Review of Chaos Communication by Feedback Control

7/31/2019 Review of Chaos Communication by Feedback Control

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Tutorials and Review

International Journal of Bifurcation and Chaos, Vol. 13, No. 2 (2003) 269285c World Scientific Publishing Company

REVIEW OF CHAOS COMMUNICATION BY

FEEDBACK CONTROL OF SYMBOLIC DYNAMICS

ERIK M. BOLLT

Department of Mathematics and Computer Science,Clarkson University, Potsdam, NY 13699-5815, USA

[email protected]://www.clarkson.edu/bolltem

Received September 19, 2001; Revised November 30, 2001

This paper is meant to serve as a tutorial describing the link between symbolic dynamicsas a description of a chaotic attractor, and how to use control of chaos to manipulate thecorresponding symbolic dynamics to transmit an information bearing signal. We use the Lorenzattractor, in the form of the discrete successive maxima map of the z-variable time-series, asour main example. For the first time, here, we use this oscillator as a chaotic signal carrier. Wereview the many previously developed issues necessary to create a working control of symboldynamics system. These include a brief review of the theory of symbol dynamics, and how theyarise from the flow of a differential equation. We also discuss the role of the (symbol dynamics)generating partition, the difficulty of finding such partitions, which is an open problem formost dynamical systems, and a newly developed algorithm to find the generating partitionwhich relies just on knowing a large set of periodic orbits. We also discuss the importance ofusing a generating partition in terms of considering the possibility of using some other arbitrarypartition, with discussion of consequences both generally to characterizing the system, and also

specifically to communicating on chaotic signal carriers. Also, of practical importance, wereview the necessary feedback-control issues to force the flow of a chaotic differential equationto carry a desired message.

Keywords: Control of chaos; communication; communication with chaos; symbolic dynamics;hyperbolicity; synchronization; feedback control.

1. Introduction

The field of controlling chaos was first popularized

by the simple OGY algorithm published in 1990by Ott, Grebogi and Yorke [Ott et al., 1990]. Theidea is that an unstable periodic orbit can be stabi-lized by parametric feedback control once ergodic-ity has caused a randomly chosen initial conditionto wander close enough to the periodicity that lin-ear control theory can be applied [Ott et al., 1990].A tremendous flood of theoretical and experimen-tal research has followed this seminal work, some

of which extends the original OGY techniques, andsome of which only has in common with OGY thesimple realization that a chaotic oscillator is es-

pecially flexible and therefore allows for very sen-sitive selection between a wide range of possiblebehaviors.

Two major methods of communication withchaotic signal carriers are (1) Synchronization, and(2) Feedback control of symbolic dynamics. Thesubject of this paper concerns the feedback con-trol symbolic dynamics approach, but we mentionsynchronization for contrast to the other school of

More information can be found on the web: http://mathweb.mathsci.usna.edu/faculty/bolltem/

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270 E. M. Bollt

methods which are expected to appear here. Thesynchronization of chaotic oscillators phenomenonwas first discovered in the early and mid-1980sby [Yamada & Fujisaka, 1983, 1984] and then[Afraimovich et al., 1986], then in 1990 by [Pecora &Carroll, 1990] the last of which started an avalancheof research. Synchronization is now known to havefundamental implications in nature well beyond the

practical engineering application of communication.To our interest here, it was shown very early on[Pecora & Carroll, 1990; Cuomo & Oppenheim,1993] that using synchronization allows a chaoticsignal carrier to transmit a message. See [Pecoraet al., 1997] for an excellent review. We also di-rect the reader to the very important practical en-gineering advances made by the European follow-ers, as typified by the works [Kennedy & Kolumban,2000; Kolumban & Kennedy, 2000; Kolumban et al.,1998] in noncoherent methods, including differentialchaos shift keying (DCSK) modulation schemes.

On the other hand, control of symbolic dynam-ics relies on a fundamental description of a chaoticoscillator by change of coordinates that brings adiscrete time map (which may arise by a flow map-ping between Poincare surfaces) on its phase spaceto an abstract Bernoulli shift map on some subshiftspace. The fundamental link between chaotic oscil-lators and symbolic dynamics was first introducedby Smale [1967]. In 1993, Hayes et al. [1993, 1994]showed that feedback control of chaotic trajectoriestogether with sensitive dependence and the sym-

bolic dynamical description of chaotic trajectoriescan be used to force a chaotic oscillator to carry in-formation. Small parametric feedback control sen-sitively alters a chaotic trajectory and hence saidchange of coordinates serves as a coding functionto transform the symbolic sequence correspondingto the original chaotic trajectory, into some othersymbolic sequence corresponding to a desired mes-sage. We would also like to point out to the readerthat at time of press, a closely related article oncommunication in chaos using symbol dynamics hasappeared [Schweizer & Schimming, 2001a, 2001b].

