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 fλ(x)=λx(1-x) on [0,1] parameterized by λ∈(0,4]. We prove that if 0<λ≤2 then the number of forts does not increase under iteration and that if λ>2 then the number of forts is not bounded under iteration. Furthermore, we focus on the case of λ>2 and give for each k=1,…,7 some critical values of λ 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 n≥1, the nth iterate fn of a mapping f:E→E, where E is a nonempty set, is defined recursively by (1)fk=f∘fk-1,f0=id,where ∘ presents the composition of functions and id denote the identity mapping; that is, id(x)≡x for all x∈E. 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 f:I→I, where I is an interval, a point x0∈I is called a monotone point of f if f is strictly monotone in a neighborhood of x0; otherwise, x0 is called a nonmonotone point or simply a fort of f. 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 S(f) denote the set of all forts of f and let N(f) denote the cardinality of S(f). It is shown in [1, 2] that(2)0=Nf0≤Nf≤Nf2≤⋯≤Nfk≤Nfk+1≤⋯;that is, the number N(fk) of forts is nondecreasing as k is increasing. One can similarly prove that (2) also holds for functions defined on the whole R. 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 (3)fx=4x1-x,x∈0,1,for example. Computing derivatives of fi, i=1,…,5, and counting the number of real zeros with odd multiplicity for the derivatives fi′ (as done in [3]), we get N(f)=1, N(f2)=3, N(f3)=7, N(f4)=15, and N(f5)=31. From the increasing tendency, without continuing the tedious computation, we have the following question: Does N(fk) have a bound or approach infinity as k 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 ≥1 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:(4)fλx=λx1-x,x∈0,1,where λ∈(0,4] 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 N(f) with f 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 N(fλk)=N(fλ)=1 for all integer k≥2 if 0<λ≤2 and that N(fλk) approaches ∞ as k→∞ if 2<λ≤4. Furthermore, for various choices of λ∈(2,4], we compute the number N(fλk) for each fixed k=2,3,…,7 in Theorem 5.
2. Preliminaries
In general, for polynomial (5)fx=∑i=0naixi,where n≥2 and an≠0. N(fk), k=1,2,…, is decided by real zeros of the derivatives fk′.
Lemma 1 (see [<xref ref-type="bibr" rid="B9">3</xref>, Lemma 2.1]).
x0 is a fort of a real polynomial f if and only if x0 is a real zero of the derivative f′ of odd multiplicity. Moreover, N(f) is odd (resp., even) if the degree n of f is even (resp., odd).
Actually, the above lemma shows how a real zero of the derivative f′ can be a fort of f. Note that fk′=(f′∘fk-1)(f′∘fk-2)⋯f′∘ff′. Then the set of zeros of fk′ is a union of the set of zeros of f′∘fk-1 and the set of zeros of (f′∘fk-2)⋯f′∘ff′. Therefore, in order to know if fk have more forts than fk-1, we need to judge if f′∘fk-1 have real zeros different from (f′∘fk-2)⋯f′∘ff′’s with odd multiplicities; the following lemma gives the answer.
Lemma 2 (see [<xref ref-type="bibr" rid="B9">3</xref>, Lemma 2.5]).
Let G and P be real polynomials and G(x)≢0. Then the composition G∘P and the derivative P′ do not have a common real zero with odd multiplicity.
Taking G=f′ and P=fk-1, by Lemmas 1 and 2, we see that fk′ has more real zeros of odd multiplicities than fk-1′ if and only if f′∘fk-1 has real zeros with odd multiplicities. Hence, in the process of computing N(fk), we only need to find out the number of real zeros for f′∘fk-1 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 Discr(f) denote the discriminant matrix of the polynomial f, which is constructed by the Sylvester matrix of f and f′ as seen in [4, Definition 1]. For each τ=1,…,n, let Dτ(f) denote the determinant of its submatrix formed by the first 2i rows and the first 2i columns. The n-tuple (D1,D2,…,Dn) is called the discriminant sequence of f and the list (6)signD1,signD2,…,signDnis called the sign list of f, where sign(x) is defined to be equal to either 1 if x>0, 0 if x=0, or -1 if x<0. Given a sign list (s1,s2,…,sn) of f, we make a new list (ε1,ε2,…,εn), called the revised sign list of f, in the following regulation:
If (si,si+1,…,si+j) is a section of the given list such that si≠0, si+1=si+2=⋯=si+j-1=0, and si+j≠0, then replace the subsection (si+1,si+2,…,si+j-1) with (εi+1,εi+2,…,εi+j-1), where εi+r=(-1)[(r+1)/2]si for r=1,2,…,j-1; otherwise, let ετ=sτ.
