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On complete chaotic maps with tent-map-like structures
Chaos, Solitons and Fractals 24 (2005) 287–299 www.elsevier.com/locate/chaos
On complete chaotic maps with tent-map-like structures Weihong Huang Nanyang Technological University, Nanyang Avenue, Nanyang 639798, Singapore Accepted 14 September 2004 Communicated by Prof. S. Yousefi
Abstract A unimodal map f : [0, 1] ! [0, 1] is said to be complete chaotic if it is both ergodic and chaotic in a probabilistic sense so as to preserve an absolutely continuous invariant measure. The sufficient conditions are provided to construct complete chaotic maps with the tent-map-like structures, that is, f(x) = 1 j1 2g(x)j, where g is an one-to-one onto map defined on [0, 1]. The simplicity and analytical characteristics of such chaotic maps simplify the calculations of various statistical properties of chaotic dynamics. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Due to the sensitive dependence on initial conditions, observation and measurement accuracy, and computation errors, predicting chaotic trajectories turns out to be an impossible and meaningless task. In many circumstances, to get a global picture of a chaotic system, long-term statistical properties such as long-run averages, observed frequencies, correlation indices have to be resorted to. The probabilistic or statistical approach is proved to be an effective approach in studying nonlinear chaotic dynamics [1]. The key element of the statistical approach is the invariant measure that describes the ‘‘steady state’’ of a nonlinear system, which is indispensable for the calculation of relevant experimental observables such as time correlations and their power spectra. Unfortunately, except for some simple maps such as the tent map and the logistic map, even though the existence of the absolute continuous invariant measure can be established for a particular chaotic map, the derivation of its exact analytical form turns out to be difficult, if not impossible. Meanwhile, the so-called inverse Frobenius–Perron problem (IFPP), i.e., finding nonlinear chaotic maps that preserve prescribed invariant densities, has become a dominant approach in exploring the relationship between the individual functional form and its related statistical features. The IFPP is traditionally treated through ‘‘topological conjugation’’ [6]. Recent advances along this endeavor include the differential equation approach [11], the transverse to conjugation approach [7,3], the branching function approach discussed in [4,8], and the stochastic approach adopted in [10]. While these studies greatly enhance our understanding of the ‘‘micro’’ relationship between the individual functional form and its related statistical features, the current research, however, aims to advance one-step further along the direction of exploring some genuine ‘‘macro’’
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characteristics exhibited in some classes of chaotic maps that share identical analytical characteristics or similar statistical properties. In particular, the class of complete chaotic maps with tent-map-like functional forms is examined. In the history of development in chaos theory, complete chaotic maps with simple structures such as the tent map defined by T(x) = 1 j1 2xj has played an important and indispensable role. Its simplicity and analytical characteristics help not only in the practical modelling in both natural and social sciences but also in pedagogical illustrations of various dynamical theories. The sufficient conditions for constructing such maps are provided and numerical simulations are presented, which verify all theoretical conclusions drawn in this article.
2. Complete chaotic maps with tent-map-like structures We start with a general description of complete chaotic maps [8] defined on a unit-interval IG[0, 1]. The particular choice of unit-interval as domain is not restrictive since any chaotic map defined on a closed interval can always be transformed to a map defined on the unit-interval through variable substitution (linear topological conjugation). A map f : I ! I is said to be unimodal if there exists a turning point ^x 2 I such that the map f can be expressed as fL ðxÞ; 0 6 x 6 ^x; f ðxÞ ¼ minffL ðxÞ; fR ðxÞg ¼ ð1Þ fR ðxÞ; ^x 6 x 6 1; where fL : ½0; ^x ! ½0; 1 and fR : ½^x; 1 ! ½0; 1 are continuous, differentiable except possibly at finite points, monotonically increasing and decreasing, respectively, and onto the unit-interval in the sense that fL(0) = fR(1) = 0 and fL ð^xÞ ¼ fR ð^xÞ ¼ 1. Definition 1. A unimodal map f : I ! I is said to be complete chaotic if it is both ergodic and chaotic in a probabilistic sense so as to preserve an absolutely continuous invariant measure. That is, for any set of A I, there exists such an absolutely continuous invariant measure, denoted as g, that the following identity (Frobenius–Perron equation) is held: gðAÞ ¼ gðf 1 ðAÞÞ
ð2Þ
A function l: I ! I is referred to as an invariant measure function if l meets the following requirements: (i) l(0) = 0 and l(1) = 1 and (ii) l 0 (x) > 0 for almost all x 2 [0, 1]. Then for any invariant measure g, there exists a unique invariant measure function l so that for any subinterval A = [a, b] I, we have g(A) = l(b)l(a). Furthermore, for a unimodal map, given any x 2 I, there exist two preimages of f, which are denoted as xL ¼ fL1 ðxÞ and xR ¼ fR1 ðxÞ, respectively. Therefore, the Frobenius–Perron equation (Eq. 2) can be rewritten as lðxÞ ¼ lðfL1 ðxÞÞ þ 1 lðfR1 ðxÞÞ
8x 2 ½0; 1:
ð3Þ
The main theme of this article is to explore the special class of unimodal complete chaotic maps, in which two branches have a tent-map-like structure, that is, fL ðxÞ ¼ 2gðxÞ; x 2 ½0; ^xÞ; f ðxÞ ¼ 1 j1 2gðxÞj ¼ ð4Þ fR ðxÞ ¼ 2ð1 gðxÞÞ; x 2 ½^x; 1; where g is an invariant measure function defined on I and ^x ¼ g1 ð1=2Þ. When g(x) = x, Eq. (4) reduces to the tent map: x 2 ½0; 1=2Þ; T L ðxÞ ¼ 2x; T ðxÞ ¼ 1 j1 2xj ¼ T R ðxÞ ¼ 2ð1 xÞ; x 2 ½1=2; 1;
ð5Þ
which is known to preserve a uniform invariant density. Fig. 1 illustrates several common-seen shapes of complete chaotic maps with the tent-map-like structures.
3. The Sufficient conditions To provide a set of sufficient conditions for the construction of complete chaotic maps that have the tent-map-like structures, we need to clarify a common mathematical term first. Although the ‘‘closed form’’ is a concept widely used as an antonym for ‘‘piecewise-defined function’’, it has never been rigorously defined in mathematics. It is generally believed that it should exclude ‘‘absolute function’’. However, any function consisting of an absolute function can al-
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Fig. 1. Illustration of tent-map-like structures: fR = 2 fL.
ways be recast to an expression that does not appear with absolute function. For instance, the tent map that is piecewise-defined in (5) can be also recast into more than one closed forms, among which are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1 ðxÞ ¼ 1 1 4x þ 4x2 ; ð6Þ 2 1 sin ðsin pxÞ; p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 3 ðxÞ ¼ 1 1 2xð1 xÞ 2x 1 xð2 xÞ:
T 2 ðxÞ ¼
ð7Þ
ð8Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eq. (6) is nothing but a simple replacement of the absolute term j1 2xj in (5) with ð1 2xÞ2 , hence we are still able to identify its piecewise-defined nature. In contrast, it is quite difficult, if not impossible, to tell just from the functional forms of T2 and T3 their ‘‘non-closed’’ characteristics. To distinguish the functions that have genuine closed format with the fake one like the tent map, we thus introduce the following definition formally. Definition 2. Any function defined on a real domain will be said to have a genuine closed form (or genuine closed format) if (i) it is expressed as a composition of smooth functions such as elementary functions, their inverses, transcendental functions etc. that are not piecewise defined and (ii) it cannot be further simplified to piecewise-defined format.
