Nonlinear and Chaotic Dynamics: An Economist's Guide Michael D. Weiss
AbsiraLI 111 recellt 'lem, re~earch 111 both matheHla· state space RI' (where R IS the real number Ime) Sup tIC; alld the applied sCWllces ha\ plOdl/ced a revo{lI pose that the economy evolves determlmstlcally In tum l1/ the uudel "tcoullnq of llonll1leal dl/1wHllcal such a way that ItS state' at any time umquely deter "1st e 111 1 U,;ed wldehl111 eCOII01l"C\ and othe, d,sc1 mmes ItS state at all later tImes Then, If the Imhal plwe, to model challge ovel tillie, the"e "I~ten" a,e pOSitIOn of the economy III R" at tIme 0 IS vo, the 1I0W kllOlV1I to be vIII"elable to a kmd oj "chaotIc," evolutIOn of the economy through tIme will be repre ''''/ll e,lIetable beh'lVlOr Till,; mi1de places tillS ,evoln sented by a path m R" startmg at Vo and traced out by t,lOll 111 1l1':Jturlcul c'olltext, dISCU,"lws som.e of Lts VI as tIme, t, moves forward ThiS path, called the I1HpllcatlOll':l tor econOHilC modell11g, and explalns orb,t generated by the ImtJal pOSItIOn Vo, represents a I/WIIII 0/ the I1l1pOltalll ",,,thematlcal Ideas all willch It "future history" of the system QuestIOns about the /I b",cd behaVIOr of the economy ovel time are really questIOns about ItS orbIts We are often mterested not so much Keywords Ll1mt, to predlctablhty, nonlmear and m the near-tenn behaVIOr of orbits as m theIr eventual chaotIc dvnamlcal systerns, strnctural stab1l1ty of behavlOl, as when we engage m long-range forecastmg econonl1c models, (ractals or study an economy's response to d new go vel nment pohcy 01' an unexpected shock after the ImtJal period of In the past two decades, the world of sCience has come adjustment has passed and the economy has settled to a fundamentally new unde1 standmg of the dynamiCS down of phenomena that vary over time Grounded m math ematlcdl discovery, yet given empIrIcal substance by Fractals, SensItive Dependence, and Chaos eVidence from a variety of diSCiplines, this new per spectIve has led to nothmg less than a re-exammatlOn SCIentIsts long have known that It IS pOSSIble for a sys of the concept of the predICtability of dynamiC be tem's state space to con tam an Isolated, unstable pomt hdvlOr OUI ImpliCIt confidence m the orderliness of p such that different Imtlal pomts near p can generate dynamlcdl systems, specifically of nontmear dynanucal orbIts WIth WIdely vdrylllg longrun behaVior (For sy,tems, has not, It turns out, been entirely Justified example, a marble bdlanced on the tiP of a cone IS Such systems are capable of behavmg m ways that are unstable m thiS sense) What WdS unexpected, how far more elTatlc and unpre(lIctable than once beheved ever, was the dIscovery that thiS type of mstdblhty FI ttmgly, the new Ideas are said to concern chaot,c can occur throughout the state space, sometImes actu dvnarrllco, or, SImply, chaos ally at every pOlllt, but often m'strangely pdtterned, fragmented subsets of the state space-subsets typ EconomIcs IS not Immune from the ImplicatIOns of thIs Ically of nomntegel dImenSIOn, called fractalo Once new understdndmg Aftel all, our subject IS replete mvestlgators knew what to look fOI, they found thiS WIth dynamIC phenomena I angmg from cattle cycles to phenomenon, termed 6cnsdwe dependence on l,n~i'wl stock mal ket cdtastrophes to the back-and-forth mter conchtlOns, to be Widespread among nonlmear dynam play of advertlsmg and product sales Ideas related to Ical systems, even among the Simplest ones Though the notIOn of chaotic behaVIOr are now part of the baSIC techmcal deflmtlOns vary, systems exhlbltlllg thiS mdthematlcal toolkIt needed for mSlghtful dynamIcal unstable behaVIOr have generally come