The following lemma tells us how to find the number of distinct zeros by means of the revised sign list.
Lemma 3 (see [<xref ref-type="bibr" rid="B8">4</xref>, Theorem 1]).
Let f be a real polynomial and suppose that the number of the sign changes in the revised sign list of f is ν. Then the number of pairs of distinct conjugate complex zeros of f equals ν. Furthermore, if the number of nonvanishing members in the revised sign list is l, then F has l-2ν distinct real zeros.
Now, we are ready to apply the above lemmas to detail the process in computing N(fk), k=1,2,…. First of all, we compute fk and f′∘fk-1 and the discriminant sequence for f′∘fk-1. Secondly, under algebraic relations among coefficients ai’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 f′∘fk-1 through the sign lists. According to the revised sign lists, we find out the number of real zeros of f′∘fk-1 with odd multiplicities and finally obtain N(fk).
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 k=7 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 λ varies under iteration and then compute N(fλk) for fλ(x) up to iteration index k=7 with different choice of λ.
Theorem 4.
The logistic mappings fλ defined in (4) have N(fλk)=N(fλ)=1 for all integer k≥2 if and only if 0<λ≤2. Otherwise, N(fλk) approaches ∞ as k→∞.
Proof.
In order to obtain the condition for N(fλk)=N(fλ)=1, from the method mentioned in the end of Section 2, we need to compute N(fλ2). Simple computation shows that(7)fλ′∘fλ=2λx2-2λx+1.Then computing the discriminant sequence of (7), we have(8)λ2,λ3λ-2.From (8), if 0<λ<2, the revised sign list is (1,-1), implying that fλ′∘fλ has one pair of complex zeros, which leads to N(fλ2)=1; if λ=2, the revised sign list is (1,0), implying that fλ′∘fλ has a double real zero, which leads to N(fλ2)=1; therefore, N(fλ)=N(fλk)=1 if and only if 0<λ≤2.
Without loss of generality, we can turn the general form fλ(x)=-λx2+λx into (9)fλx=-λx-122+λ4.So the vertex of the parabola is (10)x0,y0≔12,λ4.If 2<λ≤4, then fλ and a diagonal line intersect at two points ξ1 and ξ2, where ξ1<x0<ξ2. Obviously, ξ1 and ξ2 are fixed points of fλ; ξ1=0 and ξ2=(λ-1)/λ and fλ is strictly increasing on the subinterval [ξ1,x0]. Thus, (11)ξ1<fλ-k-1x0<fλ-kx0<x0,∀k=1,2,…,limk→+∞fλ-kx0=ξ1.Since(12)Sfλk=Sfλk-1∪x∈0,1:fλk-1∈Sfλ,we get (13)Sfλk∖Sfλk-1=Sfλ∘fλk-1∖Sfλk-1=Sfλk-1∪x∈0,1:fλk-1x∈Sfλ∖Sfλk-1=x∈0,1:fλk-1x∈Sfλ∖Sfλk-1≠∅∀k∈N+,which implies that N(fλk) approaches ∞ as k→∞. This completes the proof.
Theorem 4 shows that the number N(fλk) approaches ∞ as k→∞ for each fixed λ∈(2,4]. It is also interesting to see for each fixed k how the number N(fλk) varies as the parameter λ changes in (2,4]. The following theorem shows the change of numbers N(fλk) as λ varies for each k=2,…,7 (but larger k 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.
N(fλ2)=3 for all λ∈(2,4] and N(fλ3)=5 and 7 for λ∈(2,λ3,2] and λ∈(λ3,2,4], respectively, where λ3,2≔5+1≈3.236067977. For more details, with the convenient notations λ1,0≔0 and λ2,1≔2, numbers N(fλk), k=1,2,…,7, are given in Table 1, where (14)λ4,3≈3.831874056,λ3,4≈3.498561699,λ5,5≈3.960270127,λ3,6≈3.627557530,λ3,7≈3.738914913,λ6,8≈3.990267047,λ4,9≈3.844568792,λ4,10≈3.905706470,λ4,11≈3.937536445,λ5,12≈3.977766422,λ7,13≈3.997583118.
The number of forts of fi, i=1,2,…,7.