While this paper is meant to serve as a tuto-rial, as our main example, we will show for the firsttime how Lorenzs successive maxima map servesas a good information carrying device. In our pre-vious work the author and Dolnik [Bollt & Dolnik,1997; Dolnik & Bollt, 1998] considered a chemicalsystem. Since it has been shown that the Lorenzequations can be easily realized in circuitry [Cuomo& Oppenheim, 1993], it follows that such a system

could in principle be built. In Sec. 2, we review theLorenz system and attractor, which will serve as ourmain concrete example, and we discuss how the suc-cessive maxima map yields a one-humped map, forwhich the symbol dynamics are defined. We reviewin Sec. 3 the necessary symbolic dynamics, bothfor general diffeomorphisms of the plane, and forthe simpler chaos of one-dimensional many-to-one

maps, of which the successive maxima map of theLorenz flow is an example. Then in Sec. 4, we dis-cuss both the role and importance of using a gener-ating partition. We also discuss the consequences ofusing a nongenerating (arbitrary) partition, both indescribing the system, and also specifying the con-sequences to communicating with chaos. We brieflyreview a new algorithm to use collections of unsta-ble periodic orbits to reveal the generating parti-tion. In Sec. 5, we recall a technique, codevelopedby the author [Bollt et al., 1997; Bollt & Lai, 1998;Lai & Bollt, 1999], whereby a great deal of chan-nel noise resistance can be added to the system,and costs only a minor loss of transmitter capacity,simply by restricting trajectories to always avoida neighborhood of the generating partition. Ourfeedback control techniques discussed in Sec. 6 willfollow the methods codeveloped by the author in[Dolnik & Bollt, 1998], in which we allow for allnecessary information to be learned purely by ob-servation and measurement, in the setting of a lab-oratory experiment, and in the absence of a closedform differential equation model of the dynamical

system.

2. Lorenz Differential Equationsand Successive Maxima Map

We consider the Lorenz system [Lorentz, 1963],

x = 10(y x) ,

y = x(28 z) y ,

z = xy 8

3z ,

(1)

both because it is a benchmark example of chaoticoscillations, and because it can be physically real-ized by an electronic circuit [Cuomo & Oppenheim,1993]. Likewise those interested in optical trans-mission carriers may recall the Lorenz-like infraredNH3 laser data [Huebner et al., 1989]. E. Lorenzshowed that his equations have the property thatthe successive local maxima can be described by a


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Review of Chaos Communication 271

Fig. 1. Successive maxima map of the measure z(t) variable,of the Lorenz flow (x(t), y(t), z(t)) from Eqs. (1).

one-dimensional, one-hump map,

zn+1 = f(zn) , (2)

where we let zn be the nth local maximum of thestate variable z(t). The chaotic attractor in thephase space (x(t), y(t), z(t)) shown in Fig. 1 corre-sponds to a one-dimensional chaotic attractor in thephase space of the discrete map f(z), and hence thesymbol dynamics are particularly simple to analyze.See Fig. 2. The generating partition for defining a

Fig. 2. Lorenzs butterfly attractor. This particulartypical trajectory of Eqs. (1) gives the message BeatArmy! when interpreted relative to the generating parti-tion Eq. (3) of the one-dimensional successive maxima mapEq. (2) shown in Fig. 2.

good symbolic dynamics is the critical point zc ofthe f(z) function. A trajectory point with

z < zc(z > zc) bears the symbol 0 (1) . (3)

The partition of this one-dimensional map, of suc-cessive z(t) maxima corresponds to a traditionalPoincare surface mapping, as the two leaves of

the surface of section can be seen in Fig. 1.Each bit roughly corresponds to a rotation ofthe (x(t), y(t), z(t)) flow around the left or theright lobes of the Lorenz butterfly-shaped attrac-tor. However, the Lorenz attractor does not allowarbitrary permutations of rotations around one andthen the other lobe; the corresponding symbolic dy-namics has a somewhat restricted grammar whichmust b e learned. The grammar of the correspond-ing symbolic dynamics completely characterizes theallowed trajectories. Since the purpose of this pa-per is to show how a chaotic oscillator can be forced

to carry a message in its symbolic dynamics, we de-velop further these notions in the next section.

3. Symbolic Dynamics

In this section, we review the mathematicalchange of coordinates which allows trajectories ofa discrete dynamical system to be equivalentlydescribed as infinite bit streams. Such sym-bolic dynamical descriptions of chaos were first in-vented as a simplifying change of coordinates, in

which the proofs of many theorems become simple[Robinson, 1995]. Indeed, most of the dynamicalsystems for which mathematically rigorous proofexists concerning their chaotic properties are pre-cisely those for which a conjugacy to some sym-bolic shift exists (see examples in [Robinson, 1995;Devaney, 1989]. For our purposes in communica-tions describing controlled trajectories as messagesinvolve conversion to a bit stream as a crucial step.

