# Constructing Multi-Branches Complete Chaotic Maps That Preserve Specified Invariant Density

Authors: Weihong Huang
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

A general formulation for multi-branches complete chaotic maps that preserve a specified invariant density is provided and an implication relationship among this class of maps is revealed. Such relationship
helps to derive a whole family of complete chaotic maps that preserve not only the same invariant measure
but also the same degree of chaos in terms of Lyapunov component.


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## Body

## 1. Introduction

Much  progress has been achieved in exploring the probabilistic characteristics of nonlinear dynamics from different aspects over the last ten years. While the most general formulations  of unimodal (two-branches) complete chaotic maps that preserve general form of invariant densities have been provided in [1, 2], the characterization of such maps for some special forms of invariant densities are analyzed in [2, 3]. Interests to the unimodal chaotic maps with special analytical structural forms are explored in [4–6] and the relevant statistical dynamics are investigated in [7–9]. However, most these studies are limited to complete chaotic maps with two branches only until recently in [10], where the methodologies developed in [1, 2] are generalized to the cases with multi-branches so that most general formulations are derived. The current research intends to continue the same endeavor through deriving more compact and more simplified formulations and to reveal an important relation among them. The results we obtain do not only provide a practical approach to construct multi-branches complete chaotic maps for arbitrary specified analytical expressible invariant densities but also help in understanding the intrinsic characteristics of such maps.


