The Spectrum of Chaotic Time Series (II): Wavelet
Analysis
Goong Chen
1,2, Sze-Bi Hsu
3, Yu Huang
2,4, and Marco A. Roque-Sol
1Abstract
This paper is a continuation of Part I where the authors treated the Fourier analysis of chaotic time series generated by a chaotic interval map. Here, we perform multi-resolution analysis by using wavelet coefficients and characterize some necessary and sufficient conditions for the occurrence of chaos by the exponential growth with respect to the number of iterations n of certain sums of the wavelet coefficients.
Keywords: Topological entropy, total Variation, chaos, wavelet, multiresolution analysis.
1. Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A. Email: gchen@math.tamu.edu
2. Work completed while visiting Center for Theoretical Sciences, National Tsing Hua University, Hsinchu, Taiwan, R.O.C.
3. Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. Supported in part by a grant from NSC of Rep. of China.
Email: sbhsu@math.nthu.edu.tw
4. Department of Mathematics, Sun Yat-Sen (Zhongshan) University, Guangzhou 510275, P.R. China. Supported in part by the National Science Foundation of Guangdong Province and NSF of China. Email: stshyu@mail.sysu.edu.cn
1
Introduction
This paper is a continuation of Part I [3]. Here, again we analyze the spectrum of a time series comprised of the iterates of a deterministic chaotic map, but by using wavelets.
Wavelet transforms use a waveform (“small wave”, the mother wavelet function ψ(t)) which is scaled and translated to process, reconstruct and match the input signal ([5, 6, 8]). Such transforms can be classified into three types: the continuous wavelet transform (CWT), the discretized wavelet transform (DWT) and multiresolution-based wavelet trans-form (MRA, for multiresolution analysis). In comparison with the Fourier transtrans-form, in which signals are represented as an integral or sum of real or complex sinusoidals and the transform is only localized in frequency, wavelets are localized in both time and frequency . In a sense, CWT may be likened to the continuous Fourier transform, and DWT may be to the Fourier series. But MRA adds an extra auxiliary function, the father wavelet function
φ(t), constituting the basis for the algorithm of the fast wavelet transform. Computationally,
for a data set of size N, the complexity of the discrete wavelet transform takes O(N) time as compared to O(N log N) for the fast Fourier transform. In practical applications, for the sake of efficiency, one often prefers continuously differentiable functions with compact support as (prototype) mother wavelet functions. During the past two decades, we have seen a vast number of applications of wavelets to signal and image processing, that often supercedes the conventional Fourier transform. Thus, it’s not surprising to see earlier work such as [1, 10] in an attempt to apply wavelets to the analysis of chaos. Nevertheless, rigorous mathematical results are largely lacking, to the best of our knowledge. This motivates our work in this paper.
In this paper, in Section 2 we first review some basic properties of wavelet transforms and notation as prerequisites for the subsequent development. In Section 3, we link the wavelet transform with the total variation of a map via the Fourier transform and the Parseval identity (Theorem 3.1). We then further derive necessary and sufficient conditions for chaos in terms of MRA coefficients in Corollary 3.2. Such conditions, determined by whether certain sums of the wavelet coefficients grow exponentially with respect to n (i.e., the number of iterations), have a good deal of similarities with the main results we obtained earlier in Part I [3] based on the Fourier analysis. In Section 4, using Haar wavelets we have demonstrated that the theorems we have obtained in Section 3 are quite tight.
2
Prerequisite: notations and properties of wavelets
We inherit the theoretical background from Part I [3]. Now, for f ∈ L1(R), define its
Fourier transform by ˆ f (ω) = Z R e−i2πωtf (t)dt. (2.1)
Then by the denseness of L1(R) ∩ L2(R) in L2(R), we can define ˆf for any f ∈ L2(R) by the
usual continuity argument. From this, we have the Parseval identity
and the Plancherel formula
kf k = k ˆf k, f ∈ L2(R), (2.3) where h , i and k · k in (2.2) and (2.3) denote, respectively, the L2-inner product and the
L2-norm. A function ψ in L2(R) is called a (mother) wavelet if ψ satisfies the admissible
condition: Cψ = Z R | ˆψ(ω)|2 |ω| dω < +∞. (2.4)
If, in addition, ψ ∈ L1(R) then ˆψ is continuous and thus (2.4) implies
Z 1 −1
| ˆψ(ω)|2
|ω| dω < +∞,
further implying ˆψ(0) = 0. That is,
Z
R
ψ(t)dt = 0. (2.5)
For a given mother wavelet ψ, we can generate a doubly-indexed family of wavelets by dilation and translation:
ψs,b(t) = 1 p |s|ψ µ t − b s ¶ .
