数理解析研究所講究録
第 829 巻 1993 年 77-87

77”Indirect” Time Series Analysis for
One-Dimensional Chaos Based on
Perron-Frobenius Operator:”Generalized” Ulam-Li’s Approximation toInvariant Density
Tohru KOHDA \dag er and Kenji MURAO \dag er \dag er

Kyushu University\dagger and Miyazaki University\dagger \dagger , Japan

Abstract A unified approach to time series analysis for onedimensional discrete chaos is given which is based on the Galerkinapproximation to the Perron-Frobenius operator. The proposed

method gives approximations with high accuracy to statistics ofvarious one-dimensional chaos. Numerical results for$1/f^{\delta}$

power

spectrum of intermittent chaos also show that the observed expo-

nent of the FFT power spectrum of long trajectories as $farrow 0$ isin good agreement not with the Procaccia-Schuster’s estimate butwith our estimate.

I. Introduction
There are two kinds of time series analysis for long-time chaotic trajectories$\{x_{m}\}_{m=0}^{\infty}$

generated by a recurrence formula $x_{m+1}=\tau(x_{m})$with an ergodic transformation$\tau$: $I=[0,1]arrow I$ . One of them is

78
the “time-average technique”, in which we evaluate certain statistics of a sample long-time trajectory$\{x_{m}\}_{m=0}^{n}$with some initialvalue $x=x_{0}$ ; the other one is the “ ensemble-average technique”

under the assumption that$\tau$

is mixing with respect to an abso-

lutely continuous invariant measure, denoted by $f^{*}(x)dx$ . We givea unified approach to time series analysis for discrete chaos by suchan ensemble-average technique.
The time-average technique which is a usual $method^{[4]}$ is referred to as the(direct method’. On the contrary, the ensemble

average technique is a kind of “indirect $method_{S}’$ because there is)no need to directly calculate trajectories. Hence such an indirect

method is expected to play an important role in theoretically un-derstanding chaos. In fact, the Perron-Frobenius operator whose
fixed point is $f^{*}(x)$ permits us to theoretically calculate the ensemble average of several $statistics^{[5],[6]}$ . This operator, denotedby$P_{\tau}$, however, gives no practically calculating method because of

its infinite dimensionality. Such a situation leads us to consider
an efficient algorithm for systematically calculating statistics which

is based on the Galerkin approximation to$P_{\tau}$

on a suitable func-tion $space^{[7]-[10]}$ . This algorithm is referred to as ageneralized’)
Ulam-Li’s $method^{[7]}$ . We used the word “Ulam-Li) method” be$s$cause$Li^{[2]}$gave an affirmative answer to the Ulam’s $conjecture^{[1]}$concerning a piecewise-constant approximation for$f^{*}(x)$.

Numerical experiments demonstrate that the proposed methodcan give approximations with high accuracy to statistics of various

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one-dimensional chaos.

II. Perron-Frobenius Operator and Statisticsof Chaos
If $y=\tau(x)$ is mixing with respect to $f^{*}(x)dx$ , then for almost
initial value $x=x_{0}$ sequences$\{x_{m}\}_{m=0}^{\infty}$

can chaotically behave.

From the Birchoff individual ergodic theorem, the time average ofany$L_{1}$

function $F(x)$ along a trajectory$\{x_{m}\}_{m=0}^{\infty}$, which is definedby$\overline{F}=\lim_{Tarrow\infty}\frac{1}{T}\sum_{n=0}^{T-1}F(\tau^{n}(x))$(1),

is equal almost everywhere to the ensemble average of $F(x)$ over ,$I$defined by$<F>= \int_{I}F(x)f^{*}(x)dx$ .

(2)

The direct time series analysis is based on using$\overline{F}$. However, thesensitive dependence on initial conditions, one of chaotic $properties^{[3]}$ ,prevents us from precisely evaluating$\overline{F}$. On the other hand, the in-direct time series analysis is based on using $<F>$ . We begin withreviewing relations between typical statistics and$P_{\tau}$$P_{\tau}$. The operator

is defined by$P_{\tau}f(x)= \int_{I}\delta(x-\tau(y))h(y)dy$

For any$L_{1}$

functions of bounded variations $g(x)$ andthe important property(3).
$h(x)\rangle$$P_{\tau}$

has

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(4)

$(g(x), h(\tau(x)))=(P_{\tau}g(x), h(x))$ ,where$(g, h)= \int_{I}g(x)h(x)dx$.

(5)

The invariant density $f^{*}(x)$ which plays a key role in our indirect

method is the eigenfunction ofbelonging to the eigenvalue 1,$P_{\tau}$that is,$P_{\tau}f^{*}(x)=f^{*}(x)$.

(6)

The autocorrelation function is defined by(7)$\rho(k)=<x\tau^{k}(x)>-<x>^{2}$

The first term of the rhs of this equation is rewritten as$<x\tau^{k}(x)>=(P_{\tau}^{k}(xf^{*}(x)), x)$

where the above property ofthe eigenfunction of$P_{\tau}$

is repeatedly used. Letwith the eigenvalue$P_{\tau}$(8),$\lambda_{i}$$h_{i}(x)$befor the eigenvalue$problem^{[4]}$$P_{\tau}h_{i}(x)=\lambda_{i}h:(x)$.