## 2. Complete Chaotic Maps

LetJ=[0,1] be the unitinterval. Let F:J→J be a nonsingular map. “Nonsigularity” means that m(F-1(X))=0 whenever m(X)=0 for a measurable set X⊂J, where m denotes the Lebesgue measure (all mathematical functions in this paper are assumed to be piecewise continuous and differentiable).Definition 2.1.
A nonsingular mapF:J→J is said to be complete if(i)
there exists a partitionAF={ai}i=0r, with 0=a0<a1<⋯<ar-1<ar=1, of the unit interval J such that for each integer i (i=1,2,…,r), the restriction Fi of F to the open interval (ai-1,ai) is a monotonically continuous function that can be extended to the closed interval Ji=[ai-1,ai];(ii)
Fi(ai-1)∈{0,1},Fi(ai)∈{0,1}, and Fi(ai-1)+Fi(ai)=1, for i=1,2,…,r.We will callFi extended to Ji as ith branch of F, si=̇Fi(ai)-Fi(ai-1) as its sign index to indicate the slopping direction of F on ith subinterval Ji, i=1,…,r, and S={si}i=1r as the sign pattern of F={Fi}i=1r. In particular, we will let  I+=̇{i∣si>0} and I-=̇{i∣si<0} be the positive and negative index sets that indicate the set of branches with positive and negative slopes, respectively. For instance, the chaotic map defined by(2.1)F(x)={F1(x)=3x,x∈[0,13),F2(x)=43-3x2x∈[13,23),F3(x)=9x2-45,x∈[23,1],
is complete with AF={0,1/3,2/3,1}, S={1,-1,1},I+={1,3}, and I-={2}.Definition 2.2.
A complete mapF is said to be complete chaotic (fully chaotic) if it is(i)
ergodic with respect to the Lebesgue measure,(ii)
chaotic in the probabilistic sense, that is, an absolutely continuous invariant densityφ is preserved. In other words, for
(2.2)μ(x)=∫0xφ(x)dx,x∈J.We have
(2.3)μ(X)=μ(F-1(X))
for any set of X⊂J with m(X)≠0, where m denotes the Lebesgue measure.As illustrated in Figure1, for a complete chaotic map F defined on J=⋃i=1rJi, for any x∈J, F has exact r preimages, denoted as Fi-1(x), i=1,2,…,r. Therefore, for any x∈J1, F1(x) has r preimages in each subinterval Ji, i≥1, which are linked with x through the branching functions ωi:J1→Ji, defined byFigure 1
Illustration of branching function.(2.4)ωi(x)≐Fi-1(F1(x)),∀x∈J1,i=1,2,…,r.Depending on the sign indices ofF1 and Fi, the ith branching function ωi is a monotonic function that satisfies the following:(i)
ifFi′F1′>0, then ωi′(x)>0 for almost all x∈J1 with ωi(a0)=ai-1 and ωi(a1)=ai;(ii)
ifFi′F1′<0, then ωi′(x)<0 for almost all x∈J1 with ωi(a0)=ai and ωi(a1)=ai-1.In particular, we haveω1=x and(2.5)F1(x)=μ-1(∑j∈I+[μ(ωj(x))-μ(aj-1)]+∑j∈I-[μ(aj)-μ(ωj(x))]).Instead of working on the branching functionωi:J1→Ji, it is more convenient to manipulate some functions from J→J directly. For this purpose, we can define a special class of monotonic increasing one-to-one function on the unitinterval asDefinition 2.3.
AC1 function g:J→J is said to be a generating function if g meets the following requirements:(i)
g(0)=0 and g(1)=1;(ii)
g is continuous;(iii)
g′(x)>0 for almost all x∈J.Let𝒢 be the set of generating functions. Then for a given branching function ωi:J1→Ji,  i=1,2,…,r, there exists a unique gi∈𝒢 such that(2.6)ωi(x)={ai-1+(ai-ai-1)·gi(xa1),ifFi′F1′>0,ai-(ai-ai-1)·gi(xa1),ifFi′F1′<0.In particular, forg1(x)=x, we have ω1(x)=x.Therefore, for a complete chaotic mapF, there exists a unique set of generating functions G={gi}i=1r, where g1=x and gi∈𝒢, for i≠1. Utilizing the generating functions (which all share a common domain J) overcomes the difficulty in specifying branching functions as in (2.5) (which map J1 to other subintervals) and hence has advantage in simplifying the general formulation. It has been shown by current author in [10] that the following.Corollary 2.4.
A complete chaotic mapF={Fi}i=1r preserves a normalized absolutely continuous invariant measure μ if and only if there exists a set of generating functions G={gi}i=1r, with g1(x)=x and gi∈𝒢, for i=2,3,…,r, so that each and every branch of F can be expressed as
(2.7)Fi(x)={μ-1(∑j=1rβj·gj(gi-1(μ(x)-αi-1βi))),ifF1′>0,Fi′>0,μ-1(∑j=1rβj·gj(gi-1(αi-μ(x)βi))),ifF1′>0,Fi′<0,μ-1(1-∑j=1rβj·gj(gi-1(αi-μ(x)βi))),ifF1′<0,Fi′>0,μ-1(1-∑j=1rβj·gj(gi-1(μ(x)-αi-1βi))),ifF1′<0,Fi′<0,
where αi≐μ(ai) and βi≐αi-αi-1, for i=1,2,…,r.Example 2.5.
Upward and downward maps.
A complete map is calledupward map if si>0 for all i. We denote it as F(u)={Fi(u)}i=1r. Then (2.7) is simplified to
(2.8)Fi(u)(x)=μ-1(∑j=1rβj·gj(gi-1(μ(x)-αi-1βi))),i=1,2,…,r.
A complete map is calleddownward map if si<0 for all i. We denote it as F(d)={Fi(d)}i=1r. Then (2.7) is simplified to
(2.9)Fi(d)(x)=μ-1(1-∑j=1rβj·gj(gi-1(μ(x)-αi-1βi))),i=1,2,…,r.