Definition 2.1. Let ψ be a mother wavelet and f ∈ L2(R). The continuous wavelet
trans-form of f with respect to ψ is given by (W f )(s, b) = hf, ψs,bi = Z R fp1 |s|ψ µ t − b s ¶ dt. ¤ (2.6)
A function ψ ∈ L2(R) is said to be a dyadic wavelet if the following stability condition
holds: there exist two positive constants A, B such that
A ≤X
j∈Z
| ˆψ(2−jω)|2 ≤ B, a.e. ω. (2.7)
It is easy to see that the stability condition (2.7) implies the admissible condition (2.4). More precisely, we have the following.
Lemma 2.1. Let ψ be a dyadic wavelet, i.e., condition (2.7) holds. Then
A ln 2 ≤ Z ∞ 0 | ˆψ(ω)|2 |ω| dω, Z 0 −∞ | ˆψ(ω)|2 |ω| dω ≤ B ln 2. Moreover, if A = B in (2.7), then Cψ = Z R | ˆψ(ω)|2 |ω| dω = 2A ln 2. ¤
Definition 2.2. Let ψ be a dyadic wavelet and f ∈ L2(R). The dyadic wavelet transform
of f with respect to ψ is defined as
(Wjf )(b) = hf, 2j/2ψ(2j(· − b))i = 2j/2 Z
R
f (t)ψ(2j(t − b))dt. ¤ (2.8)
Computationally, it is impossible to analyze a signal using all coefficients from the con-tinuous wavelet transform (2.6). Thus, a discretized wavelet transform is in order. It is of great advantage to have a mother wavelet ψ such that
©
ψj,n(t) = 2j/2ψ(2jt − n)ª(j,n)∈Z2,
forms an orthonormal basis in L2(R). Multi-resolution analysis (MRA) is an effective
ap-proach for constructing such a basis. MRA consists of a sequence of closed subspaces Vj of
L2(R) with the following conditions
(1) · · · ⊂ V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 ⊂ · · · ;
(2) ∪jVj = L2(R), ∩jVj = {0}; (3) f ∈ Vj ⇔ f (2·) ∈ Vj+1;
(4) f ∈ V0 ⇔ f (· − k) ∈ V0, ∀k ∈ Z;
(5) There exists a ϕ ∈ V0 such that {ϕ(· − k)}k is an orthonormal basis for V0.
Such the function ϕ is said to be a scaling function or a father wavelet.
For a given MRA, a standard way to construct an orthonormal basis in L2(R) is given
in [6, Theorem 5.1.1].
Lemma 2.2. If a ladder of closed subspaces {Vj}j∈Z in L2(R) satisfies conditions (1)–(5)
above, then there exists an associated orthonormal wavelet basis {ψj,k | j, k ∈ Z} for L2(R)
such that
Pj+1 = Pj+ X
k
h·, ψj,kiψj,k, (2.9)
where Pj is the orthogonal projection onto Vj. A feasible way to construct the wavelet ψ is
by its Fourier transform
ˆ ψ(ω) = e−iπωm 0(π(ω + 1)) ˆϕ ³ω 2 ´ , (2.10)
where m0 is a periodic function (with period 2π) called the faltering function, given by
m0(ω) = 1 2 X k hke−ikω,
where {hk} is determined uniquely by the representation of ϕ:
(This decomposition is unique since {ϕ(2 · −k)}k is an orthonormal basis for V1 by the
assumptions.)
Similarly, the function ψ given by (2.10) has the decomposition ψ(t) =X
k
gkϕ(2t − k), (2.12)
where
gk = (−1)k−1¯h1−k. ¤
It follows from Lemma 2.2 that each f ∈ L2(R) has two representations: the first is
f (t) = X
j,k∈Z
dj,kψj,k(t), (2.13)
and the second is, for any j0 ∈ Z,
f (t) =X k∈Z cj0,kϕj0,k(t) + X j≥j0 X k∈Z dj,kψj,k(t), (2.14) where cj,k = hf, ϕj,ki, dj,k = hf, ψj,ki.
