Abstract
Using the theory of complete discrimination system and the computer algebra system MAPLE V.17, we compute the number of forts for the logistic mapping on parameterized by . We prove that if then the number of forts does not increase under iteration and that if then the number of forts is not bounded under iteration. Furthermore, we focus on the case of and give for each some critical values of for the change of numbers of forts.
1. Introduction
Iteration is the act of repeating a process with the aim of approaching a desired goal, target, or result. In mathematical sense, for a fixed integer , the th iterate of a mapping , where is a nonempty set, is defined recursively by where presents the composition of functions and denote the identity mapping; that is, for all . Being indispensable in the computer era, iteration brings many interesting but difficult problems to mathematics. Only from one-dimensional case, one can simply notice that an iterate of a linear function of any order remains linear but the degree of a polynomial may increase drastically, which shows that the nonlinear complexity is amplified by iteration.
Actually, in the one-dimensional case, the complexity of nonlinear functions is related to nonmonotonicity. For a continuous nonmonotonic self-mapping , where is an interval, a point is called a monotone point of if is strictly monotone in a neighborhood of ; otherwise, is called a nonmonotone point or simply a fort of . Obviously, a linear function does not have a fort generically. In 1980s, Zhang and Yang (see [1]) investigated the number of forts for a class of nonmonotonic functions called strictly piecewise monotone functions and simply PM functions, which are self-mapping on a compact interval and have at most finitely many forts each. Let denote the set of all forts of and let denote the cardinality of . It is shown in [1, 2] thatthat is, the number of forts is nondecreasing as is increasing. One can similarly prove that (2) also holds for functions defined on the whole . It is easy to find nonlinear functions whose number of forts, regarded as the damagers of monotonicity, increases rapidly under iteration. Consider the quadratic function for example. Computing derivatives of , , and counting the number of real zeros with odd multiplicity for the derivatives (as done in [3]), we get , , , , and . From the increasing tendency, without continuing the tedious computation, we have the following question: Does have a bound or approach infinity as tends to ? How can we compute the number of forts for nonmonotonic functions?
Polynomials, a special class of nonmonotonic functions, possess the advantage that each fort of a polynomial of degree is either a peak or a valley although the notion is not true in general. In this paper, we focus on the family of logistic mappings:where is a parameter, which is one of the simplest polynomial mappings, and a typical example used to show chaos and some complicated dynamics, for those problems. First of all, we introduce the theory of complete discrimination system (see [4]) and then use it to give a method for the computation of with polynomial in Section 2. In Section 3, we employ the method in the computer algebra system MAPLE V.17 for the family of logistic mappings. We prove in Theorem 4 that for all integer if and that approaches as if . Furthermore, for various choices of , we compute the number for each fixed in Theorem 5.
2. Preliminaries
In general, for polynomial where and . , , is decided by real zeros of the derivatives .
Lemma 1 (see [3, Lemma 2.1]). is a fort of a real polynomial if and only if is a real zero of the derivative of odd multiplicity. Moreover, is odd (resp., even) if the degree of is even (resp., odd).
Actually, the above lemma shows how a real zero of the derivative can be a fort of . Note that . Then the set of zeros of is a union of the set of zeros of and the set of zeros of . Therefore, in order to know if have more forts than , we need to judge if have real zeros different from ’s with odd multiplicities; the following lemma gives the answer.
Lemma 2 (see [3, Lemma 2.5]). Let and be real polynomials and . Then the composition and the derivative do not have a common real zero with odd multiplicity.
Taking and , by Lemmas 1 and 2, we see that has more real zeros of odd multiplicities than if and only if has real zeros with odd multiplicities. Hence, in the process of computing , we only need to find out the number of real zeros for with odd multiplicities. For this reason, we first introduce some notations of the theory of complete discrimination system (see [4, 5]) which will lead us to solve this problem.
Discriminants of polynomials are useful in determining the number of zeros for polynomials. Let denote the discriminant matrix of the polynomial , which is constructed by the Sylvester matrix of and as seen in [4, Definition 1]. For each , let denote the determinant of its submatrix formed by the first rows and the first columns. The -tuple is called the discriminant sequence of and the list is called the sign list of , where is defined to be equal to either if , if , or if . Given a sign list of , we make a new list , called the revised sign list of , in the following regulation: If is a section of the given list such that , , and , then replace the subsection with , where for ; otherwise, let .
The following lemma tells us how to find the number of distinct zeros by means of the revised sign list.