## 3. Implication Relationship

According to (2.7), the mapping at each section, Fi, i=2,3,…,r, depends on the sign index of the first branch only. Therefore, if the sign index of kth (k>1) branch is changed from sk to sk′, with formulation (2.7), only kth branch Fk needs to be changed. This observation reveals an important implication relationship among the set of complete maps that not only preserve an identical invariant measures but also share some analytical similarities.Theorem 3.1.
LetF={F1,…,Fk,…,Fr} and F̃={F1,…,F̃k,…,Fr} be two complete maps defined on an identical partition AF={ai}i=0r of J with the sign patterns differing only in kth branch (that is, sk=-s̃k). Then F and F̃ preserve an identical invariant measure μ if and only if
(3.1)F̃k(x)=Fk(ϖk(x)),
where
(3.2)ϖk(x)=μ-1(μ(ak)+μ(ak-1)-μ(x)),
is referred to as kth branch implication function.Proof.
Without loss of generality, we assume firstk≠1.
The “if” part follows directly from Corollary2.4, which suggests that, two maps F and F̃, which have identical functional forms for all branches other than kth one (k>1), can be generated with a same set of generating functions {gi}i=1r. Consider the case in which s1=s̃1>0, sk>0 and s̃k<0, the different functional forms resulted from
(3.3)Fk(x)=μ-1(∑i=1r(μ(ai)-μ(ai-1))·gi(gk-1(μ(x)-μ(ak-1)μ(ak)-μ(ak-1)))),F̃k(x)=μ-1(∑i=1r(μ(ai)-μ(ai-1))·gi(gk-1(μ(ak)-μ(x)μ(ak)-μ(ak-1)))).
To have (3.1) hold true, the monotonicity of gi, i=1,2,…,r, implies that the following identity must be held:
(3.4)μ(ϖk(x))-μ(ak-1)μ(ak)-μ(ak-1)=μ(ak)-μ(x)μ(ak)-μ(ak-1),
which yields (3.2). The cases associated with s1=s̃1<0 can be similarly verified.
The “only if” part can be seen more straightforward from the Frobenius-perron equation (2.5).
Ifsk>0 and s̃k<0, we must have
(3.5)F1(x)=μ-1(∑i∈I+-{k}[μ(ωi(x))-μ(ai-1)]+μ(ωk(x))-μ(ak-1)+∑i∈I-[μ(ai)-μ(ωi(x))]),F̃1(x)=μ-1(∑i∈Ĩ+[μ(ω̃i(x))-μ(ai-1)]+μ(ak)-μ(ω̃k(x))+∑i∈Ĩ--{k}[μ(ai)-μ(ω̃i(x))]).
The assumption thatFi(x)=F̃i(x) for all i≠k implies that ωi=ω̃i for all i≠k and that
(3.6)I+-{k}=Ĩ+,I-=Ĩ--{k},
which in turn suggest that
(3.7)μ(ωk(x))-μ(ak-1)=μ(ak)-μ(ω̃k(x)),
or, equivalently,
(3.8)ω̃k(x)=μ-1(μ(ak)+μ(ak-1)-μ(ωk(x))).
Therefore, we have
(3.9)ω̃k-1(x)=ωk-1(μ-1(μ(ak)+μ(ak-1)-μ(x))).
The facts ofFk(x)=F1(ωk-1(x)) and F̃k(x)=F1(ω̃k-1(x)), together with (3.9), thus imply (3.2) in order for the implication relationship (3.1) to hold.
The case in whichk=1 can be proved with the same principle.Remark 3.2.
Due to the fact thatϖk:Jk→Jk is a monotonically decreasing diagonal-symmetric function in the sense of
(3.10)ϖk(x)=ϖk-1(x),
we also have
(3.11)Fk(x)=F̃k(ϖk(x)),
which suggests that Fk and F̃k are mutually implied with a common implication function ϖk.Definition 3.3 (implication relation).
Two complete chaotic maps,F={Fi}i=1r and F̃={F̃i}i=1r defined on an identical partition AF={ai}i=0r of J, are said to be mutually implied to each other if there exists a subset of indices Im≐{i1,i2,…,im}, with 1≤m≤r and ik∈{1,2,…,r} for k=1,2,…,m, so that
(3.12)Fk(x)={F̃k(ϖk(x)),k∈Im,F̃k(x),k∉Im,
where ϖk is the kth branch implication function defined in (3.2).Figure2 demonstrates the mutual-implication family (mutually implication relation was first revealed in ([9]) for two branches complete chaotic maps).Figure 2
Mutually implied family.By repeatedly applying Theorem3.1, we arrive at the followingTheorem 3.4.