Also an MRA leads naturally to a fast scheme for the computation of the wavelet coefficients
cj,k and dj,k important in applications. From (2.11), we have
cj,k = Z f (t)2j2ϕ(2jt − k)dt =X l ¯hl Z f (t)2j2ϕ(2j+1t − 2k − l)dt = √1 2 X l ¯hl Z f (t)ϕj+1,2k+1(t)dt.
So we have a fast scheme for the decomposition computation:
cj,k = √1 2
X l
¯hl−2kcj+1,l. (2.15)
Similarly, from (2.12), we have
dj,k = 1 √ 2 X l ¯gl−2kcj+1,l. (2.16)
On the other hand, let Wj be the orthogonal complement of Vj in Vj+1. That is, Vj+1=
Vj L
Wj. Since {ϕj,l}l∈Z and {ψj,l}l∈Z are orthonormal bases in Vj and Wj, respectively,
ϕj+1,k ∈ Vj+1 can be decomposed uniquely as
ϕj+1,k(t) = X
where, by orthonormality, ak,l = Z ϕj,l(t)ϕj+1,l(t)dt = 2j+1/2 Z ϕ(2jt − l)ϕ(2j+1t − k)dt = 2j+1/2X m ¯hm Z ϕ(2j+1t − 2l − m)ϕ(2j+1t − k)dt = √1 2 X m ¯hm Z ϕ(t − 2l − m)ϕ(t − k)dt = √1 2 X m ¯hmδ2l+m,k= 1 √ 2¯hk−2l. Similarly, bk,l = 1 √ 2¯gk−2l.
Substituting the above two equations into (2.17) and then taking inner product with f , we have cj+1,k = 1 √ 2 X l (hk−2lcj.l+ gk−2ldj,l). (2.18)
Formula (2.18) is called the fast scheme for reconstruction computation.
3
Detecting chaos with wavelets
Having furnished the wavelet prerequisite, we now enter the main section for the study of chaotic time series by wavelets. In what follows, we always assume that f : I = [0, 1] → I is continuous and piecewise monotone with finitely many extremal points. We take f as a function on the whole line R by extending f (t) = 0 for all t /∈ I. Thus f ∈ L2(R) ∩ L1(R).
Its Fourier transform is ˆ f (w) = Z R f (t)e−i2πwtdt = Z 1 0 f (t)e−i2πwtdt. (3.1)
As in Part I [3], the total variation VI(f ) of f on I is defined as before to be the supremum of all sums
m X k=1
|f (tk) − f (tk−1)|,
with respect to all partitions {tk}m
k=0 on [0, 1] such that 0 = t0 < t1 < · · · < tm = 1. In brevity, we write VI(f ) as V (f ). The following lemma is a version of [7, Theorem 2.3.6]. Lemma 3.1. Let f : I → I be continuous with bounded total variation. Then
In particular, when f (1) = f (0) and ω = n is an integer, equation (3.2) becomes 2π|n|| ˆf (n)| ≤ V (f ).
Proof. If ω = 0, (3.2) holds obviously. Assume w 6= 0, then one may write (3.1) as
ˆ f (ω) = Z 1 0 f (t)d · e−i2πωt −i2πω ¸ . Set g(t) = e−i2πωt −i2πω.
From the definition of an integral, for any given ε > 0, there exists a sufficiently fine partition 0 = t0 < t1 < · · · < tm = 1 of the interval [0, 1], such that
| ˆf (ω) −
m X
k=1
f (tk)[g(tk) − g(tk−1)]| < ε.
Denoting by P the sum appearing within the absolute value signs above, and applying partial summation, we obtain
X = [f (1) − f (0)]g(0) − m−1 X k=1 [f (tk+1) − f (tk)]g(tk). Note here that we have used the fact that g(1) = g(0) = 1
−i2πω. Thus, | ˆf (ω)| < ε + |f (1) − f (0)||g(0)| + m−1X k=1 |f (tk+1) − f (tk)||g(tk)| ≤ ε + |f (1) − f (0)| 1 2π|ω| + 1 2πV (f ) 1 |ω|, since |g(t)| ≤ 1
2π|ω|. Letting ε −→ 0, we have obtained the desired result.
Now we consider the one-dimensional dynamical system (I, f ). That is, we consider the dynamical behavior of the iterates f°n of f as n −→ ∞.