3.1. One-dimensional maps witha single critical point

First we consider a one-humped interval map, suchas the situation of Lorenzs successive maxima map.

f : [a, b] [a, b] . (4)

Such a map has symbolic dynamics [Milnor &Thurston, 1977; de Melo & van Strein, 1992] rela-tive to a partition at the critical point xc. Choose


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272 E. M. Bollt

a two symbol partition, labeled I= {0, 1}, namingiterates of an initial condition x0 according to,

i(x0) =

0 if fi(x0) < xc1 if fi(x0) > xc

. (5)

The function h which labels each initial conditionx0 and corresponding orbit {x0, x1, x2, . . .} by an

infinite symbol sequence is,

h(x0) (x0) = 0(x0) 1(x0)2(x0) . . . . (6)

Defining the fullshift 2 = { = 0 12 . . .where 0 = 0 or 1} to be the set of all possible infi-nite symbolic strings of 0s and 1s, then any giveninfinite symbolic sequence is a singleton (a point) inthe fullshift space, (x0) 2. The usual topologyof open sets in the shift space 2 follows the metric,

d2(, ) =

i=0|i i|

2i

, (7)

which defines two symbol sequences to be close ifthey agree in the first several bits. Equation (5) isa good change of coordinates, or more precisely ahomeomorphism,1

h : [a, b]i=0

fi(xc) 2 , (8)

under conditions on f, such as piecewise |f| > 1.2

The Bernoulli shift map moves the decimal point in

Eq. (6) to the right, and eliminates the leadingsymbol,s(i) = i+1 . (9)

All of those itineraries from the map f, Eq. (4) byEq. (5), correspond to the Bernoulli shift map re-stricted to a subshift,3 s : 2

2. Furthermore,

the change of coordinates h respects the action ofthe map, it commutes, and it is a conjugacy.4

In summary, the previous paragraph simplysays that corresponding to the orbit of each ini-tial condition of the map Eq. (4), there is an in-finite itinerary of 0s and 1s, describing each iter-ates position relative to the partition in a naturalway which acts like a change of coordinates suchthat the dynamical description is equivalent. Forour purposes, controlling orbits of the map f inphase space which is an interval corresponds alsoto controlling itineraries in symbol space. The con-trol over x composed with the change of coordi-nates h can essentially be considered to be a codingalgorithm.

3.2. Learning the grammar inpractice

In a physical experiment, corresponding to the one-dimensional map such as Eq. (4), it is possible to ap-

proximately deduce the grammar of the correspond-ing symbolic dynamics by systematic recording ofthe measured variables. First note that any realmeasurement of an experiment consists of a neces-sarily finite data set. Therefore, in practice, it canbe argued that there is no such thing as a gram-mar of infinite type in the laboratory.5 So withoutloss of generality, we may consider only grammarsof finite type for our purposes. Such a subshift is aspecial case of a sophic shift [Kitchens, 1998; Lind &Marcus, 1995]. In other words, there exists a finitedigraph which completely describes the grammar.

All allowed words of the subshift, correspond-ing to itineraries of orbits of the map correspond tosome walk through the graph.

For example, the full 2-shift shift is generatedby the graph in Fig. 3(a) (but this is not theminimal graph generating 2). Likewise, the Notwo zeros in a row subshift 2 is generated byall possible infinite walks through the digraph in

1A homeomorphism between two topological spaces A and B is a one-one and onto continuous function h : A B, which maybe described loosely as topological equivalence.2Note that preimages of the critical point are removed from [a, b] for the homeomorphism. This leaves a Cantor subset ofthe interval [a, b]. This is necessary since a shift space is also closed and perfect, whereas the real line is a continuum. This

is an often overlooked technicality, which is actually similar to the well known problem when constructing the real line inthe decimal system (the ten-shift 10) which requires identifying repeating decimal expansions of repeating 9s such as forexample 1/5 = 0.199 0.2. The corresponding operation to the shift maps [Devaney, 1989] is to identify the repeating binaryexpressions 0 1 n011 0 1 n111, thus closing the holes of the shift space Cantor set.3A subshift 2 is a closed and Bernoulli shift map invariant subset of the fullshift,

2 2.4A conjugacy is a homeomorphism h between topological spaces A and B, which commutes maps on those two spaces, : A A, : B B, then h = h.5It could be furthermore argued that there is no such thing as measuring chaos in the laboratory, since the most populardefinitions of chaos [Devaney, 1989] are asymptotic requiring sensitive dependance and topological transitivity, both of whichwould require time going to infinity to confirm.


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(a) (b)

Fig. 3. The full 2-shift 2 is generated by all possible infinite walks through (a) above digraph. (b) The No two zeros in arow subshift grammar 2 is generated by all possible infinite walks through the above digraph, in which the only two verticescorresponding to 00 words have been eliminated, together with their input and output edges. Blue denotes transitions whichshift a 1 into the bit-register, and red denotes a 0 shift.