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Remark 1. Unlike ‘‘analyticity’’ requirement for a function, which demands infinitive differentiability in the whole domain, no differentiable smoothness is required in Definition 2. Therefore, an analytical function has definitely a genuine closed form. The converse, however, is not necessarily true. The following theorem provides a set of sufficient conditions for constructing complete chaotic maps with the tentmap-like structure. Theorem 1. A unimodal map constructed through 1 gðxÞ ¼ l1 ð1 lðxðxÞÞ þ lðxÞÞ; 2
ð9Þ
where (i) (x-condition) x has a genuine closed form in I and is diagonal-symmetric in the sense of xðxÞ ¼ x1 ðxÞ;
x 2 I;
ð10Þ
and (ii) (l-condition) l has a genuine closed form in [0, 2] and is odd-symmetric with respect to the point (1, 1) in the sense of lðxÞ ¼ 2 lð2 xÞ;
x 2 ½0; 1;
ð11Þ
has a tent-map-like structure given by (4) and will preserve the invariant measure l. Proof. At first, we notice that if l is the invariant measure preserved by a complete chaotic map f, Eq. (3) implies
x 2 ½0; ^xÞ; fL ðxÞ ¼ l1 ð1 lðxðxÞÞ þ lðxÞÞ; 1 1 fR ðxÞ ¼ l ð1 þ lðx ðxÞÞ lðxÞÞ; x 2 ½^x; 1; ¼ l1 ð1 jlðxðxÞÞ lðxÞjÞ 8x 2 ½0; 1:
f ðxÞ ¼
: I ! I defined by where x ( xðxÞ ¼ fR1 ðfL ðxÞÞ; xðxÞ ¼ x1 ðxÞ ¼ fL1 ðfR ðxÞÞ;
x 2 ½0; ^xÞ; x 2 ½^x; 1;
ð12Þ
ð13Þ
0 ðxÞ < 0 for almost all x 2 I) that maps one preimage of f in a segment is an one-to-one piecewise decreasing function (x is not differentiable at x ¼ ^x. to a unique counterpart in the other. Apparently, as a piecewise-defined function, x It can be noticed immediately that as long as x has a genuine closed form in I and processes the diagonal-symmetric property in the sense of xðxÞ ¼ x1 ðxÞ;
x 2 I;
ð14Þ
function defined in (13) will have a genuine closed function so that (12) can be further simplified to then the x f ðxÞ ¼ l1 ð1 jlðxðxÞÞ lðxÞjÞ 8x 2 ½0; 1:
ð15Þ
However, a tent-map-like structure defined in (4) implies that the identity fR ðxÞ ¼ 2 fL ðxÞ ¼ 2gðxÞ
ð16Þ
holds for all x 2 I, which suggests that, as a function (not as a map), the domain of fL covers whole I, so does fR. Rewrite (15) as x 2 ½0; ^xÞ 2gðxÞ ¼ l1 ð1 lðxðxÞÞ þ lðxÞÞ; : ð17Þ 2ð1 gðxÞÞ ¼ l1 ð1 þ lðxðxÞÞ lðxÞÞ; x 2 ½^x; 1 If we further impose the requirement that l (and hence l1) has a genuine closed form in I, then (17) implies that l1 ð1 þ lðxÞ lðxðxÞÞÞ þ l1 ð1 þ lðxðxÞÞ lðxÞÞ ¼ 2;
x 2 ½0; 1:
ð18Þ
Notice that the inequality of 1 + l(x) l(x(x)) 6 1 suggests 1 l(x) + l(x(x)) ? 1 for x 2 I. To meet the condition (18), l1 must be definable beyond I and should be odd-symmetric with respect to the point (1, 1) in the sense that l1(1x) + l1(1 + x) = 2 for x 2 I. Finally, we notice that if l1 is odd-symmetric, so is l. h
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be the set of continuous diagonal-symmetry functions that satisfy the x-condition. Some typical members of X Let X include x1 ðxÞ ¼ ð1 xk Þ1=k ; k > 0; 1=k 1 xk ; k > 0; x2 ðxÞ ¼ 1 þ xk x3 ðxÞ ¼ 1 ð1 ð1 xÞk Þ1=k ; 1x ; b > 0: x4 ðxÞ ¼ 1 þ bx
k > 0;
Fig. 2(a) illustrates the several typical cases (k = 1 for x1 and x2, k = 2 for x3 and b = 50 for x4). The simplest function that satisfies the x-condition is related to the symmetric function x1(x) = 1 x, which always results in a symmetric unimodal map. are be the set of invariant measure functions that satisfy the l-condition. Some common-seen members of A Let U l1 ðxÞ ¼ 1 ð1 xÞ3 ; rffiffiffi 4 x ; l2 ðxÞ ¼ arcsin p 2 l3 ðxÞ ¼ 1 ð1 xÞ1=3 ; p l4 ðxÞ ¼ 2sin2 x : 4 These invariant measure functions are illustrated in Fig. 3(a). Example 1. Some complete chaotic maps are presented in Table 1, where fijÕs are constructed with xi, i = 1, 2, 3, defined by x1 ðxÞ ¼ 1 x; 1x x2 ðxÞ ¼ ; 1þx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x3 ðxÞ ¼ 1 xð2 xÞ; and lj, i = 1, 2, defined by l1 ðxÞ ¼ 1 ð1 xÞ3 ; rffiffiffi 4 x l2 ðxÞ ¼ arcsin : p 2 The graphs of fijÕs are illustrated in Fig. 4.
/
,
, 1
Fig. 2. Diagonal-symmetricity of x in [0, 1]: x(x) = x (x).
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Fig. 3. Odd-symmetricity of l in [0, 2]: l(x) = 2 l(2 x).
Table 1 Illustrations of complete chaotic maps with tent-map-like structures fij(x)
l1
l2 3
3 1/3
x1
1j(1x) x j
x2
1 1 1þx jð1 x2 Þ3 8x3 j1=3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 jð1 xÞ3 xð2 xÞ xð2 xÞj1=3
x3
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 jð1 xÞ 1 x2 x xð2 xÞj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1þx jð1 xÞ ð3x þ 1Þð1 xÞ 2x xð2 xÞj 1j12x(2x)j
Fig. 4. Illustrations of Example 1.