Ranges of λ
Nfλ
Nfλ2
Nfλ3
Nfλ4
Nfλ5
Nfλ6
Nfλ7
λ1,0<λ≤λ2,1
1
1
1
1
1
1
1
λ2,1<λ≤λ3,2
1
3
5
7
9
11
13
λ3,2<λ≤λ3,4
1
3
7
13
21
31
43
λ3,4<λ≤λ3,6
1
3
7
13
23
37
57
λ3,6<λ≤λ3,7
1
3
7
13
23
37
59
λ3,7<λ≤λ4,3
1
3
7
13
23
39
65
λ4,3<λ≤λ4,9
1
3
7
15
29
53
93
λ4,9<λ≤λ4,10
1
3
7
15
29
53
95
λ4,10<λ≤λ4,11
1
3
7
15
29
55
101
λ4,11<λ≤λ5,5
1
3
7
15
29
55
103
λ5,5<λ≤λ5,12
1
3
7
15
31
61
117
λ5,12<λ≤λ6,8
1
3
7
15
31
61
119
λ6,8<λ≤λ7,13
1
3
7
15
31
63
125
λ7,13<λ≤4
1
3
7
15
31
63
127
Proof.
By (8), if 2<λ≤4, the revised sign list of fλ′∘fλ is (1,1), implying that it has two distinct real zeros, which shows that N(f2)=3.
Furthermore, in order to obtain N(fλ3), we compute(15)fλ′∘fλ2=2λ3x4-4λ3x3+2λ3x2+2λ2x2-2λ2x+1.As shown in Section 2, we give the discriminant sequence for (15):(16)λ6,λ11λ-2,λ14λ-22,λ15λ2-2λ-4λ-23.
Then, the revised sign list for (16) is
(1,1,1,1), if λ3,2<λ≤4, which implies that fλ′∘fλ2 has 4 distinct simple real zeros;
(1,1,1,0), if λ=λ3,2, which implies that fλ′∘fλ2 has 3 distinct real zeros, 2 of which are simple zeros and the remaining one is a double zero;
(1,1,1,-1), if λ2,1<λ<λ3,2, which implies that fλ′∘fλ2 has one pair of complex zeros and 2 distinct simple real zeros.
Here λ2,1=2 and λ3,2=5+1, as defined in the theorem. λ3,2 is the real zero of g(λ)≔λ2-2λ-4 in (2,4]. By Lemmas 1 and 2, N(fλ3)=5 if λ2,1<λ≤λ3,2 and N(fλ3)=7 if λ3,2<λ≤4.
Similarly, compute (17)fλ′∘fλ3=2λ7x8-8λ7x7+12λ7+4λ6x6+-8λ7-12λ6x5+2λ7+12λ6+2λ5+2λ4x4+-4λ6-4λ5-4λ4x3+2λ5+2λ4+2λ3x2-2λ3x+1.Then we obtain the discriminant sequence for fλ′∘fλ3: (18)λ14,λ27λ-2,λ38λ-22,λ47λ2-2λ-4λ-23,λ54λ2-2λ-4λ-24,λ59λλ-2λ2-2λ-4-16λ-25,λ62λ2-2λ-4λλ-2λ2-2λ-4-16λ-26,λ63λλ-2λ2-2λ-42-64λ2-2λ-42λ-27.Hence,
for λ2,1<λ<λ3,2, the revised sign list for fλ′∘fλ3 is (1,1,1,-1,-1,-1,1,-1), implying that fλ′∘fλ3 has 3 pairs of complex zeros and 2 distinct simple real zeros;
for λ=λ3,2, the revised sign list for fλ′∘fλ3 is (1,1,1,-1,-1,-1,0,0) and fλ′∘fλ3 can be factorized as (19)fλ′∘fλ3=164135+2932x2-32x+45+122-410-4x-12-146-25+62-210x-12+146-25+62-2104x-3+52-4x+1+52,implying that fλ′∘fλ3 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;
for λ3,2<λ<1+(3+25)1/2, where 1+(3+25)1/2 is the real zero of h(λ)≔λ(λ-2)(λ2-2λ-4)-16 in (λ3,2,4], the revised sign list for fλ′∘fλ3 is (1,1,1,1,1,-1,-1,-1), implying that fλ′∘fλ3 has a pair of complex zeros and 6 distinct simple real zeros;
for λ=1+(3+25)1/2, the revised sign list for f′∘f3 is (1,1,1,1,1,-1,-1,-1), implying that fλ′∘fλ3 has a pair of complex zeros and 6 distinct simple real zeros;
for 1+(3+25)1/2<λ<λ4,3, where (20)λ4,3≔332100+12692/3+11100+12691/3+8100+12691/31/2+1≈3.831874056is the real zero of p(λ)≔λ(λ-2)(λ2-2λ-4)2-64 in (1+(3+25)1/2,4], the revised sign list for f′∘f3 is (1,1,1,1,1,1,1,-1), implying that fλ′∘fλ3 has a pair of complex zeros and 6 distinct simple real zeros;
for λ=λ4,3, the revised sign list for fλ′∘fλ3 is (1,1,1,1,1,1,1,0), implying that fλ′∘fλ3 has 7 distinct real zeros, one of which is a double zero but the rest are all simple;
for λ4,3<λ≤4, the revised sign list for fλ′∘fλ3 is (1,1,1,1,1,1,1,1), implying that fλ′∘fλ3 has 8 distinct simple real zeros.