Fig. 3(b), in which the only two vertices correspond-ing to 00 words have been eliminated, together

with their input and output edges. Given a finitemeasured data set {xi}

Ni=0, simply by recording all

observed words of length n corresponding to ob-served orbits, and recording this list amongst allpossible 2n such words, the appropriate digraph canbe constructed as in Fig. 3. One should choose nto be the length of the observed minimal forbid-den word. Sometimes, n is easy to deduce by in-spection as would be the case if it were 00 as inFig. 3(b), but difficult to deduce for larger n. Thereexist more systematic algorithms to deduce the min-imal forbidden word length, such as the subgroupconstruction developed in [Bollt et al., 2000; Bolltet al., 2001] related to the follower-set construc-tion [Kitchens, 1998; Lind & Marcus, 1995].

Since, the data set {xi}Ni=0 is finite, then if the

true minimal forbidden word corresponding to thedynamical system is longer than data sample size,n > N, only an approximation of the grammaris possible. Therefore the corresponding observedsubshift is expected to be a subset of whatevermight be the true subshift of the model map Eq. (4)or experiment. This is generally not a serious prob-

lem for our purposes since some course-graining al-ways results in an experiment, and this sort of errorwill be small as long as the word length is chosento be reasonably large without any observed incon-sistencies. Typically, we assume that N n. Asa technical note of practical importance, we havefound link-lists to be the most efficient method torecord a directed graph together with its allowedtransitions.

In Sec. 5, we discuss how the grammar canbe deliberately restricted to offer improved channel

noise resistance.

3.3. One-dimensional maps withseveral critical points

In general, an interval map Eq. (4) may have n criti-cal points xc,j, j = 1, 2, . . . , n, and hence there maybe points x [a, b] with up to n + 1-preimages.Therefore, the symbol dynamics is naturally gener-alized [Lind & Marcus, 1995] by expanding the sym-bol set I = {0, 1, . . . , n} to define the shift space

n+1. The subshift n+1 n+1 of itinerariescorresponding to orbits of the map Eq. (4) follows

the obvious generalization of Eq. (5), i(x0) = jif xc,j < f

i(x0) < xc,j+1, j = 0, 1, . . . , n + 1,and taking xc,0 = a and xc,n+1 = b. The charac-terization of the grammar of the resulting subshiftn+1 corresponding to a map with n-turning pointsis well developed following the kneading theory ofMilnor and Thurston [1977]. See also [de Melo &van Strein, 1992].

3.4. More than one dimension

and symbolic dynamics ofdiffeomorphisms

Diffeomorphisms arise naturally by Poincare map-ping of a flow. In general, a diffeomorphism f: MM is expected for an N 1 manifold M which istransverse to a flow in RN.

Symbolic dynamics of higher dimensional sys-tems is still a highly active research area and details


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274 E. M. Bollt

here are necessarily slight. In particular, we referthe reader to see [Cvitanovic, 1988, 1991, 1995].The fundamental difference of dimensionality isthat invertible maps and hence diffeomorphisms arenecessarily simple in the interval, whereas in morethan one dimension, there may be chaos. In theinterval, only a many-to-one map allows for thefolding property which is an ingredient of chaos.

However, Smale [1967] showed that the foldingmechanism of a horseshoe allows for chaos in a pla-nar diffeomorphism.

In the development in the previous subsections,the one-sided shifts reflect the noninvertible natureof the corresponding interval maps Eq. (4). Thegeneralization of symbolic dynamics for invertiblemaps requires bi-infinite symbol sequences,

2 = { = 210 12 . . .

where 0 = 0 or 1} . (10)

The main technical difficulty of symbolic dy-namics for a map with a more than one-dimensionaldomain is to well define a partition. A notion ofMarkov partitions is well defined6 for Axiom A dif-feomorphisms [Bowen, 1975a], but such maps arenot expected to be generic. The more general no-tion of a generating partition [Rudolph, 1990] isalso well defined,7 but particularly in the case ofa nonuniformly hyperbolic dynamical system con-struction of the generating partition is an open

problem for most maps. A well regarded conjec-ture for planar diffeomorphisms, such as the Henonmap [Cvitanovic et al., 1988; Cvitanovic, 1991], isthat the generating partition should be a curve thatconnects all primary homoclinic tangencies. Seealso [Grassberger et al., 1989; Christiansen & Politi,1996, 1997; Hansen, 1992, 1993].

A main feature in the definition of a generatingpartition is that symbolic itineraries uniquely definetrajectories of the map.

4. The Role of the Partition

At the end of the previous section, we mentionedthe difficulty in finding a generating partition of thesymbolic dynamics for dynamical systems in morethan one dimension in practice, and in particular,in the sorts of dynamical systems which might arisefrom a physical application. In this section, we men-

tion two ways to address this fundamental obstacleto controlling symbolic dynamics.

4.1. Is a generating partition reallynecessary?

In [Bollt et al., 2000, 2001] we studied in rigorousmathematical detail the consequences of using anarbitrarily chosen partition, which is generally notgenerating. Our previous work cited was motivatedby the many recent experimentalists studies, whowith measured time-series data in hand, but in the

absence of a theory leading to a known generatingpartition, simply choose a threshold crossing valueof the time-series to serve as a partition of the phasespace.

