Tài liệu 72 Nonlinear Maps docx

Wornell Massachusetts Institute of Technology 72.1 Introduction 72.2 Eventually Expanding Maps and Markov Maps Eventually Expanding Maps 72.3 Signals From Eventually Expanding Maps 72.4

Steven H Isabelle, et Al “Nonlinear Maps.”

2000 CRC Press LLC <http://www.engnetbase.com>.

Nonlinear Maps

Steven H Isabelle

Massachusetts Institute of Technology

Gregory W Wornell

Massachusetts Institute of Technology

72.1 Introduction 72.2 Eventually Expanding Maps and Markov Maps

Eventually Expanding Maps

72.3 Signals From Eventually Expanding Maps 72.4 Estimating Chaotic Signals in Noise 72.5 Probabilistic Properties of Chaotic Maps 72.6 Statistics of Markov Maps

72.7 Power Spectra of Markov Maps 72.8 Modeling Eventually Expanding Maps with Markov Maps References

72.1 Introduction

One-dimensional nonlinear systems, although simple in form, are applicable in a surprisingly wide variety of engineering contexts As models for engineering systems, their richly complex behavior has provided insight into the operation of, for example, analog-to-digital converters [1], nonlinear oscillators [2], and power converters [3] As realizable systems, they have been proposed as random number generators [4] and as signal generators for communication systems [5,6] As analytic tools, they have served as mirrors for the behavior of more complex, higher dimensional systems [7,8,9] Although one-dimensional nonlinear systems are, in general, hard to analyze, certain useful classes

of them are relatively well understood These systems are described by the recursion

x[n] = f (x[n − 1]) (72.1a) y[n] = g(x[n]) , (72.1b)

initialized by a scalar initial conditionx[0], where f (·) and g(·) are real-valued functions that describe

the evolution of a nonlinear system and the observation of its state, respectively The dependence

of the sequencex[n] on its initial condition is emphasized by writing x[n] = f n (x[0]) where f n (·)

represents then-fold composition of f (·) with itself.

Without further restrictions of the form off (·) and g(·), this class of systems is too large to

easily explore However, systems and signals corresponding to certain “well-behaved” mapsf (·)

and observation functionsg(·) can be rigorously analyzed Maps of this type often generate chaotic

signals—loosely speaking, bounded signals that are neither periodic nor transient—under easily verifiable conditions These chaotic signals, although completely deterministic, are in many ways analogous to stochastic processes In fact, one-dimensional chaotic maps illustrate in a relatively simple setting that the distinction between deterministic and stochastic signals is sometimes artificial

and can be profitably emphasized or deemphasized according to the needs of an application For instance, problems of signal recovery from noisy observations are often best approached with a deterministic emphasis, while certain signal generation problems [10] benefit most from a stochastic treatment

72.2 Eventually Expanding Maps and Markov Maps

Although signal models of the form [1] have simple, one-dimensional state spaces, they can behave

in a variety of complex ways that model a wide range of phenomena This flexibility comes at a cost, however; without some restrictions on its form, this class of models is too large to be analytically tractable Two tractable classes of models that appear quite often in applications are eventually expanding maps and Markov maps

72.2.1 Eventually Expanding Maps

Eventually expanding maps—which have been used to model sigma-delta modulators [11], switching power converters [3], other switched flow systems [12], and signal generators [6,13]—have three defining features: they are piecewise smooth, they map the unit interval to itself, and they have some iterate with slope that is everywhere greater than unity Maps with these features generate time series that are chaotic, but on average well behaved For reference, the formal definition is as follows, where the restriction to the unit interval is convenient but not necessary:

DEFINITION 72.1 A nonsingular mapf : [0, 1] → [0, 1] is called eventually expanding if

1 There is a set of partition points 0= a0< a1< · · · a N = 1 such that restricted to each

of the intervalsV i = [a i−1 , a i ), called partition elements, the map f (·) is monotonic,

continuous and differentiable

2 The function 1/|f0(x)| is of bounded variation [14] (In some definitions, this smooth-ness condition on the reciprocal of the derivative is replaced with a more restrictive bounded slope condition, i.e., there exists a constant B such that|f0(x)| < B for all x.)

