convergence of the aforementioned sequence is at least geometric, with a common ratio of a. For the same conclusion holds but for a constant .
3. The 1-Dimensional Situation ()
3.1. The Discrete Dynamical System under Consideration
3.2. A Fundamental Case:
- exists and is unique,
- is such that ,
- the convergence speed is geometric, of ratio equal to a ().
3.3. General 1-D Case:
3.3.1. Fixed Points of When
- If , then is negative and then over , and after the computation of the limits of as x approaches , we can deduce the table of variations depicted in Figure 12 (and which is independent of m).Consequently, if , then has one unique solution, i.e., has one unique fixed point.
- If , then we obtain a curve similar to the fundamental case for , see Figure 13.As previously, we remark that , so .As a consequence,Again as previously, , is increasing over , and , so . Consequently, since , then , and has the opposite sign of where .Let us study the quadratic polynomial on . Its discriminant is equal to , and it has the sign table described in Figure 14.
- If , then . As , we can conclude that . So . For the same reasons, is negative and has thus only one root, which belongs to . Thus has only one fixed point in this interval.
- If , then
- If , then . So is positive outside its two roots and negative otherwise. . The largest one is outside . Let us first focus on . Since then and thus . From , one can thus deduce that . Thus, is in and . In other words,
- If is large (i.e., is close to 0), is close to . The left root of H, would be s.t. and . In such a case and there is only one fixed point in .
- If is close to 4, is close to . The left root of H, would be s.t. and . In such a case and there is one fixed point in , one in and one in .
- if , and so . There is one fixed point in and one in .
3.3.2. Fixed Points of When
4. Conclusions and Future Work
Conflicts of Interest
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