On the experimental side, there appears anincreasing interest in chaotic symbolic dynamics[Kurths et al., 1995; Lehrman & Rechester, 1997;Daw et al., 1998; Engbert et al., 1998; Mischaikowet al., 1999]. A common practice is to apply thethreshold-crossing method, i.e. to define a ratherarbitrary partition, so that distinct symbols can

be defined from measured time series. There aretwo reasons for the popularity of the threshold-crossing method: (1) it is extremely difficult tolocate the generating partition from chaotic data,and (2) threshold-crossing is a physically intuitiveand natural idea. Consider, for instance, a timeseries of temperature T(t) recorded from a turbu-lent flow. By replacing the real-valued data withsymbolic data relative to some threshold Tc, say a0 if T(t) < Tc and a 1 if T(t) > Tc, the problem

6Bowen [1970, 1975b], defined conditions for a partition of rectangles to be Markov. A topological partition {Qi} of openrectangles is Markov if,

{Qi

}have nonoverlapping interiors, such that when f(Qi)

Qj

=

, then f(Qi) stretches across Qj, in

that stretching directions are mapped to stretching directions and contracting directions are mapped to contracting directions.Said more carefully, we require that Wu(fn(z), Qi) fn(Wu(z, Qi) and fn(Ws(z, Qi) Ws(fn(z), Qi).7Given a dynamical system f : M M, a finite collection of disjoint open sets, {Bk}Kk=1, Bk Bj = (k = j), is definedto be a topological partition if the union of their closures exactly covers M : M = Kk=1Bk, [Lind & Marcus, 1995]. The setof intersection of the images and preimages of these elements ni=nf(i)(Bxi ) is in general open. For a faithful symbolicrepresentation of the dynamics, the limit n=0 ni=n f(i)(Bxi) should be a single point if nonempty. Given a dynamicalsystem f : M M on a measure space (M, F, ), a finite partition P = {Bk}Kk=1 is generating if the union of all images andpreimages of P gives the set of all -measurable sets F. In other words, the natural tree of partitions: i=fi(P), alwaysgenerates some sub--algebra, but if it gives the full -algebra of all measurable sets F, then P is called generating [Rudolph,1990].


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of data analysis can be simplified. A well chosenpartition is clearly important: for instance, Tc can-not be outside the range ofT(t) because, otherwise,the symbolic sequence will be trivial and carry noinformation about the underlying dynamics. It isthus of paramount interest, from both the theoreti-cal and experimental points of view, to understandhow misplaced partitions affect the goodness of the

symbolic dynamics such as the amount of informa-tion that can be extracted from the data.

As a model problem, we chose to analyze thetent map

f : [0, 1] [0, 1], x 1 2|x 1/2| , (11)

for which most of our proofs can be applied. Ournumerical experiments indicated that the resultswere indicative of a much wider class of dynamicalsystems, including the Henon map, and experimen-tal data from a chemical reaction.

The tent map is a one-humped map, and it isknown that the symbolic dynamics indicated by thegenerating partition at xc = 1/2, by Eq. (5) givesthe full 2-shift 2, on symbols {0, 1}. The topo-

logical entropy of {0,1}2 is ln 2. Now misplace the

partition at

p = xc + d , where d

1

2,

1

2

, (12)

is the misplacement parameter. In this case, thesymbolic sequence corresponding to a point x

[0, 1] becomes:

= 0 12 . . . ,

i(x) = a(b) if fi(x) < p(> p) ,

(13)where

Fig. 4. Tent map and a misplaced partition at x = p.

as shown in Fig. 4. The shift so obtained: {a

,b

}2 ,will no longer be a full shift because not every bi-

nary symbolic sequence is possible. Thus, {a,b}2

will be a subshift on two symbols a and b whend = 0 (p = xc). The topological entropy of the

subshift {a,b}2 , denoted by hT(d), will typically be

less than hT(0) = ln 2. Numerically, hT(d) can becomputed by using the formula [Robinson, 1995]:

hT(d) = limn

supln Nn

n, (14)

where Nn 2n is the number of (a, b) binary se-quences (words) of length n. In our computation,we choose 1024 values of d uniformly in the inter-val [1/2, 1/2]. For each value of d, we count Nn

Fig. 5. For the tent map: Numerically computed hT(d) function by following sequences of a chaotic orbit.


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276 E. M. Bollt

in the range 4 n 18 from a trajectory of 220

points generated by the tent map. The slopes ofthe plots of ln Nn versus n approximates to hT. Fig-ure 5 shows hT(d) versus d for the tent map, wherewe observe a complicated, devils staircase-like, butclearly nonmonotone behavior. For d = 0, we havehT(0) ln 2, as expected. For d = 1/2 (1/2),from Fig. 4, we see that the grammar forbids the

letter a (b) and, hence, {a,b}2 (1/2)[{a,b}2 (1/2)]

has only one sequence: = b bb ( = a aa).Hence, hT(1/2) = 0.