4. On the importance of ‘‘genuine closed form’’ It cannot be emphasized more on that the ‘‘genuine closed-form’’ requirements are critical both in (10) and (11). We will see from the following counter-examples that a piecewise-defined x function that satisfies (10) and/or a piecewisedefined invariant measure that satisfies (11), even if they are infinitely differentiable everywhere in their respectively domains, will fail to provide a complete chaotic map with a tent-map-like structure. Example 2. We first examine the case in which the l-condition is met but the x-condition is satisfied partially in the sense that identity (10) is satisfied by a piecewise-defined branching function. Let ( 0 6 x < 23 ; 1 34 x2 ; p ffiffiffiffiffiffiffiffiffiffi ffi x0 ðxÞ ¼ p2ffiffi 1 x; 23 < x 6 1; 3
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then it satisfies the identify x0 ðxÞ ¼ x1 x0 Þ is smooth and differentiable everywhere 0 ðxÞ. As illustrated in Fig. 2 (b), xð¼ in (0, 1). Referring to Example 1, if we let l1(x) = 1 (1 x)3, which satisfies the l-condition, it follows from (12) that ( f~ 1 ðxÞ ¼
l1 1 ð1 l1 ðx0 ðxÞÞ þ l1 ðxÞÞ;
0 6 x < 23 ;
l1 1 ð1
þ l1 ðx0 ðxÞÞ l1 ðxÞÞ; 23 < x 6 1: 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < f~ 1;L ðxÞ ¼ 1 14 3 64ð1 xÞ3 27x6 ; ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : f~ ðxÞ ¼ 1 3 x3 3x2 x þ 4 2 3ð1 xÞð13 4xÞ; 1;R 9 pffiffi Similarly, if we take l2 ðxÞ ¼ p4 arcsin 2x, then we have ( f~ 2 ðxÞ ¼ ( ¼
l1 2 ð1 l2 ðx0 ðxÞÞ þ l2 ðxÞÞ;
0 6 x < 23 ; 2 3
ð19Þ
< x 6 1:
0 6 x < 23 ;
2 l1 < x 6 1: 2 ð1 þ l2 ðx0 ðxÞÞ l2 ðxÞÞ; 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi f~ 2;L ðxÞ ¼ 1 þ 34 x2 xð2 xÞ 28 16 9x4 ð1 xÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f~ 2;R ðxÞ ¼ 1 þ 13 ð2rðxÞ 3Þ xð2 xÞ þ 23 ð1 xÞ rðxÞð3 rðxÞÞ;
0 6 x < 23 ; 2 3
ð20Þ
< x 6 1;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where rðxÞ ¼ 3ð1 xÞ. Fig. 5(a) shows the graphs of f~1 and f~2 , from which we can see that they do not have tent-map-like structures. Example 3. Now we shall illustrate that odd-symmetry of l alone without requirement for ‘‘genuine closed form’’ is not suffice to guarantee the resulted complete chaotic map to have a tent-map-like structure. Let l0(x) = x(2 x) be defined on I and extended odd-symmetrically to the interval [0, 2] by lðxÞ ¼
l0 ðxÞ ¼ xð2 xÞ; 2 l0 ð2 xÞ ¼ 2 xð2 xÞ;
0 6 x < 1; 1 6 x 6 2:
As can be seen from Fig. 3 (b), l so defined is differentiable everywhere in [0, 2]. Since l1 0 ðxÞ ¼ 1 x1(x) = 1 x, we obtain a symmetric unimodal map: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f~ 3 ðxÞ ¼ 1 j1 2xj ( pffiffiffiffiffiffiffiffiffiffiffiffiffi f~ 3;L ðxÞ ¼ 1 1 2x; ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi f~ 3;R ðxÞ ¼ 1 2x 1;
0 6 x < 1=2; 1=2 6 x 6 1:
Fig. 5. Complete chaotic maps that do not have tent-map-like structures.
ð21Þ pffiffiffiffiffiffiffiffiffiffiffi 1 x, if we take
ð22Þ
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Similarly, if we let x2 ðxÞ ¼ 1x , we have 1þx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p 1 f~ 4 ðxÞ ¼ 1 1þx jx4 6x2 þ 1j; ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f~ 4;L ðxÞ ¼ 1 1þx x4 6x2 þ 1; ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f~ 4;R ðxÞ ¼ 1 1þx 6x2 x4 1;
pffiffiffi 0 6 x < 2 1; pffiffiffi 2 1 6 x 6 1:
ð23Þ
Functional graphs of f~ 3 and f~ 4 are shown in Fig. 5(b). We see again that none of f~ 3 and f~ 4 has a tent-map-like structure. 5. Uniform invariant density It is not clear whether the x-condition and the l-condition together are the necessary conditions for a complete chaotic map to have a tent-map-like structure. Nevertheless, each individual condition is indeed a necessary condition if the other is assumed to hold true. In other words, to construct a complete chaotic map with a tent-map-like structure, either condition is indispensable if the other is satisfied. For instance, if the x-condition holds alone, a complete chaotic map given by (12) can only be expressed as f ðxÞ ¼ l1 ð1 jlðxðxÞÞ lðxÞjÞ; which does not have a tent-map-like structure in general. The simplest invariant measure which satisfies the l-condition is the Lebesgue measure, that is, l(x) = x. The class of complete chaotic maps preserving the Lebesgue measure (uniform invariant density) form the groundstones for pseudorandom number generators [2] and hence will be referred to as Lebesgue maps. For the Lebesgue measure, l(x) = x, the l-condition is automatically satisfied, so the x-condition becomes a necessary condition. In fact, it follows from (12) that 2gðxÞ ¼ 1 þ x xðxÞ;
x 2 ½0; ^xÞ
2ð1 gðxÞÞ ¼ 1 x þ x1 ðxÞ;
x 2 ½^x; 1:
Since the domain of g is I, the above two identities request that both x and x1 in terms of algebraic functions must have closed-forms in I, which lead to x1 ðxÞ ¼ xðxÞ ¼ xðxÞ;
x 2 ½0; 1;
which is exactly the x-condition. function suggests that a Lebesgue map f must satisfy the derivative Remark 2. The down-sloping condition for the x requirement f 0 ðxÞ P 1; for almost all x 6 ^x;
ð24Þ
where ^x ¼ g1 ð1=2Þ. In particular, when xðxÞ ¼ 1 x, f(x) = T(x), that is, the tent map is a special case of Lebesgue map, in which the structure is symmetric. Table 2 lists some examples of such Lebesgue maps with related x functions. Fig. 6 illustrate these Lebesgue maps. Table 2 Lebesgue maps with tent-map-like structures i
fi
2
pffiffiffi 1 j1 2 xj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 j1 x xð2 xÞj
3
1 j1 xð3þxÞ 1þx j
4
1 j1
xð3xÞ 2x j
5
pffiffiffiffiffiffiffiffiffiffiffi 1 jx 1 xj pffiffiffiffiffiffiffiffiffiffiffi 1 j2 1 x 1j
1
6
^xi 1/4
pffiffiffi 1 2=2 pffiffiffi 21 pffiffiffi 2 2 pffiffiffi 2=2 3/4
xi
pffiffiffi ð1 xÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xð2 xÞ 1x 1þx x 1 2x pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2
1 ð1
pffiffiffiffiffiffiffiffiffiffiffi 2 1 xÞ
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Fig. 6. Lebesgue maps with the tent-map-like structures.
6. On the sufficient conditions The abundance of complete chaotic maps with tent-map-like structures becomes clear after we examine the two critical sufficient conditions stated in Theorem 1 more closely. The following facts sheds light on the direction of constructing the invariant measure functions that satisfy the l condition, the set of which is denoted by A. let l 2 A, i = 1, 2, and l = al1 + (1 a)l2, where a 2 [0, 1], l 2 A. That is, if two invariant i(i) Average rule for A: i
measure functions satisfy the l-condition, then their linear combinations also satisfy the l-condition; i = 1, 2, then their direct compositions l12 = l1(l2(x)) and l21 = l2(l1(x)) also let l 2 A, (ii) composition rule for A: i That is, direct compositions of all invariant measure functions that satisfy the l-condition, still satbelong to A. isfy the l-condition. For x functions satisfying the x-condition, we can verify the following rule straightforwardly. For any invariant measure function h and any x function that satisfies the xLemma 1. The composition rule for X: condition, the function constructed by ~ xðxÞ ¼ h1 ðxðhðxÞÞÞ ~ branches at h1 ð^xÞ. also satisfies the x-condition. Furthermore, if x branches at ^x, that is, xð^xÞ ¼ ^x, then x The abundance of diagonal-symmetric x functions is further manifested by the following constructive properties of invariant measure functions, the set of which is denoted as H. Lemma 2. Properties of invariant measure functions: 1. if 2. if 3. if 4. if
h 2 H, then h1 2 H; hi 2 H, i = 1, 2, then their compositions defined by h12 = h1(h2(x)) and h21 = h2(h1(x)) also belong to H; hi 2 H, i = 1, 2, then their average function given by h1 + (1 )h2, with 0 6 6 1, also belongs to H; hi 2 H, i = 1, 2, then their product given by hk1 hl2 , with k,l > 0, also belongs to H.
