CONSTRUCTING CHAOTIC TRANSFORMATIONS WITH CLOSED FUNCTIONAL FORMS

One of the most unique characteristics demonstrated in chaotic dynamical systems is the stable frequency resulted from long-run trajectories, the so-called invariant density. The study of relationship between the analytical structure of a chaotic transformation and its invariant density has been one of the most difficult tasks. Little progress has been made along the endeavor of deriving the analytical structure of invariant density directly from a chaotic transformation [4, 11]. Instead, the inverse Frobenius-Perron approach, that is, finding nonlinear chaotic transformations that preserve prescribed invariant densities, dominates the research trend [2, 3, 9, 10]. Recently, attempts have been made in deviating this approach by exploring generic “macro” characteristics exhibited in some classes of chaotic transformations that share certain similar analytical properties in the invariant densities preserved [1, 5, 6] or in the transformation itself [7, 8]. The current study advances one step further along the latter direction through exploring the class of complete chaotic transformations that are smooth and can be expressed with closed functional forms. Complete chaotic transformations with closed functional form such as the logistic map (quadratic transformation) and cusp-shaped map have played an important and indispensable role in the historical development of nonlinear dynamic theory. Their simplicity and analytical characteristics help not only in the practical modeling in both natural and social sciences but also in pedagogical illustrations of various dynamical theories. In practical application, to fit a random-like time series (either empirical or experimental)


Introduction
One of the most unique characteristics demonstrated in chaotic dynamical systems is the stable frequency resulted from long-run trajectories, the so-called invariant density. The study of relationship between the analytical structure of a chaotic transformation and its invariant density has been one of the most difficult tasks. Little progress has been made along the endeavor of deriving the analytical structure of invariant density directly from a chaotic transformation [4,11]. Instead, the inverse Frobenius-Perron approach, that is, finding nonlinear chaotic transformations that preserve prescribed invariant densities, dominates the research trend [2,3,9,10]. Recently, attempts have been made in deviating this approach by exploring generic "macro" characteristics exhibited in some classes of chaotic transformations that share certain similar analytical properties in the invariant densities preserved [1,5,6] or in the transformation itself [7,8]. The current study advances one step further along the latter direction through exploring the class of complete chaotic transformations that are smooth and can be expressed with closed functional forms.
Complete chaotic transformations with closed functional form such as the logistic map (quadratic transformation) and cusp-shaped map have played an important and indispensable role in the historical development of nonlinear dynamic theory. Their simplicity and analytical characteristics help not only in the practical modeling in both natural and social sciences but also in pedagogical illustrations of various dynamical theories. In practical application, to fit a random-like time series (either empirical or experimental) 2 Chaotic transformations with closed forms that exhibits stable frequency in long-run with a deterministic dynamic system, a system with closed functional form is always the first choice, since no matter the parameter estimation, the comparative study of parameters or the bifurcation analysis can all be carried out more efficiently than it would otherwise be taken for a piecewise transformation.
The paper is organized as follows. Section 2 defines the concept SCC transformation. Section 3 then proposes an approach in terms of sufficient conditions to construct smooth chaotic transformations with closed form. The same methodology developed applies to the construction of analytical chaotic transformations. Section 4 is devoted to a more detailed analysis of sufficient conditions.