Many of our techniques were somewhat combi-natorial, relying on a simple idea that a dense set ofmisplacement values (in particular if d is dyadic,of the form d = p/2n), allows us to study the re-lated problem of counting distinctly colored pathsthrough an appropriate graphic presentation of theshift, in which vertices have been relabeled accord-ing to where the misplacement occurs. See Fig. 6.

One of our principal results was a rigorous proofthat the entropy can be a nonmonotone and devilsstaircase-like function of the misplacement param-eter. As such, the consequence of a misplacedpartition can be severe, including significantly re-duced topological entropies and a high degree ofnonuniqueness. Of importance to the experimental-ist who wishes to characterize a dynamical systemby observation of a bit stream generated from themeasured time-series, we showed that interpreting

Fig. 6. Graphic presentation for the Bernoulli fullshift andsome dyadic misplacements. Placing the partion at p =(1/2) + d, (1/2) d 1/2, and d as dyadic, such as d = 0,1/16, 1/8, 3/16, or, 1/4, as shown, corresponds to arelabeling of the originally right resolving presentation of thefullshift. The 16 4-bit words are arranged above monotoni-cally with the kneading order of a one-hump map, and so therelabeling occurs in a predictable fashion: shifts to a on theleft, and to b on the right, and the details of the relabelingdepends on the chosen values of d, some of which are shown.

any results obtained from threshold-crossing typeof analysis should be exercised with extreme cau-tion. While apparently by computer experiment,entropy is continuous with partition placement, anarbitrarily chosen partition is expected not to beclose to the true partition, and also there can besevere nonuniqueness problems.

Specifically, we proved that the splitting prop-

erties of a generating partition are lost in a severeway. We defined a point x to be p-undistinguishedif there exists a point y = x such that the p-named a b word according to Eq. (13) does notdistinguish the points, (x) = (y). We defineda point x to be uncountably p-undistinguished, ifthere exists uncountable many such y. We proveda theorem in [Bollt et al., 2001] that states that ifp = q/2n = 1/2, then the set of uncountably p-undistinguished initial conditions is dense in [0, 1].In other words, the inability of symbolic dynamicsfrom the bad nongenerating partition to distin-guish the dynamics of points is severe. We describedthe situation as being similar to that of trying to in-terpret the dynamical system by watching a projec-tion of the true dynamical system. In this scenario,some shadow, or projection of the points corre-spond to uncountably many suspension points. Inour studies [Bollt et al., 2000; Bollt et al., 2001]we also gave many further results both describingthe mechanism behind the indistinguishability, andfurther elucidating the problem.

Now we return to the problem of communicat-

ing with chaos. Is it p ossible to transmit a mes-sage on a chaotic carrier despite using other thanthe generating partition? In Eqs. (5) and (6) we

describe the function, : [0, 1] {0,1}2 and in

Eq. (13), the function : [0, 1] {a,b}2 . Hence,

we imply a function p : {0,1}2

{a,b}2 which is

uncountably-many-to-one at some points. Nonethe-less, each initial condition x [0, 1] has an a bsequence, and if we know how to control arbitrar-ily (the 0 1 corresponding sequence) the orbit ofx by feedback control, we are correspondingly con-trolling its a b sequence. While each x has onecorresponding code-image (x), since is a welldefined function, it is not necessarily a problemfor the purpose of communication that (x) can beuncountably-many-to-one. So in the situation thatthe transmitter plant knows the dynamical systemvery well, meaning how to control trajectories arbi-trarily (knowing how to control its true underlying

{0,1}2 symbolic dynamics) to infinite precisionthen


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the transmitter is also controlling the correspond-

ing {a,b}2 symbol dynamics. The receiver need only

know the key, which requires knowing the partitionp in Eq. (13). If on the other hand, the transmit-ter does not know the underlying generating parti-tion, then the poor continuity properties of the function make the situation impractical, to say theleast. Of course, the assumption of infinite precisionis ridiculous for an experiment. It turns out that ifa small channel error occurs, then the received bitwill be misinterpreted meaning a received errorcan occur. The solution for the generating partitiondiscussed in Sec. 5 is also much more difficult whennot using the generating partition. A final conse-quence of trying to use a nongenerating partitionwould be reduced topological entropy, and hencereduced transmitter capacity, which is an issue alsofurther discussed in Sec. 5.

In fact, a recent application in chaotic cryp-

tography makes use of misplacing the partitiondeliberately as a cryptic key [Alvarez et al., 1999].