One direct implication of the above properties is, for any h 2 H, given a b > 0, h(xb) and hb(x) are invariant measure functions as well. With the above rules, we are able to conclude that, for any complete chaotic map f that has a tent-map-like structure, its conjugate defined by f~ h ðxÞ ¼ h1 ðf ðhðxÞÞÞ preserves the tent-map-like structure if and only if h is an invariant measure function that satisfies hðxÞ ¼ 2 hð2 xÞ;
x 2 ½0; 1:
ð25Þ
The very observation explains why in general a conjugate of the tent map does not preserve a similar structure. In fact, given any h 2 H, we can infer from ( h1 ð2hðxÞÞ; x 2 ½0; ^xÞ; 1 T~ h ðxÞ ¼ h ðT ðhðxÞÞÞ ¼ h1 ð2ð1 hðxÞÞÞ; x 2 ½^x; 1; that a h that satisfies (25) is indispensable for T~ h to have a tent-map-like structure.
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Example 4. As listed in Table 2, the map pffiffiffi f1 ¼ 1 j1 2 xj preserves a uniform invariant density. (i) Taking h1(x) = 1 (1 x)3, h1 ðxÞ ¼ h1 1 ðf1 ðh1 ðxÞÞÞ will preserve an invariant density u1 defined by ~ 1 ðxÞ ¼ h01 ðxÞ ¼ 3ð1 xÞ2 : u It is easy to verify that h1 ðxÞ ¼ 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 2 xð3 3x þ x2 Þ:
~ 2 defined by (ii) Taking h2 ðxÞ ¼ 2sin2 ðp4 xÞ, h2 ðxÞ ¼ h1 2 ðf1 ðh2 ðxÞÞÞ will preserve an invariant density u ~ 2 ðxÞ ¼ h02 ðxÞ ¼ u
p p sin x: 2 2
Simple mathematical manipulations give us
Fig. 7. Illustrations of Example 4.
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 8 pffiffiffi > 2 sin p4 x; < h2L ðxÞ ¼ p4 arcsin h2 ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : h ðxÞ ¼ 4 arcsin 1 pffiffi2ffi sin p x: 2R p 4 As illustrated in Fig. 7(c), h2R(x) = 2 h2L(x). And hence, h2 is also a complete chaotic map with a tent-map-like structure so that it can be recast as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 p h2 ðxÞ ¼ 1 1 arcsin 2 sin x: p 4 Computer simulations for h1 and h2 are provided in Fig. 7 (all realized densities are simulated with Maple 9.0 with 50 000 iterations, 100 intervals, and an initial value of x0 = p/10). Furthermore, the above discussion leads us to conclude that, for any complete chaotic map f with a tent-map-like structure that preserves an invariant measure l, its conjugation given by h = l(f(l1(x))) is a Lebesgue map with a tent-map-like structure. For instance, the map f32(x) 1 j1 2x(2 x)j has a tent-map-like structure (Example 1) p=ffiffiffiffiffiffiffi and preserves the invariant measure lðxÞ ¼ p4 arcsin x=2. It can be verified that l(f32(l1(x))) is nothing but the tent map.
7. Lyapunov exponents The simplicity and analytical characteristics of complete chaotic maps with the tent-map-like structure simplify the calculations of various statistical properties of chaotic dynamics. Due to the limited scope, we shall illustrate this point with the Lyapunov exponent (LC). The Lyapunov exponent of a chaotic process defined by K ¼ lim
N !1
Z 1 N 1 X ln jf 0 ðxt Þj ¼ uðxÞ ln jf 0 ðxÞjdx N t¼0 0
ð26Þ
characterizes the degree of chaosity through measuring the average divergence rate of nearby trajectories. The simplicity in the functional form of complete chaotic maps with the tent-map-like structure map provides us a great advantage in the evaluation K both analytically and numerically. Substituting (4) into (26), we obtain Z 1 uðxÞ ln g0 ðxÞdx: ð27Þ K ¼ ln 2 þ 0
On the other hand, if g is constructed through (9), it follows from (15) that lðf ðxÞÞ ¼ 1 þ jlðxÞ lðxðxÞÞj
for x 2 ½0; 1;
which leads to jf 0 ðxÞjuðf ðxÞÞ ¼ uðxÞ uðxðxÞÞx0 ðxÞ: Therefore, K¼
Z
1
uðxÞ ln 0
uðxÞ uðxðxÞÞx0 ðxÞ dx ¼ uðf ðxÞÞ
Z
1
uðxÞ lnðuðxÞ uðxðxÞÞx0 ðxÞÞdx
0
Z
1
uðxÞ ln uðf ðxÞÞdx:
0
Now that the ergodicity of the complete chaotic map f implies that Z 0
1
uðxÞ ln uðf ðxÞÞdx ¼
Z
1
uðxÞ ln uðxÞdx; 0
we finally have Z 1 uðxðxÞÞ 0 x ðxÞ dx: uðxÞ ln 1 K¼ uðxÞ 0
ð28Þ
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Formula (28) enables us to explore LC under different structures of x and u. For instance, consider the case in which f is symmetric, that is, x(x) = 1 x, we have K¼
Z 0
1
uð1 xÞ uðxÞ ln 1 þ dx: uðxÞ
ð29Þ
Further consider two special cases: 1. u is R symmetric as well, i.e., u(x) = u(1 x), then we arrive at a maximum possible Lyapunov exponent 1 K ¼ 0 ln 2uðxÞdx ¼ ln 2, a conclusion drawn in [6]. 2. u is odd-symmetric, i.e., u(x) = 1 u(1 x), we have Z 1 Z 1 1 uðxÞ dx ¼ uðxÞ ln 1 þ uðxÞ lnðuðxÞÞdx ¼ entropy of u: K¼ uðxÞ 0 0 Numerical calculation with (28)p isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi much ffisimpler than that with (26). For instance, referring to Example 1, for l(x) = 1 (1 x)3 and xðxÞ ¼ 1 xð2 xÞ, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðxÞ ¼ 1 jð1 xÞ3 xð2 xÞ xð2 xÞj1=3 : It can be verified that the expression of LC either in (26) or in (27) turns out to be very complex. However, substituting u(x) = 3(1 x)2 and x into (28) leads us to the following simple result: rffiffiffiffiffiffiffiffiffiffiffi Z 1 1 x 2 dx 0:4339: 3ð1 xÞ ln 1 þ K¼ 1 x 2 x 0 Moreover, with (28) in hand, we are able to evaluate comparative statics with respect to the effects of various parameters on the Lyapunov component directly.
8. Conclusion We have provided in theory a set of sufficient conditions so that a class of complete chaotic maps with the tent-maplike structures can be constructed explicitly with closed formulas. Examples and numerical simulations are provided, which have been shown to be consistent with the theoretical findings. Our results greatly extend the applicability of the general formulation of unimodal complete chaotic maps given by (12).
Acknowledgment The author wish to thank Shahria Yousefi for his constructive suggestions. Help comments from an anonymous referee is greatly appreciated. The usual disclaimer applies.
References [1] Lasota A, Mackey MC. Probabilistic properties of deterministic systems. Cambridge: Cambridge University Press; 1985. [2] Boyarsky A, Go´ra P. Laws of chaos: invariant measures and dynamical systems in one dimension. Reading, MA: AddisonWesley; 1994. [3] Csorda´s G, Gyo¨gyi A, Sze´pfalusy P, Te´l T. Statistical properties of chaos demonstrated in a class of one-dimensional maps. Chaos 1993;3:31–49. [4] Ershov SH, Malinetskii GG. The solution of the inverse problem for the Frobenius–Perron equation. USSR Comput Maths Math Phys 1988;28:136–41. [6] Grossmann S, Thomae S. Invariant distributions and stationary correlation functions of one-dimensional discrete process. Z Naturforschung 1977;32:1353–63. [7] Gyo¨gyi G, Sze´pfalusy P. Fully developed chaotic 1-d maps. Z Phys B—Condens Matter 1984;55:179–86. [8] Pingel D, Schmelcher P, Diakonos FK. Theory and examples of the inverse Frobenius–Perron problem for complete chaotic Maps. Chaos 1999;9:357–66.