Two mutually implied complete chaotic maps preserve an identical invariant measure.An important characteristic of the implication relationship is that it is invariant with smooth conjugations; that is, ifFa and Fb are two complete maps mutually implied by a subset of implication functions ϖk, for all k∈Im, then for any one-to-one piecewise C1 function h, F̂a≐h-1∘Fa∘h, and F̂b≐h-1∘Fb∘h are mutually implied by ϖ̂k, k∈Im, where ϖ̂k≐h-1(ϖk(h(x))), for all k∈Im. An application of mutual implication relationship in constructing an opposite for a two-segmental complete map is offered in [9].The implication relationship (the set of implication functions) is uniquely determined by the partitionAF={ai}i=0r and the invariant measure preserved μ, but is independent of the generating functions G. For any complete map F={Fi}i=1r preserving an invariant measure μ, there exists a unique implied family of F, denoted as Ψ(F), in which each and every map can be derived from the other through replacing some branches with their implied counterparts. On the other hand, for each and every sign pattern S={si}i=1r, there associates one and only one map in an implied family. The size of this family is thus always 2r. Moreover, for any θ={θi}i=1r∈Ψ(F), either θi=Fi or θi=Fi∘ϖi, for i=1,2,…,r. And hence, the family Ψ(F) is uniquely specified by 2r mappings and will be denoted as Ψ(F)=[Fiu,Fid]i=1r, or,(3.13)Ψ(F)=[F1u,F1u∘ϖ1⋮⋮Fiu,Fiu∘ϖi⋮⋮Fru,Fru∘ϖr]=[F1d∘ϖ1,F1d⋮⋮Fid∘ϖi,Fid⋮⋮Frd∘ϖr,Frd].Due to the uniqueness of implied family, just substituting (3.2) into (2.8) yields the following.Theorem 3.5.
A complete mapF={Fi}i=1r preserves a normalized absolutely continuous invariant measure μ if and only if there exists a set of generating functions G={gi}i=1r, with g1(x)=x and gi∈𝒢, for i=2,3,…,r, so that each and every branch of F can be expressed as
(3.14)Fi(x)={μ-1(∑j=1rβj·gj(gi-1(μ(x)-αi-1βi))),ifFi′>0,μ-1(∑j=1rβj·gj(gi-1(αi-μ(x)βi))),ifFi′<0,i=1,2,…,r,
where αi≐μ(ai) and βi≐αi-αi-1, for i=1,2,…,r.We further show that any member of an implication family shares the same degree of chaos.Theorem 3.6.
All members of an implication familyΨ(F) have an identical Lyapunov exponent.Proof.
Due to the facts that two conjugated complete maps have an identical Lyapunov exponent and that the implication relationship is invariant of topological conjugation, we can proceed with the complete maps that preserve the uniform invariant density (μ(x)≡x).
For a complete mapF defined on a partition AF={ai}i=0r of J, its Lyapunov exponent, denoted as Λ(F), is given by Λ(F)=∑i=1rλi(Fi), where
(3.15)λi(Fi)≐∫ai-1ailn(|Fi′(x)|)φ(x)dx,∀i∈{1,2,…,r}.
It is thus sufficient to prove that the identityλi(Fi)=λi(F̃i) holds true for all i∈{1,2,…,r}, where F̃i=Fi(ϖi).
Indeed, sinceμ(x)≡x (and hence φ(x)=1), we have ϖi(x)=ai+ai-1-x so that
(3.16)λi(Fi)=∫ai-1ailn(|Fi′(ϖi)ϖi′|)dx=∫ai-1ailn(|Fi′(ai+ai-1-x)|)dx=λi(Fi),
which completes the proof.Forgi(x)=x, i=1,2,…,r,  (3.14) simplifies to(3.17)Fi(x)={μ-1(μ(x)-αi-1βi),ifFi′>0,μ-1(αi-μ(x)βi),ifFi′<0,i=1,2,…,r,
which is nothing but a topological conjugate of complete piecewise linear chaotic map defined by(3.18)Ti(x)={x-αi-1αi-αi-1,ifTi′>0,αi-xαi-αi-1,ifTi′<0,i=1,2,…,r,
with AT={αi}i=1r.It is well known thatT={Ti}i=1r preserves the Lebesgue measure. This observation reveals the fact that the class of complete chaotic maps that are conjugated to the complete piecewise linear chaotic maps consists of only a subset of all complete chaotic maps that preserve the same absolute continuous invariant measures.For the special case in whichgi(x)=x for all i, it is immediately observed that(3.19)λ(F)=-∑i=1rβilnβi.Finally, we just point out that (3.14) is not the unique way to constructing complete chaotic maps with given μ,AF, and G. For instance, we may substitute (3.2) into (2.9) alternatively so as to obtain another implied family given by F̂={F̂i}i=1r, where(3.20)F̂i(x)={μ-1(1-∑j=1rβj·gj(gi-1(αi-μ(x)βi))),ifFi′>0,μ-1(1-∑j=1rβj·gj(gi-1(μ(x)-αi-1βi))),ifFi′<0,i=1,2,…,r.F̂ defined in (3.20) is identical to F defined in (3.14) if and only if gi(x)=x for all i. However, it is easy to see that F̂i is nothing but to implement formulation (3.14) with a set of new generating functions Ĝ={ĝi}i=1r with ĝi(x)=1-gi(1-x) for all i. Therefore, we will concentrate on the formulation (3.14) only. Moreover, we can further show that two families Ψ(F) and Ψ(F̂) have an identical Lyapunov exponent.