4.2. The skeleton of periodic orbitsand generating partitions

Davidchack et al. [Ruslan et al., 2000] have recentlyshown that symbolic dynamics can be learned solelyby observation of a large collection of the unstableperiodic orbits of the dynamical system. It has beensaid by Cvitanovic [1988, 1991, 1995] that periodicorbits are the skeleton of chaos, meaning that the

complete collection of periodic orbits reveals the er-godic properties of a dynamical system.There are several recent advancements both

in computer hardware, and general algorithms[Davidchack et al., 1999; Schmelcher & Diakonos,1998, 1997; Diakonos et al., 1998] that make it rea-sonable to collect the hundreds of thousands of pe-riodic orbits of periods up to say 25 to 30 for wellknown maps such as Henon or Ikeda, with a certainamount of confidence that all have been collected.Interval Newtons method and bisection methodcan be used to guarantee that no periodic orbits

are missed, say for Henon map successfully for pe-riods up to 30 [Galais, 1999, 2001, 2002]. Since thedefinition of a generating partition requires that ev-ery initial condition be named uniquely by the sym-bol dynamics, in particular, a well defined partitionmust label each of these numerically collected pe-riodic orbits uniquely. This is the concept of split-ting which is necessary to well define a generatingpartition. In [Ruslan et al., 2000], we devised a

so-called proximity function to create a simplealgorithm whereby the complete list of collectedorbits could be checked against a proposed trial par-tition, which is then refined in a systematic anddecreasing way relative to the proximity functionwhich moves the partition as little as possible oneach trial. By creating partitions which are consis-tent for all periodic orbits through period-m, say,

then on each refinement to the next order period,m + 1, we find that the corrections to the partitionposition b ecome increasingly small. In particular,see [Ruslan et al., 2000] for a color picture of theIkeda map partition, first found by this method.Perhaps the most interesting feature of this tech-nique, is that once lists of periodic orbits have beencollected, the technique is essentially dimension in-dependent, since one is simply painting and thenrepainting lists of numbers in a specified order.The interested reader should investigate an alter-native algorithm by [Lefranc et al., 1994; Boulantet al., 1997a, 1997b] to find the generating parti-tion, also by using periodic orbits, but by methodswhich are theoretically deeply rooted in the theoryof templates and the knotted structure of periodicorbits of flows in three dimensions.

5. Dealing with Noise by CodeRestrictions

It is no surprise that channel noise can disrupt a

signal transmission, but unchecked, controlled sym-bolic dynamics would be particularly suseptible tobit errors even with small amplitude channel noise.It turns out that there is a very simple way togreatly reduce the vulnerability of a transmitted sig-nal to misinterpretation due to channel noise, by atechnique developed by the author and colleagues[Bollt et al., 1997; Bollt & Lai, 1998; Lai & Bollt,1999].

If the measured variable x is targeted to a pointnear the generating partition at xc, then a smallnoise amplitude in this measurement can frequently

cause the opposite bit than the intended transmis-sion to be interpreted by the receiver. For example,in Fig. 2, if z(t) is controlled to just to the left ofthe partition line, at approximately zc 38.5754,then even a small noise amplitude may cause thatmeasurement to be misinterpreted as being to theright of the partition. In addition to the belowdescribed scheme, one could use an error correct-ing code, such as the Viterbi algorithm [Forney,


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278 E. M. Bollt

1975; Heller & Jacobs, 1971; Lin & Costello, 1982;Michelson & Levesque, 1985; Peterson & Weldon,1998; Pless, 1998] in the case of Gaussian noise tofurther improve noise resistance, rather than trans-mitting plain ascii with the symbol masking as wehave described below for clarity.

5.1. Introducing a code restrictionCommunicating on unstablechaotic saddles

The solution to avoid channel noise is simple: avoidtargeting a neighborhood of the partition. Supposex0 is targeted to x1 = xc + to input, say, a 1-bitinto the bit register, then a channel noise of am-plitude can push x1 below xc, and the receiverinterprets a 0-bit. Simply, never targeting a gap(xc , xc + ) prevents this problem. But thenin dynamical systems, it is not possible to simply

remove a region from the phase space; it is neces-sary to dynamically remove the region by removingit and all of its preimages. The remaining set,

M() = [0, 1] i=0fi((xc , xc + )) ,

> 0 , (15)

leaves a Cantor set, which for many one-humpedmaps, including that in Fig. 2, is nonempty for small

Fig. 7. An unstable chaotic saddle M(s) remains after a gapis removed around the generating partition, as per Eq. (15).This Cantor set has favorable channel noise resistance proper-ties as a message carrier, but costs to the transmitter capacityis only slight, as shown in Fig. 8.

Fig. 8. Topological entropy hT() as a function of noise gap. At each fixed value of , the topological entropy of theunstable chaotic saddles M() in Eq. (15) are shown, oneof which is depicted in Fig. 7. Notice that initially, even afairly large noise gap costs only a modest decrease in hT(),relative to maximal at = 0.

enough . See Fig. 7. Such invariant sets are calledunstable chaotic saddles embedded in the full in-terval invariant set. We have found that they arehighly useful for communicating with chaos, sincethey are noise resistant, easy to stabilize, and costonly a relatively small amount of transmitter ca-pacity, as we discuss b elow. Also important, is thefact that these sets can be characterized as a sub-shift embedded in the subshift of the full interval,

and hence they can be specified by a simple coderestriction.