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[10] Diakonos FK, Pingel D, Schmelcher P. A stochastic approach to the construction of one-dimensional chaotic maps with prescribed statistical properties. Phys Lett A 1999;264:162–70. [11] Shinji K. The inverse problem of Flobenius–Perron equations in 1D difference systems. Prog Theor Phys 1991;86:991– 1002.
On complete chaotic maps with tent-map-like structures Weihong Huang Nanyang Technological University, Nanyang Avenue, Nanyang 639798, Singapore Accepted 14 September 2004 Communicated by Prof. S. Yousefi
Abstract A unimodal map f : [0, 1] ! [0, 1] is said to be complete chaotic if it is both ergodic and chaotic in a probabilistic sense so as to preserve an absolutely continuous invariant measure. The sufficient conditions are provided to construct complete chaotic maps with the tent-map-like structures, that is, f(x) = 1 j1 2g(x)j, where g is an one-to-one onto map defined on [0, 1]. The simplicity and analytical characteristics of such chaotic maps simplify the calculations of various statistical properties of chaotic dynamics. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Due to the sensitive dependence on initial conditions, observation and measurement accuracy, and computation errors, predicting chaotic trajectories turns out to be an impossible and meaningless task. In many circumstances, to get a global picture of a chaotic system, long-term statistical properties such as long-run averages, observed frequencies, correlation indices have to be resorted to. The probabilistic or statistical approach is proved to be an effective approach in studying nonlinear chaotic dynamics [1]. The key element of the statistical approach is the invariant measure that describes the ‘‘steady state’’ of a nonlinear system, which is indispensable for the calculation of relevant experimental observables such as time correlations and their power spectra. Unfortunately, except for some simple maps such as the tent map and the logistic map, even though the existence of the absolute continuous invariant measure can be established for a particular chaotic map, the derivation of its exact analytical form turns out to be difficult, if not impossible. Meanwhile, the so-called inverse Frobenius–Perron problem (IFPP), i.e., finding nonlinear chaotic maps that preserve prescribed invariant densities, has become a dominant approach in exploring the relationship between the individual functional form and its related statistical features. The IFPP is traditionally treated through ‘‘topological conjugation’’ [6]. Recent advances along this endeavor include the differential equation approach [11], the transverse to conjugation approach [7,3], the branching function approach discussed in [4,8], and the stochastic approach adopted in [10]. While these studies greatly enhance our understanding of the ‘‘micro’’ relationship between the individual functional form and its related statistical features, the current research, however, aims to advance one-step further along the direction of exploring some genuine ‘‘macro’’
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.021
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characteristics exhibited in some classes of chaotic maps that share identical analytical characteristics or similar statistical properties. In particular, the class of complete chaotic maps with tent-map-like functional forms is examined. In the history of development in chaos theory, complete chaotic maps with simple structures such as the tent map defined by T(x) = 1 j1 2xj has played an important and indispensable role. Its simplicity and analytical characteristics help not only in the practical modelling in both natural and social sciences but also in pedagogical illustrations of various dynamical theories. The sufficient conditions for constructing such maps are provided and numerical simulations are presented, which verify all theoretical conclusions drawn in this article.
2. Complete chaotic maps with tent-map-like structures We start with a general description of complete chaotic maps [8] defined on a unit-interval IG[0, 1]. The particular choice of unit-interval as domain is not restrictive since any chaotic map defined on a closed interval can always be transformed to a map defined on the unit-interval through variable substitution (linear topological conjugation). A map f : I ! I is said to be unimodal if there exists a turning point ^x 2 I such that the map f can be expressed as fL ðxÞ; 0 6 x 6 ^x; f ðxÞ ¼ minffL ðxÞ; fR ðxÞg ¼ ð1Þ fR ðxÞ; ^x 6 x 6 1; where fL : ½0; ^x ! ½0; 1 and fR : ½^x; 1 ! ½0; 1 are continuous, differentiable except possibly at finite points, monotonically increasing and decreasing, respectively, and onto the unit-interval in the sense that fL(0) = fR(1) = 0 and fL ð^xÞ ¼ fR ð^xÞ ¼ 1. Definition 1. A unimodal map f : I ! I is said to be complete chaotic if it is both ergodic and chaotic in a probabilistic sense so as to preserve an absolutely continuous invariant measure. That is, for any set of A I, there exists such an absolutely continuous invariant measure, denoted as g, that the following identity (Frobenius–Perron equation) is held: gðAÞ ¼ gðf 1 ðAÞÞ
ð2Þ
A function l: I ! I is referred to as an invariant measure function if l meets the following requirements: (i) l(0) = 0 and l(1) = 1 and (ii) l 0 (x) > 0 for almost all x 2 [0, 1]. Then for any invariant measure g, there exists a unique invariant measure function l so that for any subinterval A = [a, b] I, we have g(A) = l(b)l(a). Furthermore, for a unimodal map, given any x 2 I, there exist two preimages of f, which are denoted as xL ¼ fL1 ðxÞ and xR ¼ fR1 ðxÞ, respectively. Therefore, the Frobenius–Perron equation (Eq. 2) can be rewritten as lðxÞ ¼ lðfL1 ðxÞÞ þ 1 lðfR1 ðxÞÞ
8x 2 ½0; 1:
ð3Þ
The main theme of this article is to explore the special class of unimodal complete chaotic maps, in which two branches have a tent-map-like structure, that is, fL ðxÞ ¼ 2gðxÞ; x 2 ½0; ^xÞ; f ðxÞ ¼ 1 j1 2gðxÞj ¼ ð4Þ fR ðxÞ ¼ 2ð1 gðxÞÞ; x 2 ½^x; 1; where g is an invariant measure function defined on I and ^x ¼ g1 ð1=2Þ. When g(x) = x, Eq. (4) reduces to the tent map: x 2 ½0; 1=2Þ; T L ðxÞ ¼ 2x; T ðxÞ ¼ 1 j1 2xj ¼ T R ðxÞ ¼ 2ð1 xÞ; x 2 ½1=2; 1;
ð5Þ
which is known to preserve a uniform invariant density. Fig. 1 illustrates several common-seen shapes of complete chaotic maps with the tent-map-like structures.
3. The Sufficient conditions To provide a set of sufficient conditions for the construction of complete chaotic maps that have the tent-map-like structures, we need to clarify a common mathematical term first. Although the ‘‘closed form’’ is a concept widely used as an antonym for ‘‘piecewise-defined function’’, it has never been rigorously defined in mathematics. It is generally believed that it should exclude ‘‘absolute function’’. However, any function consisting of an absolute function can al-
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289
Fig. 1. Illustration of tent-map-like structures: fR = 2 fL.
ways be recast to an expression that does not appear with absolute function. For instance, the tent map that is piecewise-defined in (5) can be also recast into more than one closed forms, among which are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1 ðxÞ ¼ 1 1 4x þ 4x2 ; ð6Þ 2 1 sin ðsin pxÞ; p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 3 ðxÞ ¼ 1 1 2xð1 xÞ 2x 1 xð2 xÞ:
T 2 ðxÞ ¼
ð7Þ
ð8Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eq. (6) is nothing but a simple replacement of the absolute term j1 2xj in (5) with ð1 2xÞ2 , hence we are still able to identify its piecewise-defined nature. In contrast, it is quite difficult, if not impossible, to tell just from the functional forms of T2 and T3 their ‘‘non-closed’’ characteristics. To distinguish the functions that have genuine closed format with the fake one like the tent map, we thus introduce the following definition formally. Definition 2. Any function defined on a real domain will be said to have a genuine closed form (or genuine closed format) if (i) it is expressed as a composition of smooth functions such as elementary functions, their inverses, transcendental functions etc. that are not piecewise defined and (ii) it cannot be further simplified to piecewise-defined format.