(9)

If we can expand $xf^{*}(x)$ as$xf^{*}(x)= \sum_{i=1}^{\infty}\eta_{i}h_{i}(x)$

,

(10)

then we have$\rho(k)=\sum_{i=2}^{\infty}u_{l}\lambda_{i)}^{k}$

(11)

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the Fourier Transform of which gives the power spectrum $S(\nu)$(12)$S( \iota/)=\sum_{\mathfrak{i}=2}^{\infty}u_{i}\frac{1-\lambda_{\mathfrak{i}}^{2}}{(1-\lambda_{i}z)(1-\lambda_{i}z^{-1})}$

where$\lambda_{1}=1,$ $u_{i}=\eta_{\dot{t}}(x, h_{i}))$and(13)$z=exp(j2\pi\nu)$with $0<\nu<1$ . Oono and $Takahashi^{[5],[6]}$ demonstrated that theFredholm theory of$P_{\tau}$plays an important role in discussions of thepower spectrum. It is, however, difficult to find exact solutions ofeigenvalues and eigenfunctions of$P_{\tau}$, primalily because$P_{\tau}$

has theinfinite dimensionality. Such a situation led us to consider an effi-cient algorithm of the indirect method.

III. Galerkin Approximations to Perron-Frobenius

OperatorLet$\triangle$

function

divide$I$

be a function space which is spanned by a vector basis$\ell(x)arrow$. The constructing method of

into $N$ subintervals$\triangle$

is as follows. We

with partition points$\{I_{n}\}$$\{c_{i}\}^{\dot{N}_{=0}}$

sat-isfying $0=c_{0}<c_{1}<c_{2}<\cdots<c_{N}=1$ such that
$I= \bigcup_{n=1}^{N}I_{n}$,$I_{n}=[c_{n-1}, c_{n}]$

(14)

.

Our Galerkin approximations depend on the appropriate selectionsof$\{c_{i}\}_{i=0}^{N}$and of$\ell(x)^{[7]-[10]}arrow$. A simple but efficient procedure, how-ever, is omitted here for selecting$\ell_{nk}(x)$such$as^{[10]}$$\{c_{i}\}_{\mathfrak{i}=0}^{N}$.

Next, we take bases

82$\ell_{nk}(x)=p_{nk}(x)s(x)\chi_{n}(x)$In the above equation,$\chi_{n}(x)$,$0\leq k\leq K$,$1\leq n\leq N$.

is the characteristic function of(15)$I_{n}$

andis the -th order Legendre polynomial which is orthogonal$p_{nk}(x)$)

$k$

$s$

to each other on
2. When$I_{n}$. For most of practical usages, we use $K=$

has a bounded invariant density, the function $s(x)$ ,$\tau$

referred to as a supplementary function, is taken to be 1. On theother hand,$\tau$

has an unbounded invariant density, $s(x)$ is chosen

to be a singular function which approximates to singularities of theunbounded invariant density and the inner product $(g, h)$ must bealso replaced by the weighted inner product(16)$(g, h)_{w}= \int_{I}g(x)h(x)w(x)dx$

with the weighting function$w(x)=s^{-2}(x)$ .

Each component(17)$\ell_{nk}(x)$is an appropriately chosen piecewise poly-

nomial of at most $K$ degree whose combination approximates to$xf^{*}(x)$by the Galerkin $method^{[7]}$ such as(18)$xf^{*}(x)\simeq f^{t}\ell^{arrow}(x))$

where the superscript denotes the transpose of the vector . Using
$f$

$t$

$\ell^{arrow}(x)$, we get$<x\tau^{k}(x)>\simeq f^{t}(P_{\tau}^{k}\ell^{arrow}(x), x)$.

Furthermore, using the Galerkin method withproximate to$P_{\tau}\ell(x)arrow$such as(19)$\ell(x)arrow$on$\triangle$, we ap-83(20)$P_{\tau}\ell^{arrow}(x)\simeq\hat{P}_{\tau}^{t_{\ell(X)}^{arrow}}$which leads us to readily obtain$<x\tau^{k}(x)>\simeq f^{t}(\hat{P}_{\tau}^{t})^{k}(\ell(x), x)arrow$

,

(21)

where the $N(K+1)\cross N(K+1)$ matrix$\hat{P}_{\tau}$

is referred to as theGalerkin-approximated matrix of the Perron-Frobenius operator where$N$

and $K$ are integers to be given below. The explicit form ofis given$in^{[7]}$eigenvalue. Let$\lambda_{i}$$\hat{\lambda}_{1}$be the -th right eigenvector of$i$$\hat{P}_{\tau}$with thefor the easily tractable eigenvalue problem$\hat{P}_{\tau}h_{i}=\hat{\lambda}_{\mathfrak{i}}h;$Let$h_{i}$$\hat{P}_{\tau}$

.