Lemma 3 (see [4, Theorem 1]). Let be a real polynomial and suppose that the number of the sign changes in the revised sign list of is . Then the number of pairs of distinct conjugate complex zeros of equals . Furthermore, if the number of nonvanishing members in the revised sign list is , then has distinct real zeros.
Now, we are ready to apply the above lemmas to detail the process in computing , . First of all, we compute and and the discriminant sequence for . Secondly, under algebraic relations among coefficients ’s in the discriminant sequence, discuss the sign of each component of the discriminant sequence and list the sign lists. Then, compute the revised sign lists for through the sign lists. According to the revised sign lists, we find out the number of real zeros of with odd multiplicities and finally obtain .
The above idea can be implemented in the computer algebra system MAPLE V.17, and we will use this method for the logistic mappings up to iteration index in next section.
3. Number of Forts
In this section, we first draw a conclusion for the logistic mappings which describe that the numbers of forts can be preserved or approach as varies under iteration and then compute for up to iteration index with different choice of .
Theorem 4. The logistic mappings defined in (4) have for all integer if and only if . Otherwise, approaches as .
Proof. In order to obtain the condition for , from the method mentioned in the end of Section 2, we need to compute . Simple computation shows thatThen computing the discriminant sequence of (7), we haveFrom (8), if , the revised sign list is , implying that has one pair of complex zeros, which leads to ; if , the revised sign list is , implying that has a double real zero, which leads to ; therefore, if and only if .
Without loss of generality, we can turn the general form into So the vertex of the parabola is If , then and a diagonal line intersect at two points and , where . Obviously, and are fixed points of ; and and is strictly increasing on the subinterval . Thus, Sincewe get which implies that approaches as . This completes the proof.
Theorem 4 shows that the number approaches as for each fixed . It is also interesting to see for each fixed how the number varies as the parameter changes in . The following theorem shows the change of numbers as varies for each (but larger can be considered if the computational capacity of our computer is better). It gives a sequence of parameter values at which new forts arise.
Theorem 5. for all and and for and , respectively, where . For more details, with the convenient notations and , numbers , , are given in Table 1, where
Proof. By (8), if , the revised sign list of is , implying that it has two distinct real zeros, which shows that .
Furthermore, in order to obtain , we computeAs shown in Section 2, we give the discriminant sequence for (15):Then, the revised sign list for (16) is (i), if , which implies that has 4 distinct simple real zeros;(ii), if , which implies that has 3 distinct real zeros, 2 of which are simple zeros and the remaining one is a double zero;(iii), if , which implies that has one pair of complex zeros and 2 distinct simple real zeros.Here and , as defined in the theorem. is the real zero of in . By Lemmas 1 and 2, if and if .
Similarly, compute Then we obtain the discriminant sequence for : Hence, (i)for , the revised sign list for is , implying that has 3 pairs of complex zeros and 2 distinct simple real zeros;(ii)for , the revised sign list for is and can be factorized as implying that has a pair of complex zeros and 4 distinct real zeros and two of the 4 distinct real zeros are simple and the rest are both double zeros;(iii)for , where is the real zero of in , the revised sign list for is , implying that has a pair of complex zeros and 6 distinct simple real zeros;(iv)for , the revised sign list for is , implying that has a pair of complex zeros and 6 distinct simple real zeros;(v)for , where is the real zero of in , the revised sign list for is , implying that has a pair of complex zeros and 6 distinct simple real zeros;(vi)for , the revised sign list for is , implying that has 7 distinct real zeros, one of which is a double zero but the rest are all simple;(vii)for , the revised sign list for is , implying that has 8 distinct simple real zeros.It follows that if , if , and if .
We similarly compute and obtain the discriminant sequence Similar discussion gives for various as shown in Table 1.
It is more complicated to compute discriminant sequences of and because the two discriminant sequences contain and components, respectively, and the biggest component in the discriminant sequence of is a polynomial in the single variable of degree with 130 terms. Using a similar discussion as for , , we obtain parameter values , , , , , , , and as well as the numbers and on intervals between them as shown in Table 1. This completes the proof.
Although we are not able to compute for all those parameter values for changes of in Theorem 5, those data of Table 1 for show that the number can reach its maximum if , . We naturally have the following.
Question 1. Does reach the maximum for any integer ?
Additionally, the well-known Feigenbaum sequence (see [6, 7]) is , , , , , , at each of which a period-doubling bifurcation happens in the logistic mapping. This suggests the following.
Question 2. Is there any relation between our sequence and the Feigenbaum sequence ?
A related work can be found from [8], but the question is not answered yet.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work is supported by FP7-PEOPLE-2012-IRSES-316338.