For a given mother wavelet ψ ∈ L2(R), the continuous wavelet transform of f°n with respect to ψ is W (f°n)(s, b) = hf°n, ψs,bi = Z R f°n(t)p1 |s|ψ µ t − b s ¶ dt. (3.3)
Here and in the following, we view f°n as a function on the whole real line R by setting
Theorem 3.1. Let f : I −→ I be continuous and piecewise monotone with finitely many
extremal points and ψ be a mother wavelet. In addition, if ψ satisfies
Aψ ≡ Z R | ˆψ(ω)| |ω| dω < +∞, (3.4) then sup s6=0,b∈R {|s|−1/2|W (f°n)(s, b)|} ≤ Aψ 2π(1 + V (f °n)), ∀n = 1, 2, 3, . . . . (3.5)
Proof. It suffices to prove that
sup s6=0,b∈R{|s|
−1/2|W f (s, b)|} ≤ Aψ
2π(1 + V (f )). (3.6)
From the Parseval identity, we obtain (W f )(s, b) = Z R fp1 |s|ψ µ t − b s ¶ dt = Z R ˆ f (ω)p|s| ˆψ(sω)ei2πbωdω.
By Lemma 2.2, (3.4) and the facts that f (0), f (1) ∈ [0, 1], the above leads to
|(W f )(s, b)| ≤ Z R | ˆf (ω)|p|s|| ˆψ(sω)|dω ≤ Z R |f (1) − f (0)| + V (f ) 2π|ω| p |s|| ˆψ(sω)|dω ≤ p |s| 2π (1 + V (f )) Z R | ˆψ(ω)| |ω| dω = p |s|Aψ 2π (1 + V (f )). Therefore, we have established (3.6).
Replace f in (3.6) with f°n, we get (3.5).
Condition (3.4) can be eliminated, as shown in the following
Corollary 3.1. Under the same assumptions as in Theorem 3.1 except (3.4), we have sup s6=0,b∈R {|s|−1/2|W (f°n)(s, b)|} ≤ A(V (f°n) + 1), ∀n = 1, 2, 3, . . . , where A = kθ(t)k∞, and θ(t) ≡R0tψ(τ )dτ .
Proof. We follow the proof of [2, Lemma 2.3.2, p. 24] that for any interval [c, d] in R,
Z d c
g(t)ψ(t)dt ≤ kθ(t)k∞(V[c,d](g) + kgk∞),
where g(t) = f (st + b) and V[c,d](g) denote the variation of g on [c, d]. Thus,
|(W f )(s, b)| = | Z R fp1 |s|ψ µ t − b s ¶ dt| = |√s s Z (1−b)/s −b/s f (sτ + b)ψ(τ )dτ | ≤√skθ(t)k∞(V[−b s,1−bs ](g) + kgk∞) ≤√skθ(t)k∞(V (f ) + kf k∞) ≤√skθ(t)k∞(V (f ) + 1). We thus obtain the result by using f°n for f .
Example 3.1. The Haar wavelet may be said to be the earliest wavelet, dating back to Haar’s work in the early 20th century. It is defined as
ψ(t) = 1 0 ≤ t < 1 2, −1 1 2 ≤ t ≤ 1, (= χ[0,1/2)(t) − χ[1/2,1](t)), 0 otherwise,
where χJ denote the characteristic function of an interval J. We have ˆ
ψ(ω) = 1
i2πω(1 − e
iπω)2.
A direct computation shows that condition (3.4) is satisfied.
We will use Haar’s wavelet shortly as a major example. ¤
Example 3.2. For a family of derivatives of a Gaussian wavelet
ψ(t) = (−1) m+1 p Γ(m + 1/2) dm dtm(e −t2 2),
by direct computation, one obtains ˆ ψ(ω) = −i m p Γ(m + 1/2)ω me−ω2 2 , , m = 1, 2, · · ·
Thus, again, condition (3.4) holds for m = 1, 2, · · · . ¤
Corollary 3.2. Under the assumptions of Theorem 3.1, if there exist sn 6= 0 and bn ∈ R such that lim n→+∞ 1 nln |sn| −1/2|W (f°n)(s n, bn)| > 0, (3.7) then lim n→+∞ 1 n ln V (f °n) > 0.
Thus, the system (I, f ) has positive topological entropy and is chaotic in the sense of Li-Yorke.
Proof. The first part comes directly from Theorem 3.1. The second part follows from the
first part and the results in [2, 3, 7].