Complete chaotic transformations with closed forms
We start with a general description of complete chaotic transformations defined on a unit-interval I= [0,1]. The particular choice of unit-interval as domain is not restrictive since any chaotic transformation defined on a closed interval can always be transformed to a transformation defined on the unit-interval through variable substitution (linear topological conjugation).
A chaotic transformation F : I → I is said to be unimodal if there exists a participation point (sectioning point) x c ∈ I such that the transformation F can be expressed as A unimodal transformation F defined in (2.1) is said to be complete chaotic (fully chaotic) if it is (i) ergodic with respect to the Lebesgue measure and (ii) chaotic in the probabilistic sense, that is, an absolutely continuous invariant density ϕ is preserved.
A function defined on a real domain is said to possess a closed form (a closed-format, a non-piecewise-defined form) if it is expressed as a composition of elementary functions, their inverses, and smooth functions that are not piecewise-defined. In mathematics, there is no rigourously and precisely defined antonym for "piecewise-defined function" since all functions can be assumed to be piecewisely defined. The widely adopted terminology-"closed form"-is regarded as a most closed one. However, the vaguely defined terminology "closed form" does not include "absolute function" in general. However, we will abandon such exclusion since as to be illustrated that any function consisting of an absolute term can be recast to an expression without absolute term. For example, |1 − 2x| can be rewritten as (1 − 2x) 2 .
A complete chaotic transformation F(x) = ( f L (x), f R (x)) xc is said to have a closed form if (i) f L and f R have a common functional form, that is, f L ≡ f R ≡ f , and (ii) f itself has a closed form.
Weihong Huang 3 Typical examples of complete chaotic transformations that appear with closed forms include (i) the logistic map which preserves an invariant density ϕ Q (x) = 1/(π x(1 − x)); (ii) the cusp-shaped map modeling the intermittency [4] C which preserves an invariant density ϕ C (x) = 2x; (iii) the tent map which preserves a uniform invariant density, that is, ϕ T (x) = 1.
It can be noted immediately that Q has a genuine closed format in the sense that it can never be recast to a piecewise-defined format, while the transformations C and T, respectively, defined in (2.3) and (2.4) are piecewise-defined in nature ( f L = f R ) since they can be further simplified to To distinguish the complete chaotic transformations that have a genuine closed format like Q with the "fakes" like C and T, we thus introduce the following definitions formally. (i) f cannot be alternatively recast in piecewise-defined format, that is, It needs to emphasize that it is quite difficult, if not impossible, to tell just from the functional form alone whether a complete chaotic transformation is SCC or not. For example, T can be also recast as with no absolute term at all: In the definition of SCC, only first-order differentiable smoothness at the sectioning point x c is imposed instead of a more strict requirement of "analyticity of f ," which demands infinitive differentiability for all x ∈ (0,1). Therefore, an analytical complete chaotic transformation is definitely an SCC transformation. The converse, however, is not necessarily true, as shown in the next example.

Example 2.2 (analytical transformations and nonanalytical SCC transformations).
The following complete chaotic transformations have genuine closed formats and are differentiable at the respective sectioning points: The absolutely continuous invariant densities preserved by F i 's are , respectively. The graphs of F i and their invariant densities ϕ i for i = 1,2,3,4 are shown in Figure 2.1. While F 1 , F 2 , and F 3 are analytical in (0,1), F 4 is not because the derivatives do not exist at the two reflectional points in (0,1).

Remark 2.3.
It is interesting to see that F 1 and F 2 defined in (2.6) have a similar structure as the logistic map. In fact, any complete chaotic transformation with closed form can be expressed in a logistic-map-like format: where h is an invariant measure function defined on the unit-interval I.

Construction of NCC transformations
With a general unimodal chaotic transformation F defined by (2.1), for any given point x ∈ I, there exist two preimages: f −1 L (x) and f −1 R (x), the functional relation between which can be described by an auxiliary function ω that transforms one preimage of F in a segment to a unique counterpart in the other [10]: Apparently, ω is differentiable except at finite number of points. Moreover, it is monotonically downward-sloping so that For a complete chaotic transformation F preserving an absolute continuous invariant density ϕ, denote μ(x) = x 0 ϕ(x)dx as the related absolute continuous invariant measure, then it is well known that the following Frobenius-Perron equation holds: that is, Chaotic transformations with closed forms With the auxiliary function ω : I → I defined in (3.1), (3.4) can be recast as It can be noticed immediately that as long as ω has a closed form in I and processes the diagonal-symmetric property in the sense of the ω function defined in (3.1) will become a closed function so that (3.5) is simplified into And hence, as long as neither μ nor μ −1 is piecewise-defined function, F given by (3.7) is an NCC transformation.
The condition (3.6) will be referred to as the ω-condition. Denote Ω as the set of ω functions that meet ω-condition. Typical members of Ω include It deserves to warn that although μ (and hence μ −1 ) is monotonic over its domain I, it is not necessarily defined (or keeping the same monotonicity) beyond I. And hence, it should not be inferred from (3.8 , which would lead us to a contradictory conclusion: To meet the condition (3.8), domain of μ −1 should be expanded beyond I and should be symmetric with respect to unity in the sense that (3.9) Finally, to ensure that F (x c ) = 0, μ −1 itself must be differentiable smooth at x = 1, so is ω at x c .  The above observations can be summarized into the following proposition.
is an SCC transformation that preserves the invariant measure μ if the following conditions are satisfied: (a) (ω-condition) ω has a closed form and is differentiable at x c , that is, ω (x c ) < 0, and diagonal-symmetric in the sense of (3.6), (b) (μ-condition) μ −1 has a closed form and is differentiable at x = 1, that is, μ −1 (1) = 0, and symmetric with respect to the point x = 1 in the sense of (3.9).
We can check immediately that the invariant measure μ 0 (x) = (2/π)arcsin( √ x) preserved by the logistic map Q satisfies the μ-condition, and ω 0 (x) = 1 − x implied by the symmetry of Q also meets the ω-condition.
Denote M as the set of invariant measures that fulfill the μ-condition.
n is even. Before elaborating more on the ω-condition and μ-condition, we offer some SCC transformations that have simple analytical structures. Also provided are the Lyapunov exponents (LCs) that measure the degree of chaoticity of the related SCC transformations. In [8], a simplified formula for calculating LCs is provided: Formula (3.11) enables us to explore LC under different structures of ω and ϕ. For instance, consider the case in which the transformation is symmetric, that is, ω(x) = 1 − x, we have First, we consider the symmetric SCC transformations, which are constructed with ω(x) = 1 − x.