A simple fact from the kneading theory[Milnor & Thurston, 1977] provides that the codesof the unit interval are ordered monotonically in theunit interval when choosing the Gray-Code order. That is x < y implies (x) (y). In Fig. 3(a),the 4-bit words are arranged left to right accord-ing to . Therefore, an -neighborhood ofxc canbe characterized by a range of codes, and approxi-mated by some n-bit code restriction. For example,

the Lorenz map allows the 4-bit words 0.100 and1.100, which can be seen in Fig. 3(a), and definesan -neighborhood of xc, (xc , xc + ). To avoidthis neighborhood, it is necessary to never transmitthe phrase, 1 followed by 00, as can be deducedin Fig. 3(b). This particular code restriction stabi-lizes the chaotic saddle whose subshift is generatedin Fig. 3(b).


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282 E. M. Bollt

parameter perturbation kn+1.

pn+1 = pn+1 pn

=

fdes(xn) fp0(xn) (pn p0)f

p

(xn,p0)

f1

p

(xn,p0)

.

(27)

The derivative f/p characterizes the rate ofthe static map variations. To estimate this quan-tity, one must experimentally iterate the one-dimensional maps obtained for several successivePoincare sections, for three values of the fixed ad-justable parameter p. Each time a simulation witha new value of p is started, the first intersectionwith Poincare surface is neglected, because it is notpart of static 1-D map. In our previous chemical re-action application [Dolnik & Bollt, 1998], we havedealt with unavoidable noise fluctuations, by appli-cation of cubic smoothing spline fits [Hutchinson,1986]. We estimate the derivatives f/p from theequation,

f

p

(x,p0)

=fp+(x) fp0(x)

. (28)

Direct application of the above formulas wouldrequire random access to initial conditions, whichis generally not possible in learning the real physi-cal experiment. We proposed a method [Dolnik &Bollt, 1998] of learning these quantities on-the-flyby appropriate manipulations of a running experi-ment. The dynamic rate of map variations, f1/p,can be learned experimentally. As described in fig-ures in more detail in [Dolnik & Bollt, 1998], wewait for two iterates at the nominal value p0, be-tween experimental parameter variations, to makesure that the transients settle onto the 1-D attrac-tor, to a high degree of accuracy. The observed nextpiercing of the Poincare surface, after a parametervariation, is the dynamic response f1(x). Each two-iterate-time interval of fixed p(t) = p0 are followedby a one iterate time interval ofp(t) = p0. Theresponse function x(t) must be read accordingly.After sampling this dynamic response value f1(x),

we reset the flow rate back to the nominal valuefor two reference events. This procedure can be re-peated, or a perturbation with the same amplitudebut of opposite sign can be applied. By periodicallyrepeating this process with positive and negativeperturbations, we learn dynamic responses f1p0(x)for values ofx ergodically scattered throughout the

Fig. 9. The z(t) time-series from the Lorenz Equations, Eqs. (1), can be controlled to any desired message, when successivemaxima are read relative to the generating partition, which is the horizontal line. This particular z(t) time-series is fromthe (x(t), y(t), z(t)) trajectory shown in Fig. 1. The underlined bits denote noninformation bearing buffer bits which arenecessary either due to nonmaximal topological entropy of the underlying attractor, or further code restrictions which wereadded for noise resistance, as discussed in Sec. 5. The code used was the no-two zeros in a row rule, which requires a buffer1-bit be inserted after every 0-bit appears. The message is encoded in standard ASCII, and requires 63-bits, but with theextra 33 buffer bits, the total transmission is slowed to 96-bits. This particularly restrictive code has a topological entropy ofhT = ln((1 +

5)/2) 0.4812, but gives a particularly wide noise gap.


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interval. The derivative f1/p is estimated sim-ilarly to the derivative f/p, from the differencequotient:

f1

p

(x,p0)

=f1p0+(x) fp0(x)

. (29)

More details are given in [Dolnik & Bollt, 1998]

concerning these necessary technical considerationsto control a one-dimensional map derived by suc-cessive maxima of a measured time-series of a dif-ferential equation, or real experiment. We concludethis section by presenting for the first time a plot inFig. 9 of a controlled trajectory of the Lorenz equa-tions which have been forced to follow a path whosez(t) time-series corresponds to a popular message atthe authors home institution.

Acknowledgments

The author was supported by the NSF underGrant No. DMS-9704639 during 19972000, andcurrently under No. DMS-0071314. I would liketo thank, Ying-Cheng Lai, Milos Dolnik, CelsoGrebogi, Karol Zyczkowski, Ted Stanford, RuslanDavidchack, Mukesh Dhamala, Eric Kostelich, JimMeiss and Aaron Klebanoff, with whom I have hadthe pleasure to work on research in the areas ofcommunication with chaos, or symbolic dynamics,or control of chaos, and some of this research isreviewed here.

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