(22)

be the maximum eigenvalue of$\hat{P}$. It is easily shown that$\hat{\lambda}_{1}$when the supplementary function $s(x)=1$ , namely, when both thepolynomial bases and the unweighted inner product are used. But$\wedge\lambda_{1}\simeq 1$

when $s(x)\neq 1$ , that is, when both the singular bases andthe weighted inner product are used. For the latter case, numericalexperiments show
$10^{-6}$$\hat{\lambda}_{1}$is nearly equal to 1 with the eror less thanfor $K=2$ and $N=32$ . An approximate solution to theinvariant density given by(23)$\hat{f}^{*}(x)=h_{1}^{t}l^{arrow}(x)$where$h_{1}$

is normalized such that(24)$\int_{I}\hat{f}^{*}(x)dx=1$

It is easily shown that Eq.(23) is an approximate solution to Eq.(6)by the Galerkin method and that$\hat{f}^{*}(x)$when $K=0$ gives the84
results by the well-known Ulam-Li’s method. Figure 1 shows con-vergence rates of approximate solution$\hat{f}^{*}(x)$by our $method^{[7_{1}}$ .

Fig. 1 Convergence rates of approximate solutions$\hat{f}^{*}(x)$(theproposed method) to the bounded invariant densities $f^{*}(x)$ for

mixing chaos in several examples.

IV. Numerical Examples
Example 1Let$\tau(x)=\{\begin{array}{l}ax^{z}+(a+b-ab)/b0\leq x\leq x_{p}=(1-1/b)^{l/z}-b(x^{z}-1)x_{p}<x\leq 1\end{array}$This map can generate periodic chaos for suitable parameters. Figures 2 and 3 show $f^{*}(x)$ and the power spectrum$\tilde{S}_{T}(\nu)$for periodic

chaos of period 6 which are calculated by our $method^{[8],[9]}$ . In thiscalculation, we take$\{\tau^{n}(0)\}_{n=1}^{30}$as the partition points$\{c_{t}\}_{i=1}^{N-1}$so85
that edges of the support of $f^{*}(x)$ will coincide with the partition

points. In the calculation of$\tilde{S}_{T}(\nu)$, the finite discrete Fourier trans-

form of $\{\rho(k)\}_{k=0}^{T-1}(T=1,024\cross 6)$ is used instead of using Eq.(12).
On the other hand, $S_{T,m}(\nu)$ is obtained by averaging $m=200$ dis-crete Fourier transforms of trajectories of length$\tilde{S}_{T}(\nu)$$T$. The spectrum

is in good agreement with $S_{T,m}(\nu)$ except for fluctuations inthe latter.

Fig. 2 Approximated invariant density$\hat{f}^{*}(x)$(by our indirectmethod) for periodic chaos of period 6in example 1.

Fig. 3 Power spectra$\tilde{S}_{T}(\nu)$(by our indirect method) and$\tilde{S}_{T,m}(u)$(by the direct method) for periodic chaos of period 6 in example 1.

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Example 2Let$\tau(x)=\{\begin{array}{l}x+ux^{z}(x-x_{p})/(1-x_{p})\end{array}$$0\leq_{p^{X}}\leq_{X^{X_{p}}}x<\leq 1$

where $\tau(x_{p})=1,$ $u>0,1<z<2$ . This map generates inter-mittent chaos with the power spectrum$1/f^{\delta}$. Figure 4 shows thepower spectrum $S(\nu)$ by our $method^{[10]}$ (the smooth solid line) and$S_{T,m}(\nu)$with $T=2^{15}$ and $m=100$ by the direct method (the fluc-tuated line), each of which is in good agreement each other in wide
frequency range. In applying our method, we used $s(x)=x^{-(z-1)}$

because$\tau$

has the unbounded invariant density with a $(z-1)$ -thorder pole at $x=0$ . In this figure, the broken line shows the Procaccia and Schuster’s estimate [11] of the spectrum when
$0$

$\nu$

goes towhich does not coincide well with the former two.

屋$\underline{\infty OO}0$log(f)Fig. 4 Comparison of power spectra calculated by using threedifferent methods for intermittent chaos in example 2.

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References[1] S.Ulam, Problems in modern mathematics, Intersience Pub.
(1960).
[2] T.Y.Li, J.Approx. Theo., 17, pp.177-186(1976).
[3] S.Grossmann and S.Thomae, Z. Naturforsch., 32a, $1353- 1363$(1977).

[4] S.J.Chang and J.Wright, Phys.Rev.A, 23, $1419- 1433$ (1981).

[5] Y.Oono and Y.Takahashi, Progr.Theo.Phys., 63-5, 1804-1807(1980).

[6] Y.Takahashi and Y.Oono, Progr.Theo.Phys., 71-4, 851-854(1980).

[7] T.Kohda and K.Murao, Trans. IEICE, J65-A-6, 505-512(1982).
[8] K.Murao and T.Kohda, Trans. IEICE, J67-A-5, 511-518(1984).
[9] K.Murao and T.Kohda, Trans. IEICE, J68-A-7, 405-406(1985).
[10] T.Kohda and K.Murao, Trans. IEICE, E73-6, 793-800 (1990).

[11] I.Procaccia and H.Schuster, Phys.Rev., A, $28- 2,1210- 1212$(1983).