3 There exists a realλ > 1 and a integer m such that

dx d f m (x)

≥ λ

wherever the derivative exists This is the eventually expanding condition

Every eventually expanding map can be expressed in the form

f (x) =XN

i=1

where eachf i (·) is continuous, monotonic, and differentiable on the interior of the ith partition

element and the indicator functionχ i (x) is defined by

χ i (x) =



1 x ∈ V i ,

This class is broad enough to include for example, discontinuous maps and maps with discontinuous

or unbounded slope Eventually expanding maps also include a class that is particularly amenable to analysis—the Markov maps

Markov maps are analytically tractable and broadly applicable to problems of signal estimation, signal generation, and signal approximation They are defined as eventually expanding maps that are piecewise-linear and have some extra structure

DEFINITION 72.2 A mapf : [0, 1] → [0, 1] is an eventually expanding, piecewise-linear, Markov map if f is an eventually expanding map with the following additional properties:

1 The map is piecewise-linear, i.e., there is a set of partition points 0= a0< a1< · · · <

a N = 1 such that restricted to each of the intervals V i = [a i−1 , a i ), called partition

elements, the mapf (·) is affine, i.e., the functions f i (·) on the right side of (72.2) are of the form

f i (x) = s i x + b i

2 The map has the Markov property that partition points map to partition points, i.e., for eachi, f (a i ) = a jfor somej.

Every Markov map can be expressed in the form

f (x) =XN

i=1

(s i x + b i ) χ i (x) , (72.4) wheres i 6= 0 for all i Fig.72.1shows the Markov map

f (x) =



(1 − a)x/a + a 0 ≤ x ≤ a (1 − x)/(1 − a) a < x ≤ 1 , (72.5)

which has partition points{0, a, 1}, and partition elements V1= [0, a) and V2= [a, 1).

FIGURE 72.1: An example of a piecewise-linear Markov map with two partition elements

Markov maps generate signals with two useful properties: they are, when suitably quantized, indistinguishable from signals generated by Markov chains; they are close, in a sense, to signals generated by more general eventually expanding maps [15] These two properties lead to applications

of Markov maps for generating random numbers and approximating other signals The analysis underlying these types of applications depends on signal representations that provide insight into the structure of chaotic signals

72.3 Signals From Eventually Expanding Maps

There are several general representations for signals generated by eventually expanding maps Each provides different insights into the structure of these signals and proves useful in different applications First, and most obviously, a sequence generated by a particular map is completely determined by (and is thus represented by) its initial conditionx[0] This representation allows certain signal

estimation problems to be recast as problems of estimating the scalar initial condition Second, and less obviously, the quantized signaly[n] = g(x[n]), for n ≥ 0 generated by (72.1) withg(·) defined

by

uniquely specifies the initial conditionx[0] and hence the entire state sequence x[n] Such quantized

sequencesy[n] are called the symbolic dynamics associated with f (·) [7] Certain properties of a map, such as the collection of initial conditions leading to periodic points, are most easily described

in terms of its symbolic dynamics Finally, a hybrid representation ofx[n] combining the initial

condition and symbolic representations

H[N] = {g(x[0]), , g(x[N]), x[N]}

is often useful

72.4 Estimating Chaotic Signals in Noise

The hybrid signal representation described in the previous section can be applied to a classical signal processing problem—estimating a signal in white Gaussian noise For example, suppose the problem

is to estimate a chaotic sequencex[n], n = 0, , N − 1 from the noisy observations

r[n] = x[n] + w[n], n = 0, , N − 1 (72.7)

wherew[n] is a stationary, zero-mean white Gaussian noise sequence with variance σ2

w, andx[n]

is generated by iterating (72.1) from an unknown initial condition Because w[n] is white and

Gaussian, the maximum likelihood estimation problem is equivalent to the constrained minimum distance problem

minimize

x[n] : x[i] = f (x[i − 1]) ε[N] =XN

k=0

(r[k] − x[k])2 (72.8) and to the scalar problem

minimize

x[0] ∈ [0, 1] ε[N] =

N

X

k=0



r[k] − f k (x[0])2 (72.9)

Thus, the maximum-likelihood problem can, in principle, be solved by first estimating the initial condition, then iterating (72.1) to generate the remaining estimates However, the initial condition is often difficult to estimate directly because the likelihood function (72.9), which is highly irregular with fractal characteristics, is unsuitable for gradient-descent type optimization [16] Another solution divides the domain off (·) into subintervals and then solves a dynamic programming problem [17]; however, this solution is, in general, suboptimal and computationally expensive