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Remark 1. Unlike ‘‘analyticity’’ requirement for a function, which demands infinitive differentiability in the whole domain, no differentiable smoothness is required in Definition 2. Therefore, an analytical function has definitely a genuine closed form. The converse, however, is not necessarily true. The following theorem provides a set of sufficient conditions for constructing complete chaotic maps with the tentmap-like structure. Theorem 1. A unimodal map constructed through 1 gðxÞ ¼ l1 ð1 lðxðxÞÞ þ lðxÞÞ; 2
ð9Þ
where (i) (x-condition) x has a genuine closed form in I and is diagonal-symmetric in the sense of xðxÞ ¼ x1 ðxÞ;
x 2 I;
ð10Þ
and (ii) (l-condition) l has a genuine closed form in [0, 2] and is odd-symmetric with respect to the point (1, 1) in the sense of lðxÞ ¼ 2 lð2 xÞ;
x 2 ½0; 1;
ð11Þ
has a tent-map-like structure given by (4) and will preserve the invariant measure l. Proof. At first, we notice that if l is the invariant measure preserved by a complete chaotic map f, Eq. (3) implies
x 2 ½0; ^xÞ; fL ðxÞ ¼ l1 ð1 lðxðxÞÞ þ lðxÞÞ; 1 1 fR ðxÞ ¼ l ð1 þ lðx ðxÞÞ lðxÞÞ; x 2 ½^x; 1; ¼ l1 ð1 jlðxðxÞÞ lðxÞjÞ 8x 2 ½0; 1:
f ðxÞ ¼
: I ! I defined by where x ( xðxÞ ¼ fR1 ðfL ðxÞÞ; xðxÞ ¼ x1 ðxÞ ¼ fL1 ðfR ðxÞÞ;
x 2 ½0; ^xÞ; x 2 ½^x; 1;
ð12Þ
ð13Þ
0 ðxÞ < 0 for almost all x 2 I) that maps one preimage of f in a segment is an one-to-one piecewise decreasing function (x is not differentiable at x ¼ ^x. to a unique counterpart in the other. Apparently, as a piecewise-defined function, x It can be noticed immediately that as long as x has a genuine closed form in I and processes the diagonal-symmetric property in the sense of xðxÞ ¼ x1 ðxÞ;
x 2 I;
ð14Þ
function defined in (13) will have a genuine closed function so that (12) can be further simplified to then the x f ðxÞ ¼ l1 ð1 jlðxðxÞÞ lðxÞjÞ 8x 2 ½0; 1:
ð15Þ
However, a tent-map-like structure defined in (4) implies that the identity fR ðxÞ ¼ 2 fL ðxÞ ¼ 2gðxÞ
ð16Þ
holds for all x 2 I, which suggests that, as a function (not as a map), the domain of fL covers whole I, so does fR. Rewrite (15) as x 2 ½0; ^xÞ 2gðxÞ ¼ l1 ð1 lðxðxÞÞ þ lðxÞÞ; : ð17Þ 2ð1 gðxÞÞ ¼ l1 ð1 þ lðxðxÞÞ lðxÞÞ; x 2 ½^x; 1 If we further impose the requirement that l (and hence l1) has a genuine closed form in I, then (17) implies that l1 ð1 þ lðxÞ lðxðxÞÞÞ þ l1 ð1 þ lðxðxÞÞ lðxÞÞ ¼ 2;
x 2 ½0; 1:
ð18Þ
Notice that the inequality of 1 + l(x) l(x(x)) 6 1 suggests 1 l(x) + l(x(x)) ? 1 for x 2 I. To meet the condition (18), l1 must be definable beyond I and should be odd-symmetric with respect to the point (1, 1) in the sense that l1(1x) + l1(1 + x) = 2 for x 2 I. Finally, we notice that if l1 is odd-symmetric, so is l. h
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be the set of continuous diagonal-symmetry functions that satisfy the x-condition. Some typical members of X Let X include x1 ðxÞ ¼ ð1 xk Þ1=k ; k > 0; 1=k 1 xk ; k > 0; x2 ðxÞ ¼ 1 þ xk x3 ðxÞ ¼ 1 ð1 ð1 xÞk Þ1=k ; 1x ; b > 0: x4 ðxÞ ¼ 1 þ bx
k > 0;
Fig. 2(a) illustrates the several typical cases (k = 1 for x1 and x2, k = 2 for x3 and b = 50 for x4). The simplest function that satisfies the x-condition is related to the symmetric function x1(x) = 1 x, which always results in a symmetric unimodal map. are be the set of invariant measure functions that satisfy the l-condition. Some common-seen members of A Let U l1 ðxÞ ¼ 1 ð1 xÞ3 ; rffiffiffi 4 x ; l2 ðxÞ ¼ arcsin p 2 l3 ðxÞ ¼ 1 ð1 xÞ1=3 ; p l4 ðxÞ ¼ 2sin2 x : 4 These invariant measure functions are illustrated in Fig. 3(a). Example 1. Some complete chaotic maps are presented in Table 1, where fijÕs are constructed with xi, i = 1, 2, 3, defined by x1 ðxÞ ¼ 1 x; 1x x2 ðxÞ ¼ ; 1þx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x3 ðxÞ ¼ 1 xð2 xÞ; and lj, i = 1, 2, defined by l1 ðxÞ ¼ 1 ð1 xÞ3 ; rffiffiffi 4 x l2 ðxÞ ¼ arcsin : p 2 The graphs of fijÕs are illustrated in Fig. 4.
/
,
, 1
Fig. 2. Diagonal-symmetricity of x in [0, 1]: x(x) = x (x).
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Fig. 3. Odd-symmetricity of l in [0, 2]: l(x) = 2 l(2 x).
Table 1 Illustrations of complete chaotic maps with tent-map-like structures fij(x)
l1
l2 3
3 1/3
x1
1j(1x) x j
x2
1 1 1þx jð1 x2 Þ3 8x3 j1=3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 jð1 xÞ3 xð2 xÞ xð2 xÞj1=3
x3
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 jð1 xÞ 1 x2 x xð2 xÞj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1þx jð1 xÞ ð3x þ 1Þð1 xÞ 2x xð2 xÞj 1j12x(2x)j
Fig. 4. Illustrations of Example 1.