Example 3.2 (symmetric SCC transformations). With ω(x)
(3.13) The shapes of μ −1 i 's are illustrated in Figure 3.1(a). The functional graphs of η i 's together with some typical histograms generated from numerical simulations are presented in Figure 3.3. With a same symmetric structure, SCC transformations that are flatter in the middle range and steeper on the two ends are less chaotic in the sense of lower LC values.
Next, we will examine the relationship between the shape of ω and the related Lyapunov component for a given invariant density. The graphs of ω i 's are illustrated in Figure 3.2(a). The functional graphs of θ i 's together with some typical histograms generated from numerical simulations are presented in Figure 3.4. Even with the invariant density, complete chaotic transformations that are skew to one end are less chaotic in the sense of lower LC values.

More on the sufficient conditions
It is not clear whether the ω-condition and the μ-condition together are the necessary conditions for a unimodal complete chaotic transformation to have a smooth closed form. Nevertheless, each individual condition is indeed a necessary condition if the other is assumed to hold true. In other words, to construct an SCC transformation, either condition is indispensable if the other is satisfied. If we release the μ-condition and impose  the ω-condition alone, we can only obtain a closed complete chaotic transformation which usually but not necessarily contains an absolute value function in its final appearance. However, it cannot be emphasized more that the "differentiability" and the "closedform" requirements are critical both in the ω-condition and in the μ-condition. The following counterexamples show that how a piecewise-defined ω function that satisfies (3.6) or a piecewise-defined invariant measure μ that satisfies (3.9) (even if they are infinitively differentiable elsewhere) results in a nonsmooth complete chaotic transformation.
As illustrated in Figure 3.1(b), although μ −1 4 appears with a closed format and is symmetric with respect to x = 1 as well, it is not differentiable at x = 1. This is because μ −1 4 is essentially a piecewise-defined function over the range [0,2] since it can be recast as μ −1 If we assume ω(x) = 1 − x, formulation (3.10) yields a closed but not smooth complete chaotic transformation given by then, as illustrated in Figure 3.2(b), ω 4 is not only diagonal-symmetric in [0,1] but also smooth and differentiable anywhere in (0,1). Nevertheless, ω 4 violates the "closed-form" requirement (ω 4 = ω 4 .) It follows from (3.5) that The graph of θ 4 is shown in Figure 4.2. It is interesting to see that θ 4 is smooth at the sectioning point x c so as to give an illusion of SCC transformation. Also seen in Figure 4.2 is a typical realized density from computer simulations that is consistent with the invariant density ϕ θ4 (x) = 1/(2 To form the invariant measures that satisfy μ-condition, the following lemma suggests two constructive approaches.

Lemma 4.3. For an analytical two-to-one complete transformation f defined on
is a member of M. On the other hand, if h is an invariant measure function that is odd- is a member of M.