Without condition (3.4), instead, if we assume that {ψj,k; j, k ∈ Z} forms an orthonormal wavelet basis for L2(R) which is derivable by Lemma 2.2, we can establish not only
neces-sary conditions but also sufficient conditions for chaos of the system (I, f ) in terms of the corresponding wavelet coefficients. There are two ways to decompose f in terms of the basis for L2(R). One is to extend f as a function on the whole line R by setting f (t) = 0 outside
the interval [0, 1] and then decompose it as (2.13) or (2.14). This approach is often not appreciated in the point of view of wavelet analysis since it introduces an artificial “jump” at the boundary of I, and is reflected in the wavelet coefficients. Another is to modify the orthonormal basis for L2(R) such that it forms an orthonormal basis for L2(0, 1) and then
decompose f in terms of the modified basis. There are several different ways to achieve this modification. But the basis for L2(0, 1) obtained becomes more complicated and as a result it
is difficult to compute the corresponding wavelet coefficients analytically. Here since we are concerned only with the dynamics, the artificial jump at the boundary caused by the former approach does not affect our analysis. Therefore, we have decided to take the extension by setting f = 0 outside the interval I.
By Lemma 2.2, f has decomposition (2.13) or (2.14). In addition, if the scaling function
ϕ and the mother function ψ have compact supports, which, without loss of generality,
we may assume that they are contained in (0, 1), then for any given j0 > 0, j ≥ j0 and
n ≥ 2j, the support of ψ
j,n has no intersection with [0, 1]. Nor does ϕj0,n for n ≥ 2j0. Thus
decomposition (2.14) can be written as
f (t) =X j≥j0 2j−1 X k=0 dj,kψj,k(t) + 2Xj0−1 k=0 cj0,kϕj0,k(t), (3.8) for any j0 > 0.
Theorem 3.2. Let ϕ and ψ be real and have compact supports in [0, 1]. If f ∈ W1,1(0, 1),
then there exists a constant A > 0 such that
V (f ) ≥ A sup j≥j0 2j/2 2j−1 X k=0 |hf, ψj,ki|. (3.9)
Proof. Let
θ(t) =
Z t
0
ψ(τ )dτ.
Then from (2.5) θ has support in [0, 1]. For any j ≥ j0 doing integration by parts, we have 2j−1 X k=0 |hf, ψj,ki| = 2j−1 X k=0 | Z 1 0 f (t)2j/2ψ(2jt − k)dt| = 2j−1 X k=0 | Z 1 0 f0(t)2−j/2θ(2jt − k)dt| ≤ 2−j/2 2j−1 X k=0 | Z 1 0 |f0(t)||θ(2jt − k)dt|.
Since the support of θ is contained in [0, 1], it follows that
2j−1 X k=0 |hf, ψj,ki| ≤ 2−j/2sup t∈R |θ(t)| Z 1 0 |f0(t)|dt = 2−j/2A−1 Z 1 0 |f0(t)|dt | {z } V (f ) , where A−1 = sup t∈R|θ(t)|. Thus we have (3.9).
Theorem 3.3. Let ϕ and ψ have compact supports in [0, 1]. If, in addition, V (ψ) < +∞,
then there exists a constant B > 0 such that
V (f ) ≤ B X j≥j0 2j−1 X k=0 2j/2|hf, ψ j,ki| + 2j0/2 2Xj0−1 k=0 |hf, ϕj0,ki| . (3.10)
Proof. It follows from (3.8) that
V (f ) ≤ X j≥j0 2j−1 X k=0 2j/2|hf, ψ j,ki|V (ψj,k) + 2j0/2 2Xj0−1 k=0 |hf, ϕj0,ki|V (ϕj0,k), in which V (ψj,k) = Z 1 0 2j/22j|ψ0(2j− k)|dt ≤ 2j/2V (ψ). And similarly, V (ϕj0,k) ≤ 2 j0/2V (ϕ).
The inequality (3.10) follows from the above three inequalities.
Corollary 3.3. Assume that ϕ and ψ have compact supports in [0, 1] with finite total
(1) If f ∈ W1,∞ and there exists an increasing integer sequence j n→ ∞ and lim n→∞ 1 nln 2jn/2 2Xjn−1 k=0 |hf°n, ψjn,ki| > 0, then lim n→∞ 1 n ln V (f °n) > 0,
and f has chaotic oscillations.