4. On the importance of ‘‘genuine closed form’’ It cannot be emphasized more on that the ‘‘genuine closed-form’’ requirements are critical both in (10) and (11). We will see from the following counter-examples that a piecewise-defined x function that satisfies (10) and/or a piecewisedefined invariant measure that satisfies (11), even if they are infinitely differentiable everywhere in their respectively domains, will fail to provide a complete chaotic map with a tent-map-like structure. Example 2. We first examine the case in which the l-condition is met but the x-condition is satisfied partially in the sense that identity (10) is satisfied by a piecewise-defined branching function. Let ( 0 6 x < 23 ; 1 34 x2 ; p ffiffiffiffiffiffiffiffiffiffi ffi x0 ðxÞ ¼ p2ffiffi 1 x; 23 < x 6 1; 3
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293
then it satisfies the identify x0 ðxÞ ¼ x1 x0 Þ is smooth and differentiable everywhere 0 ðxÞ. As illustrated in Fig. 2 (b), xð¼ in (0, 1). Referring to Example 1, if we let l1(x) = 1 (1 x)3, which satisfies the l-condition, it follows from (12) that ( f~ 1 ðxÞ ¼
l1 1 ð1 l1 ðx0 ðxÞÞ þ l1 ðxÞÞ;
0 6 x < 23 ;
l1 1 ð1
þ l1 ðx0 ðxÞÞ l1 ðxÞÞ; 23 < x 6 1: 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < f~ 1;L ðxÞ ¼ 1 14 3 64ð1 xÞ3 27x6 ; ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : f~ ðxÞ ¼ 1 3 x3 3x2 x þ 4 2 3ð1 xÞð13 4xÞ; 1;R 9 pffiffi Similarly, if we take l2 ðxÞ ¼ p4 arcsin 2x, then we have ( f~ 2 ðxÞ ¼ ( ¼
l1 2 ð1 l2 ðx0 ðxÞÞ þ l2 ðxÞÞ;
0 6 x < 23 ; 2 3
ð19Þ
< x 6 1:
0 6 x < 23 ;
2 l1 < x 6 1: 2 ð1 þ l2 ðx0 ðxÞÞ l2 ðxÞÞ; 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi f~ 2;L ðxÞ ¼ 1 þ 34 x2 xð2 xÞ 28 16 9x4 ð1 xÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f~ 2;R ðxÞ ¼ 1 þ 13 ð2rðxÞ 3Þ xð2 xÞ þ 23 ð1 xÞ rðxÞð3 rðxÞÞ;
0 6 x < 23 ; 2 3
ð20Þ
< x 6 1;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where rðxÞ ¼ 3ð1 xÞ. Fig. 5(a) shows the graphs of f~1 and f~2 , from which we can see that they do not have tent-map-like structures. Example 3. Now we shall illustrate that odd-symmetry of l alone without requirement for ‘‘genuine closed form’’ is not suffice to guarantee the resulted complete chaotic map to have a tent-map-like structure. Let l0(x) = x(2 x) be defined on I and extended odd-symmetrically to the interval [0, 2] by lðxÞ ¼
l0 ðxÞ ¼ xð2 xÞ; 2 l0 ð2 xÞ ¼ 2 xð2 xÞ;
0 6 x < 1; 1 6 x 6 2:
As can be seen from Fig. 3 (b), l so defined is differentiable everywhere in [0, 2]. Since l1 0 ðxÞ ¼ 1 x1(x) = 1 x, we obtain a symmetric unimodal map: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f~ 3 ðxÞ ¼ 1 j1 2xj ( pffiffiffiffiffiffiffiffiffiffiffiffiffi f~ 3;L ðxÞ ¼ 1 1 2x; ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi f~ 3;R ðxÞ ¼ 1 2x 1;
0 6 x < 1=2; 1=2 6 x 6 1:
Fig. 5. Complete chaotic maps that do not have tent-map-like structures.
ð21Þ pffiffiffiffiffiffiffiffiffiffiffi 1 x, if we take
ð22Þ
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Similarly, if we let x2 ðxÞ ¼ 1x , we have 1þx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p 1 f~ 4 ðxÞ ¼ 1 1þx jx4 6x2 þ 1j; ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f~ 4;L ðxÞ ¼ 1 1þx x4 6x2 þ 1; ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f~ 4;R ðxÞ ¼ 1 1þx 6x2 x4 1;
pffiffiffi 0 6 x < 2 1; pffiffiffi 2 1 6 x 6 1:
ð23Þ
Functional graphs of f~ 3 and f~ 4 are shown in Fig. 5(b). We see again that none of f~ 3 and f~ 4 has a tent-map-like structure. 5. Uniform invariant density It is not clear whether the x-condition and the l-condition together are the necessary conditions for a complete chaotic map to have a tent-map-like structure. Nevertheless, each individual condition is indeed a necessary condition if the other is assumed to hold true. In other words, to construct a complete chaotic map with a tent-map-like structure, either condition is indispensable if the other is satisfied. For instance, if the x-condition holds alone, a complete chaotic map given by (12) can only be expressed as f ðxÞ ¼ l1 ð1 jlðxðxÞÞ lðxÞjÞ; which does not have a tent-map-like structure in general. The simplest invariant measure which satisfies the l-condition is the Lebesgue measure, that is, l(x) = x. The class of complete chaotic maps preserving the Lebesgue measure (uniform invariant density) form the groundstones for pseudorandom number generators [2] and hence will be referred to as Lebesgue maps. For the Lebesgue measure, l(x) = x, the l-condition is automatically satisfied, so the x-condition becomes a necessary condition. In fact, it follows from (12) that 2gðxÞ ¼ 1 þ x xðxÞ;
x 2 ½0; ^xÞ
2ð1 gðxÞÞ ¼ 1 x þ x1 ðxÞ;
x 2 ½^x; 1:
Since the domain of g is I, the above two identities request that both x and x1 in terms of algebraic functions must have closed-forms in I, which lead to x1 ðxÞ ¼ xðxÞ ¼ xðxÞ;
x 2 ½0; 1;
which is exactly the x-condition. function suggests that a Lebesgue map f must satisfy the derivative Remark 2. The down-sloping condition for the x requirement f 0 ðxÞ P 1; for almost all x 6 ^x;
ð24Þ
where ^x ¼ g1 ð1=2Þ. In particular, when xðxÞ ¼ 1 x, f(x) = T(x), that is, the tent map is a special case of Lebesgue map, in which the structure is symmetric. Table 2 lists some examples of such Lebesgue maps with related x functions. Fig. 6 illustrate these Lebesgue maps. Table 2 Lebesgue maps with tent-map-like structures i
fi
2
pffiffiffi 1 j1 2 xj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 j1 x xð2 xÞj
3
1 j1 xð3þxÞ 1þx j
4
1 j1
xð3xÞ 2x j
5
pffiffiffiffiffiffiffiffiffiffiffi 1 jx 1 xj pffiffiffiffiffiffiffiffiffiffiffi 1 j2 1 x 1j
1
6
^xi 1/4
pffiffiffi 1 2=2 pffiffiffi 21 pffiffiffi 2 2 pffiffiffi 2=2 3/4
xi
pffiffiffi ð1 xÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xð2 xÞ 1x 1þx x 1 2x pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2
1 ð1
pffiffiffiffiffiffiffiffiffiffiffi 2 1 xÞ
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Fig. 6. Lebesgue maps with the tent-map-like structures.
6. On the sufficient conditions The abundance of complete chaotic maps with tent-map-like structures becomes clear after we examine the two critical sufficient conditions stated in Theorem 1 more closely. The following facts sheds light on the direction of constructing the invariant measure functions that satisfy the l condition, the set of which is denoted by A. let l 2 A, i = 1, 2, and l = al1 + (1 a)l2, where a 2 [0, 1], l 2 A. That is, if two invariant i(i) Average rule for A: i
measure functions satisfy the l-condition, then their linear combinations also satisfy the l-condition; i = 1, 2, then their direct compositions l12 = l1(l2(x)) and l21 = l2(l1(x)) also let l 2 A, (ii) composition rule for A: i That is, direct compositions of all invariant measure functions that satisfy the l-condition, still satbelong to A. isfy the l-condition. For x functions satisfying the x-condition, we can verify the following rule straightforwardly. For any invariant measure function h and any x function that satisfies the xLemma 1. The composition rule for X: condition, the function constructed by ~ xðxÞ ¼ h1 ðxðhðxÞÞÞ ~ branches at h1 ð^xÞ. also satisfies the x-condition. Furthermore, if x branches at ^x, that is, xð^xÞ ¼ ^x, then x The abundance of diagonal-symmetric x functions is further manifested by the following constructive properties of invariant measure functions, the set of which is denoted as H. Lemma 2. Properties of invariant measure functions: 1. if 2. if 3. if 4. if
h 2 H, then h1 2 H; hi 2 H, i = 1, 2, then their compositions defined by h12 = h1(h2(x)) and h21 = h2(h1(x)) also belong to H; hi 2 H, i = 1, 2, then their average function given by h1 + (1 )h2, with 0 6 6 1, also belongs to H; hi 2 H, i = 1, 2, then their product given by hk1 hl2 , with k,l > 0, also belongs to H.