## 4. Numerical Examples

Example 4.1.
Consider a 3-to-1 complete chaotic map withμ(x)=2arcsin(x)/π (φ(x)=1/(πx(1-x))), A={0,1/4,3/4,1}, and G={x,x,x}.
Withα0=0, α1=μ(1/4)=1/3, α2=μ(3/4)=2/3, and α3=1, we have βi=αi-αi-1=1/3 for all i.
For this simple generating functions, we can apply directly formulation (3.17) to get
(4.1)F1(u)(x)=F2(d)(x)=F3(u)(x)=f(x)≐x(4x-3)2,F1(d)(x)=F2(u)(x)=F3(d)(x)=g(x)≐(1-x)(1-4x)2.
Moreover,λ(F)=-∑i=13βilnβi=ln3.
The three branching functions are given by
(4.2)ϖ1(x)=μ-1(α1+α0-μ(x))=14+12x-123x(1-x),ϖ2(x)=μ-1(α2+α1-μ(x))=1-x,ϖ3(x)=μ-1(α3+α2-μ(x))=14+12x+123x(1-x).
It is interesting to verify that
(4.3)f(ϖi(x))=g(x),g(ϖ3(x))=f(x)
for i=1,2,3.
The above conclusions are illustrated in Figure3. In particular, Figure 3(d) provides a typical realized frequency from f(x)=x(4x-3)2.Illustration of Example4.1.
(a)
Invariant measureμ and its inverse(b)
Branching functions(c)
F resulted from G={x,x,x}(d)
ϕ(x) and frequency of f(x)Example 4.2.
Consider a 3-to-1 complete chaotic map withμ(x)=x2(φ(x)=2x) and AF={0,1/3,2/3,1}.
Withα0=0, α1=1/9, α2=4/9, and α3=1, we have β1=1/9, β2=1/3, and β3=5/9, respectively.
The branching functions turn out to be
(4.4)ϖ1(x)=μ-1(α1+α0-μ(x))=19-x2,ϖ2(x)=μ-1(α2+α1-μ(x))=59-x2,ϖ3(x)=μ-1(α3+α2-μ(x))=139-x2.
(i) Consider first the set of generating functions,G(1)≐{x,x,x}, then (3.17) yields
(4.5)F1(u)(x)=3x,F1(d)(x)=F1(u)(ϖ1(x))=1-9x2,F2(u)(x)=3x2-13,F2(d)(x)=F2(u)(ϖ2(x))=1312-27x2,F3(u)(x)=95x2-45,F3(d)(x)=F3(u)(ϖ3(x))=355-5x2.
It is straightforward to verify that λ(F)≃0.93689=-∑i=1rβilnβi.
(ii) Alternatively, if we consider the set of generation functionsG(2)≐{x,x2,x}, (3.14) yields
(4.6)F̅1(u)(x)=μ-1(β1·g1(g1-1(μ(x)-α0β1))+β2·g2(g1-1(μ(x)-α0β1))+β3·g3(g1-1(μ(x)-α0β1)))=x6+27x2,F̅2(u)(x)=μ-1(β1·g1(g2-1(μ(x)-α1β2))+β2·g2(g2-1(μ(x)-α1β2))+β3·g3(g2-1(μ(x)-α1β2)))=1323(9x2-1)+9x2-1,F̅3(u)(x)=μ-1(β1·g1(g3-1(μ(x)-α2β3))+β2·g2(g3-1(μ(x)-α2β3))+β3·g3(g3-1(μ(x)-α2β3)))=15(9x2-4)(3x2+2).
Accordingly,  we have
(4.7)F̅1(d)(x)=F1(u)(ϖ1(x))=(1-3x2)(1-9x2),F̅2(d)(x)=F2(u)(ϖ2(x))=134-9x2+23(4-9x2),F̅3(d)(x)=F3(u)(ϖ3(x))=153(1-x2)(19-9x2).
Straight calculation reveals thatλ(F)=0.89123.
The above conclusions are illustrated in Figure4.Illustration of Example4.2.
(a)
Invariant measureμ and its inverse(b)
Branching functions(c)
F resulted from G={x,x,x}(d)
F̅ resulted from G={x,x2,x}



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