(2) If f ∈ W1,∞ and there exists an increasing integer sequence j
n→ +∞ such that lim n→∞ 2Xjn−1 k=0 |hf°n, ψ jn,ki| > 0, then lim n→∞ 1 n ln V (f °n) > 0,
and f has chaotic oscillations.
(3) Conversely, if f ∈ W1,∞ and lim n→∞ 1 n ln V (f °n) > 0, then X j≥j0 2j−1 X k=0 2j/2|hf°n, ψj,ki| + 2j0/2 2Xj0−1 k=0 |hf°n, ϕj0,ki| grows exponentially as n −→ ∞. ¤
Example 3.3. Consider the tent map (as in Part I [3]):
f (t) = ½ 2t if 0 ≤ t < 1 2, −2(t − 1) if 1 2 ≤ t ≤ 1.
This map is well known to be chaotic. Here we apply Corollary 3.3 part (1) to this map. We choose the Haar wavelet as in Example 3.1:
ψ(t) = χ[0,1
2)(t) − χ[12,1](t). (3.11)
For j > 0, consider the wavelet coefficients of f°n, where by Part I [3],
f°n(t) = ½ 2nt − 2(l − 1), if 2(l−1) 2n ≤ t < 2l−12n , −2nt + 2l, if 2l−1 2n ≤ t ≤ 22ln, (3.12) l = 1, 2, · · · , 2n−1.
For k = 0, 1, · · · , 2j − 1, we compute the wavelet coefficients of f°n with respect to the wavelet basis {ψj,k}: dn j,k ≡ Z 1 0 f°n(t)ψ j,k(t)dt = 2j/2 Z 1 0 f°n(t)ψ(2jt − k)dt = 2−j/2 Z 2j−k −k f°n(2−j(τ + k))ψ(τ )dτ = 2−j/2 Z 1 0 f°n(2−j(τ + k))ψ(τ )dτ,
by change of variables and noting that supp(ψ) = [0, 1], where “supp” means “the support of”. Substituting (3.11) into the above equation, we have
dn j,k = 2 j 2 "Z 2−j(k+1/2) 2−jk f°n(t)dt − Z 2−j(k+1) 2−j(k+1/2) f°n(t)dt # . (3.13)
Taking j = n, from (3.12), it follows that when k is even
dn n,k = 2 n 2 "Z 2−n(k+1/2) 2−nk (2nt − k)dt − Z 2−n(k+1) 2−n(k+1/2) (2nt − k)dt # = 2n22−(n+1) µ −1 2 ¶ ,
and when k is odd,
dn n,k = 2 n 2 "Z 2−n(k+1/2) 2−nk (−2nt + k)dt − Z 2−n(k+1) 2−n(k+1/2) (−2nt + k)dt # = 2n22−(n+1) µ 1 2 ¶ . Therefore 2n−1 X k=0 |dn n,k| = 2 n 2−2 > 0,
which, by Corollary 3.3, implies
lim n→∞
1
n ln V (f
°n) > 0.
In fact, combining with Theorem 3.2, we have lim
n→∞ 1
nln V (f
4
The growth rate of the total variation of f
°nin terms
of the wavelet coefficient of the Haar wavelet
Let f : [0, 1] → [0, 1] be continuous and piecewise monotone, and ψ be a mother wavelet with compact supports in [0, 1]. For j ≥ j0 with respect to the scaling function ϕ, denote by
dn
j,k the wavelet coefficients of f°n. That is,
dn
j,k = hf°n, ψj,ki = Z 1
0
f°n(t)2j/2ψ(2jt − k)dt.