It follows that N(fλ4)=7 if λ2,1<λ≤λ3,2, N(fλ4)=13 if λ3,2<λ≤λ4,3, and N(fλ4)=15 if λ4,3<λ≤4.
We similarly compute fλ′∘fλ4 and obtain the discriminant sequence (21)λ30,λ59λ-2,λ86λ-22,λ111λ2-2λ-4λ-23,λ134λ2-2λ-4λ-24,λ155λ4-4λ3+8λ-16λ-25,λ174λ2-2λ-4λ4-4λ3+8λ-16λ-26,λ191λ6-6λ5+4λ4+24λ3-16λ2-32λ-64λ2-2λ-42λ-27,λ206λ6-6λ5+4λ4+24λ3-16λ2-32λ-64λ2-2λ-42λ-28,λ219λ2-2λ-4λ10-10λ9+28λ8+16λ7-160λ6+64λ5+192λ4+384λ3-256λ2-1024λ+1024λ-29,λ230λ4-4λ3+8λ-16λ10-10λ9+28λ8+16λ7-160λ6+64λ5+192λ4+384λ3-256λ2-1024λ+1024λ-210,λ239λ2-2λ-4λ8-8λ7+16λ6+16λ5-64λ4+128λ-256λ4-4λ3+8λ-162λ-211,λ246λ8-8λ7+16λ6+16λ5-64λ4+128λ-256λ2-2λ-42λ4-4λ3+8λ-162λ-212,λ251λ4-4λ3+8λ-16λ14-14λ13+72λ12-136λ11-144λ10+1088λ9-1600λ8-640λ7+4096λ6-512λ5-4096λ4-12288λ3+16384λ2+8192λ-16384λ2-2λ-43λ-213,λ254λ6-6λ5+4λ4+24λ3-16λ2-32λ-64λ14-14λ13+72λ12-136λ11-144λ10+1088λ9-1600λ8-640λ7+4096λ6-512λ5-4096λ4-12288λ3+16384λ2+8192λ-16384λ2-2λ-44λ-214,λ255λ12-12λ11+48λ10-40λ9-192λ8+384λ7+64λ6-1024λ4-512λ3+2048λ2+4096λ6-6λ5+4λ4+24λ3-16λ2-32λ-642λ2-2λ-45λ-215.Similar discussion gives N(fλ5) for various λ as shown in Table 1.
It is more complicated to compute discriminant sequences of fλ′∘fλ5 and fλ′∘fλ6 because the two discriminant sequences contain 32 and 64 components, respectively, and the biggest component in the discriminant sequence of fλ′∘fλ5 is a polynomial in the single variable λ of degree 1152 with 130 terms. Using a similar discussion as for fλ′∘fλk, k=1,2,3,4, we obtain parameter values λ3,6, λ3,7, λ6,8, λ4,9, λ4,10, λ4,11, λ5,12, and λ7,13 as well as the numbers N(fλ6) and N(fλ7) on intervals between them as shown in Table 1. This completes the proof.
Although we are not able to compute for all k those parameter values for changes of N(fλk) in Theorem 5, those data of Table 1 for k=1,2,…,7 show that the number N(fλk) can reach its maximum 2k-1 if λ>λk,l, l=0,1,2,3,5,8,13. We naturally have the following.
Question 1. Does N(fλk) reach the maximum 2k-1 for any integer k≥1?
Additionally, the well-known Feigenbaum sequence (see [6, 7]) is λ1=3, λ2≈3.449490, λ3≈3.544090, λ4≈3.564407, λ5≈3.568750, λ6=3.569690,…, 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 {λk,l} and the Feigenbaum sequence {λk}?
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.
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