Although the maximum likelihood problem described above need not, in general, have a computa-tionally efficient recursive solution, it does have one when, for example, the mapf (·) is a symmetric

tent map of the form

f (x) = β − 1 − β|x| , x ∈ [−1, 1] (72.10)

with parameter 1< β ≤ 2 [5] This algorithm solves for the hybrid representation of the initial condition from which an estimate of the entire signal can be determined The hybrid representation

is of the form

H[N] = {y[0], , y[N], x[N]} ,

where eachy[i] takes one of two values which, for convenience, we define as y[i] = sgn (x[i]) Since

eachy[n] can independently takes one of two values, there are 2 Nfeasible solutions to this problem

and a direct search for the optimal solution is thus impractical even for moderate values ofN.

The resulting algorithm has computational complexity that is linear in the length of the observation,

N This efficiency is the result of a special separation property, possessed by the map [10]: given

y[0], , y[i − 1] and y[i + 1], , y[N] the estimate of the parameter y[i] is independent of y[i + 1], , y[N] The algorithm is as follows Denoting by ˆφ[n|m] the ML estimates of any

sequenceφ[n] given r[k] for 0 ≤ k ≤ m, the ML solution is of the form,

ˆx[n|n] = β2− 1

β2n r[n] + β2n− 1
ˆx[n|n − 1]

ˆy[n|N] = sgn ˆx[n|n] (72.12)

whereˆx[n|n−1] = f ( ˆx[n−1|n−1]), the initialization is ˆx[0|0] = r[0], and the function L β ( ˆx[n|n]),

defined by

L β (x) =







x x ∈ (−1, β − 1)

−1 x ≤ −1 ·

serves to restrict the ML estimates to the intervalx ∈ (−1, β−1) The smoothed estimates ˆx ML [n|N]

are obtained by converting the hybrid representation to the initial condition and then iterating the estimated initial condition forward

72.5 Probabilistic Properties of Chaotic Maps

Almost all waveforms generated by a particular eventually expanding map have the same average behavior [18], in the sense that the time average

¯h(x[0]) = lim n→∞1n n−1X

k=0

h(x[k]) = lim

n→∞

1

n

n−1

X

k=0

hf k (x[0]) (72.15)

exists and is essentially independent of the initial conditionx[0] for sufficiently well-behaved

func-tionsh(·) This result, which is reminiscent of results from the theory of stationary stochastic

processes [19], forms the basis for a probabilistic interpretation of chaotic signals, which in turn leads to analytic methods for characterizing their time-average behavior

To explore the link between chaotic and stochastic signals, first consider the stochastic process generated by iterating (72.1) from a random initial conditionx[0], with probability density function

p0(·) Denote by p n (·) the density of the nth iterate x[n] Although, in general, the members of the

sequencep n (·) will differ, there can exist densities, called invariant densities, that are time-invariant,

i.e.,

p0(·) = p1(·) = = p n (·) = p(·) (72.16) 1

When the initial conditionx[0] is chosen randomly according to an invariant density, the resulting

stochastic process is stationary [19] and its ensemble averages depend on the invariant density Even

when the initial condition is not random, invariant densities play an important role in describing the time-average behavior of chaotic signals This role depends on, among other things, the number of invariant densities that a map possesses

A general one-dimensional nonlinear map may possess many invariant densities For example, eventually expanding maps withN partition elements have at least one and at most N invariant

densities [20] However, maps can often be decomposed into collections of maps, each with only one invariant density [19], and little generality is lost by concentrating on maps with only one invariant density In this special case, the results that relate the invariant density to the average behavior of chaotic signals are more intuitive

The invariant density, although introduced through the device of a random initial condition, can also be used to study the behavior of individual signals Individual signals are connected to ensembles of signals, which correspond to random initial conditions, through a classical result due to Birkhoff, which asserts that the time average ¯h(x[0]) defined by Eq (72.15) exists wheneverf (·) has

an invariant density When thef (·) has only one invariant density, the time average is independent

of the initial condition for almost all (with respect to the invariant densityp(·)) initial conditions

and equals

lim

n→∞

1

n

n−1

X

k=0

h(x[k]) = lim

n→∞

1

n

n−1

X

k=0

hf k (x[0])=

Z

h(x)p(x)dx (72.17)

where the integral is performed over the domain off (·) and where h(·) is measurable.