Then by Theorem 3.2, we have
V (f°n) ≥ A sup j≥j0 2j/2 2j−1 X k=0 |dnj,k|, (4.1)
where A > 0 is defined in Theorem 3.2. We denote by ˜dn
j,k the coefficients of f with respect to the basis { ˜ψj,k(t) ≡ 2j/2ψ(2jt −
k −1
2)}j,k∈Z. That is, for k = 0, 1, · · · , 2j − 2,
˜ dn j,k = Z 1 0 f°n(t)2j/2ψ µ 2jt − k − 1 2 ¶ dt, d˜n j,2j−1 = 0. (4.2)
Please note that for k ≥ 2j − 1 the support of ˜ψ
j,k has empty intersection with [0, 1]. Note that the basis { eψj,k}j,k∈Zis constructed from the translation and dilation of eψ(t) = ψ(t−1/2). The coefficients ˜dn
j,k in (4.2) will help us establish Theorem 4.1 below. By the same approach as in the proof of Theorem 3.2, we also have
V (f°n) ≥ A sup j≥j0 2j/2 2j−1 X k=0 | ˜dn j,k|. (4.3) If we denote Wψ(f°n) = lim j→∞2 j/2 2j−1 X k=0 (|dn j,k| + | ˜dnj,k|), (4.4) then from (4.1) and (4.3), we obtain
2V (f°n) ≥ AW ψ(f°n). Thus we have lim n→∞ 1 n ln V (f °n) ≥ lim n→∞ 1 nln Wψ(f °n). (4.5)
A natural question here is if there exists a wavelet such that inequality (4.5) becomes equality. We have the following.
Theorem 4.1. Let f : [0, 1] → [0, 1] be continuous and piecewise monotone, and ψ be the
Haar wavelet given by (3.11). Then
lim n→∞ 1 n ln V (f °n) = lim n→∞ 1 nln Wψ(f °n), (4.6) where Wψ(f°n) is defined by (4.4).
Proof. Since the mother wavelet we choose here is the Haar wavelet, we have (3.13). Thus
dn j,k = 2 j 2 "Z 2−j(k+1/2) 2−jk f°n(t)dt − Z 2−j(k+1) 2−j(k+1/2) f°n(t)dt # = 22j(f°n(ξk) − f°n(ξ0 k))2−j−1, k = 0, 1, · · · , 2j − 1, for some ξk ∈ [2−j, 2−j µ k + 1 2 ¶ ], ξ0 k ∈ [2−j µ k + 1 2 ¶ , 2−j(k + 1)]. By the same argument, we have for the coefficients ˜dn
j,k, ˜ dn j,k = 2 j 2 "Z 2−j(k+1) 2−j(k+1/2) f°n(t)dt − Z 2−j(k+3/2) 2−j(k+1) f°n(t)dt # = 22j(f°n(ξ0 k) − f°n(ξk+1))2−j−1, k = 0, 1, · · · , 2j − 2. Therefore 2j/2 2 j−1 X k=0 |dn j,k| + 2j−2 X k=0 | ˜dn j,k| = 1 2 2 j−1 X k=0 |f°n(ξ k) − f°n(ξk0)| + 2j−2 X k=0 |f°n(ξ0 k) − f°n(ξk+1)| −→ 1 2V (f °n), as j → ∞.
We have (4.6) and the proof is complete.
Remark 4.1. It follows from Theorem 4.1 that a one-dimensional dynamical system (I, f ) has chaotic behavior if and only if there exists an orthonormal wavelet basis (Haar wavelet)
{ψj,k}j,k∈Z such that either
2jn/2 2Xjn−1 k=0 |dn jn,k| or 2jn/2 2Xjn−1 k=0 | ˜dn jn,k| grows exponentially as jn → ∞.
Remark 4.2. It is well known that the Haar wavelet is disadvantageous in wavelet analysis since it lacks smoothness. Here, rather, we see from Theorem 4.1 that for interval maps Haar’s wavelet can be efficiently used to detect the growth of the total variation of iterates
of f . ¤
Finally, we give a result pertaining to topological conjugate systems.
Theorem 4.2. Let f and g be continuous maps from I into itself and be piecewise monotone,
and ψ be the Haar wavelet. If f and g have topological conjugacy, then
lim n→∞ 1 nln Wψ(f °n) = lim n→∞ 1 nln Wψ(g °n).
In other words, the quantity
lim n→∞ 1 nln Wψ(f °n) is a topological invariant.
Proof. This follows from Theorem 4.1 and the fact that
lim n→∞ 1 nln V (f °n) = lim n→∞ 1 nln V (g °n),
since f and g are topologically conjugate and piecewise monotone.
A time series f°n generated by a deterministic chaotic interval map f is known to behave chaotically on some subintervals while non-chaotically on other subintervals. In this case, the use of wavelet analysis would offer distinct advantages over Fourier analysis due to certain wavelets multi-resolution capability. Nevertheless, concrete examples with explicitly calculated wavelet coefficients are nearly impossible to construct due to the complexities involved. Thus, one must still rely on numerics to perform simulations.
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