One direct implication of the above properties is, for any h 2 H, given a b > 0, h(xb) and hb(x) are invariant measure functions as well. With the above rules, we are able to conclude that, for any complete chaotic map f that has a tent-map-like structure, its conjugate defined by f~ h ðxÞ ¼ h1 ðf ðhðxÞÞÞ preserves the tent-map-like structure if and only if h is an invariant measure function that satisfies hðxÞ ¼ 2 hð2 xÞ;
x 2 ½0; 1:
ð25Þ
The very observation explains why in general a conjugate of the tent map does not preserve a similar structure. In fact, given any h 2 H, we can infer from ( h1 ð2hðxÞÞ; x 2 ½0; ^xÞ; 1 T~ h ðxÞ ¼ h ðT ðhðxÞÞÞ ¼ h1 ð2ð1 hðxÞÞÞ; x 2 ½^x; 1; that a h that satisfies (25) is indispensable for T~ h to have a tent-map-like structure.
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Example 4. As listed in Table 2, the map pffiffiffi f1 ¼ 1 j1 2 xj preserves a uniform invariant density. (i) Taking h1(x) = 1 (1 x)3, h1 ðxÞ ¼ h1 1 ðf1 ðh1 ðxÞÞÞ will preserve an invariant density u1 defined by ~ 1 ðxÞ ¼ h01 ðxÞ ¼ 3ð1 xÞ2 : u It is easy to verify that h1 ðxÞ ¼ 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 2 xð3 3x þ x2 Þ:
~ 2 defined by (ii) Taking h2 ðxÞ ¼ 2sin2 ðp4 xÞ, h2 ðxÞ ¼ h1 2 ðf1 ðh2 ðxÞÞÞ will preserve an invariant density u ~ 2 ðxÞ ¼ h02 ðxÞ ¼ u
p p sin x: 2 2
Simple mathematical manipulations give us
Fig. 7. Illustrations of Example 4.
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 8 pffiffiffi > 2 sin p4 x; < h2L ðxÞ ¼ p4 arcsin h2 ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : h ðxÞ ¼ 4 arcsin 1 pffiffi2ffi sin p x: 2R p 4 As illustrated in Fig. 7(c), h2R(x) = 2 h2L(x). And hence, h2 is also a complete chaotic map with a tent-map-like structure so that it can be recast as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 p h2 ðxÞ ¼ 1 1 arcsin 2 sin x: p 4 Computer simulations for h1 and h2 are provided in Fig. 7 (all realized densities are simulated with Maple 9.0 with 50 000 iterations, 100 intervals, and an initial value of x0 = p/10). Furthermore, the above discussion leads us to conclude that, for any complete chaotic map f with a tent-map-like structure that preserves an invariant measure l, its conjugation given by h = l(f(l1(x))) is a Lebesgue map with a tent-map-like structure. For instance, the map f32(x) 1 j1 2x(2 x)j has a tent-map-like structure (Example 1) p=ffiffiffiffiffiffiffi and preserves the invariant measure lðxÞ ¼ p4 arcsin x=2. It can be verified that l(f32(l1(x))) is nothing but the tent map.
7. Lyapunov exponents The simplicity and analytical characteristics of complete chaotic maps with the tent-map-like structure simplify the calculations of various statistical properties of chaotic dynamics. Due to the limited scope, we shall illustrate this point with the Lyapunov exponent (LC). The Lyapunov exponent of a chaotic process defined by K ¼ lim
N !1
Z 1 N 1 X ln jf 0 ðxt Þj ¼ uðxÞ ln jf 0 ðxÞjdx N t¼0 0
ð26Þ
characterizes the degree of chaosity through measuring the average divergence rate of nearby trajectories. The simplicity in the functional form of complete chaotic maps with the tent-map-like structure map provides us a great advantage in the evaluation K both analytically and numerically. Substituting (4) into (26), we obtain Z 1 uðxÞ ln g0 ðxÞdx: ð27Þ K ¼ ln 2 þ 0
On the other hand, if g is constructed through (9), it follows from (15) that lðf ðxÞÞ ¼ 1 þ jlðxÞ lðxðxÞÞj
for x 2 ½0; 1;
which leads to jf 0 ðxÞjuðf ðxÞÞ ¼ uðxÞ uðxðxÞÞx0 ðxÞ: Therefore, K¼
Z
1
uðxÞ ln 0
uðxÞ uðxðxÞÞx0 ðxÞ dx ¼ uðf ðxÞÞ
Z
1
uðxÞ lnðuðxÞ uðxðxÞÞx0 ðxÞÞdx
0
Z
1
uðxÞ ln uðf ðxÞÞdx:
0
Now that the ergodicity of the complete chaotic map f implies that Z 0
1
uðxÞ ln uðf ðxÞÞdx ¼
Z
1
uðxÞ ln uðxÞdx; 0
we finally have Z 1 uðxðxÞÞ 0 x ðxÞ dx: uðxÞ ln 1 K¼ uðxÞ 0
ð28Þ
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Formula (28) enables us to explore LC under different structures of x and u. For instance, consider the case in which f is symmetric, that is, x(x) = 1 x, we have K¼
Z 0
1
uð1 xÞ uðxÞ ln 1 þ dx: uðxÞ
ð29Þ
Further consider two special cases: 1. u is R symmetric as well, i.e., u(x) = u(1 x), then we arrive at a maximum possible Lyapunov exponent 1 K ¼ 0 ln 2uðxÞdx ¼ ln 2, a conclusion drawn in [6]. 2. u is odd-symmetric, i.e., u(x) = 1 u(1 x), we have Z 1 Z 1 1 uðxÞ dx ¼ uðxÞ ln 1 þ uðxÞ lnðuðxÞÞdx ¼ entropy of u: K¼ uðxÞ 0 0 Numerical calculation with (28)p isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi much ffisimpler than that with (26). For instance, referring to Example 1, for l(x) = 1 (1 x)3 and xðxÞ ¼ 1 xð2 xÞ, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðxÞ ¼ 1 jð1 xÞ3 xð2 xÞ xð2 xÞj1=3 : It can be verified that the expression of LC either in (26) or in (27) turns out to be very complex. However, substituting u(x) = 3(1 x)2 and x into (28) leads us to the following simple result: rffiffiffiffiffiffiffiffiffiffiffi Z 1 1 x 2 dx 0:4339: 3ð1 xÞ ln 1 þ K¼ 1 x 2 x 0 Moreover, with (28) in hand, we are able to evaluate comparative statics with respect to the effects of various parameters on the Lyapunov component directly.
8. Conclusion We have provided in theory a set of sufficient conditions so that a class of complete chaotic maps with the tent-maplike structures can be constructed explicitly with closed formulas. Examples and numerical simulations are provided, which have been shown to be consistent with the theoretical findings. Our results greatly extend the applicability of the general formulation of unimodal complete chaotic maps given by (12).
Acknowledgment The author wish to thank Shahria Yousefi for his constructive suggestions. Help comments from an anonymous referee is greatly appreciated. The usual disclaimer applies.
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