Birkhoff ’s theorem leads to a relative frequency interpretation of time-averages of chaotic signals

To see this, consider the time-average of the indicator function˜χ [s−,s+] (x), which is zero everywhere

but in the interval[s − , s + ] where it is equal to unity Using Birkhoff’s theorem with Eq (72.17) yields

lim

n→∞

1

n

n−1

X

k=0

˜χ [s−,s+] (x[k]) =

Z

˜χ [s−,s+] (x)p(x)dx (72.18)

=

Z

[s−,s+] p(x)dx (72.19)

where Eq (72.20) follows from Eq (72.19) when is small and p(·) is sufficiently smooth The

time-average (72.18) is exactly the fraction of time that the sequencex[n] takes values in the interval

[s − , s + ] Thus, from (72.20), the value of the invariant density at any points is approximately

proportional to the relative frequency with whichx[n] takes values in a small neighborhood of the

point Motivated by this relative frequency interpretation, the probability that an arbitrary function

h(x[n]) falls into an arbitrary set A can be defined by

P r {h(x) ∈ A} = lim n→∞ n1

n−1

X

k=0

˜χ A (h(x[k])) (72.21)

Using this definition of probability , it can be shown that for any Markov map, the symbol sequence

y[n] defined in Section72.3is indistinguishable from a Markov chain in the sense that

P r {y[n]|y[n − 1], , y[0]} = P r {y[n]|y[n − 1]} , (72.22) holds for alln [21] The first order transition probabilities can be shown to be of the form

P r(y[n]|y[n − 1]) = V y[n]

s y[n] V y[n−1] ,

where thes i are the slopes of the mapf (·) as in Eq (72.4) and|V y[n]| denotes the length of the intervalV y[n] As an example, consider the asymmetric tent map

f (x) =



(1 − x)/(1 − a) a < x ≤ 1 ,

with parameter in the range 0 < a < 1 and a quantizer g(·) of the form (72.6) The previous results establish thaty[n] = g(x[n]) is equivalent to a sample sequence from the Markov chain with

transition probability matrix

[P ] ij =



a 1 − a

a 1 − a



,

where[P ] ij = P r{y[n] = i|y[n − 1] = j} Thus, the symbolic sequence appears to have been

generated by independent flips of a biased coin with the probability of heads, say, equal toa When

the parameter takes the valuea = 1/2, this corresponds to a sequence of independent equally likely

bits Thus, a sequence of Bernoulli random variables can been constructed from a deterministic sequencex[n] Based on this remarkable result, a circuit that generates statistically independent bits

for cryptographic applications has been designed [4]

Some of the deeper probabilistic properties of chaotic signals depend on the integral (72.17), which

in turn depends on the invariant density For some maps, invariant densities can be determined explicitly For example, the tent map (72.10) withβ = 2 has invariant density

p(x) =



1/2 −1 ≤ x ≤ 1

0 otherwise

as can be readily verified using elementary results from the theory of derived distributions of functions

of random variables [22] More generally, all Markov maps have invariant densities that are piecewise-constant function of the form

n

X

i=1

wherec i are real constants that can be determined from the map’s parameters [23] This makes Markov maps especially amenable to analysis

72.6 Statistics of Markov Maps

The transition probabilities computed above may be viewed as statistics of the sequencex[n] These

statistics, which are important in a variety of applications, have the attractive property that they are defined by integrals having, for Markov maps, readily computable, closed-form solutions This property holds more generally—Markov maps generate sequences for which a large class of statistics can be determined in closed form These analytic solutions have two primary advantages over empirical solutions computed by time averaging: they circumvent some of the numerical problems that arise when simulating the long sequences of chaotic data that are necessary to generate reliable averages; and they often provide insight into aspects of chaotic signals, such as dependence on a parameter, that could not be easily determined by empirical averaging

Statistics that can be readily computed include correlations of the form

R f ;h0,h1, ,h r[k1, , k r] = lim

L→∞

1

L

L−1

X

n=0

h0(x[n])h1(x[n + k1]) · · · h r (x[n + k r ])(72.24)

= Z

h0(x[n])h1(x[n + k1]) · · · h r (x[n + k r ]) p(x) dx ,(72.25)

where theh i (·)0s are suitably well-behaved but otherwise arbitrary functions, the k0

i s are nonnegative

integers, the sequence x[n] is generated by Eq (72.1), andp(·) is the invariant density This

class of statistics includes as important special cases the autocorrelation function and all higher-order moments of the time-series Of primary importance in determining these statistics is a linear transformation called the Frobenius-Perron (FP) operator, which enters into the computation of these correlations in two ways First, it suggests a method for determining an invariant density Second, it provides a “change of variables” within the integral that leads to simple expressions for correlation statistics

The definition of the FP operator can be motivated by using the device of a random initial condition

x[0] with density p0(x) as in Section72.5 The FP operator describes the time evolution of this initial probability density More precisely, it relates the initial density to the densitiesp n (·) of the random

variablesx[n] = f n (x[0]) through the equation

p n (x) = P f n p0(x) (72.26)

whereP n

f denotes then-fold self-composition of P f This definition of the FP operator, although phrased in terms of its action on probability densities, can be extended to all integrable functions This extended operator, which is also called the FP operator, is linear and continuous Its properties are closely related to the statistical structure of signals generated by chaotic maps (see [9] for a thorough discussion of these issues) For example, the evolution equation (72.26) implies that an invariant density of a map is a fixed point of its FP operator, that is, it satisfies

This relation can be used to determine explicitly the invariant densities of Markov maps [23], which may in turn be used to compute more general statistics

Using the change of variables property of the FP operator, the correlation statistic (72.25) can be expressed as the ensemble average

R f ;h0,h1, ,h r[k1, , k r]= (72.28) Z

h r (x)P k r −k r−1 f

n

h r−1 (x) · · · P k2−k1

f

n

h1(x)P k1

f {h0(x)p(x)}o· · ·odx (72.29)

Although such integrals are, for general one-dimensional nonlinear maps, difficult to evaluate, closed-form solutions exist when f (·) is a Markov map— a development that depends on an explicit

expression for FP operator

The FP operator of a Markov map has a simple, finite-dimensional matrix representation when it operates on certain piecewise polynomial functions Any function of the form

h(·) =XK

i=0

N

X

j=1

a ij x i χ j (x)

can be represented by anN(K + 1) dimensional coordinate vector with respect to the basis



θ1(x), θ2(x), , θ N(K+1) =1

n

χ1(x), , χ N (x), xχ1(x), , xχ N (x), , x K χ1(x), , x K χ N (x)o .(72.30)

The action of the FP operator on any such function can be expressed as a matrix-vector product: when the coordinate vector ofh(x) is h, the coordinate vector of q(x) = P f h(x) is

q = PKh,

where Pkis the squareN(K + 1) dimensional, block upper-triangular matrix

PK =









P00 P01 · · · P0K

0 P11 P12 · · · P1K

0 0 · · · PKK







and where each nonzeroN × N block is of the form

Pij =



j i



P0Bj−iSj forj ≥ i (72.32)

TheN × N matrices B and S are diagonal with elements B ii = −b i andS ii = 1/s i, respectively,

while P0= P00is theN × N matrix with elements

[P0]ij =



1/ s j i ∈ I j ,

The invariant density of a Markov map, which is needed to compute the correlation statistic (72.25), can be determined as the solution of an eigenvector problem It can be shown that such invariant densities are piecewise constant functions so that the fixed point equation (72.27) reduces to the matrix expression

P0p= p

Due to the properties of the matrix P0, this equation always has a solution that can be chosen to have

nonnegative components It follows that the correlation statistic (72.29) can always be expressed as

R f ;h0,h1, ,h r [k1, , k r]= gT1Mg2 (72.34)

where M is a basis correlation matrix with elements

[M]ij =

Z

θ i (x)θ j (x) dx (72.35)

and giare the coordinate vectors of the functions

g2(x) = P k r −k r−1

f

n

h r−1 (x) · · · P k2−k1

f

n

h1(x)P k1

f {h0(x)p(x)}o· · ·o . (72.37)

By the previous discussion, the coordinate vectors g1and g2can be determined using straightforward matrix-vector operations Thus, expression (72.34) provides a practical way of exactly computing the integral (72.29), and reveals some important statistical structure of signals generated by Markov maps

72.7 Power Spectra of Markov Maps

An important statistic in the context of many engineering applications is the power spectrum The power spectrum associated with a Markov map is defined as the Fourier transform of its autocorre-lation sequence

R xx [k] =

Z

x[n]x[n + k]p(x)dx (72.38)

or unbounded slope Eventually expanding maps also include a class that is particularly amenable to analysis—the Markov maps