to be called modeling Agncultural economIsts need to gam an "chaotIc" understandmg of these Ideas Just as they would any other slgmficant mathematICal contributIOn to their For chaotIC systems, any errOl m speclfYlllg an mltlal field ThiS article IS mtended to assist m thIS educa pomt, even the most mmute error (due to, say, com tIOnal process puter roundmg m the thousandth deCimal place), can g1Ve I1se to an orbit whose 10ng:lUn behaVIOr bears,no What exactly has chdos theory revealed? To address resemblance to that of the orbit of the {ntended Imtlal thIS questIOn, let us conSIder an economy, subject to pomt Smce, m the real world, we can never specify a change over time, whose state at time t can be pomt With mathematically perfect preCISIOn, It follows descl1bed by a vectOl, v" of (say) 14 numbers (money that practical longrun predictIOn of the state of a cha supply at time t, mflatlOn rate,at time t, and so on) otIc system IS ImpOSSIble Formally, thIS vector IS d pomt m the 14-dlmenslOnal Attractors WeISS IS an economist In the CommodIty ELOnomlcs DIVISion ERS The author thanks John McClelland and other partiCipants I~ the ERS Chaos Theon Semmar COl many stImulatmg dlSCllSSlon,> on For a dynamlCaJ system, perhaps the most baSIC ques chaotiC dynamiC'> Callos Arnade, Rlchclrd Heifner. and an anony tlOn IS "where does the system go, and what does It do mous lefelee furmshed helpful reView comments when It gets there?JJ In the earlIer view of dynamical 2 THE JOURNAL OF AGRICULTURAL ECONOMICS RESEARCHIYOL 4,1. NO 3. SUMMER 1991 systems, the place whel e the system went, the pomt nor the Imagmg techmques available at the tIme per set m the state space to whIch orbIts converged (called mItted him to explore h,S mtUltIOns fully an attraetor), was usually assumed to be a geo metrICally uncomphcated object such as a closed curve Followmg Pomcare's work and that of the AmerICan or a sIngle pomt EconomIc modelel s, for e,ample, mathematicIan G D Blrkhoff m the early part of thiS have often Imphcltly assumed that a dynamIC economIC century, and despIte contmumg mterest m the SovIet process WIll ultimately achIeve eIther an eqUlhbnum, a Umon, Lhe subject of dynamical systems fell mto lela cychc pattern, or some other orderly behaVIOr How tlve obSCUrity DUllng th,S perIod, there was some ever, another dIscovery of chaos theory has been that dwarene"s among mathemdtlclans, SCientIsts, and the attractor of a nonlInear system can be a b,zarre, engmeers that nonhnear systems were capable of erra fractal set wIthIn "h,ch the system's state can flIt tiC behaVIOr However, examples of such behaVIOr endlessly m a chaotIC, seemmgly random manner were Ignored, claSSified as "nOIse," or dIsmIssed as aberratIOns The Idea that these phenomena were ,Just as an economy can have two OJ more eqUlhbna, a chal actenstlc of nonhnear dynamICal systems and that dynamIcal system can have two 01 more attractors In It was the well-behaved, textbook examples that were such a case, the set of all Imtlal pomts whose orbIts the speCial cases had not yet taken root converge to a partIcular attractOl IS called a bas", of attractlOn A recent findmg has been that the bound Then, m the 1960's and 1970's, there was a flurry of ary between competmg basms of attractIOn can be a activity m dynamIcal systems by both mathematicIans fractal even when the attractors themselves are unex and SCIentists workmg entirely mdependently Mathe ceptIOnal sets A type of senSItIVIty to the mltlal comh matlClan Stephen Smale turned hiS attentIOn to the tIOn can operate here too the shghtest movement subject and used the techmques of modern differential away from an Imtlal pOInt Iymg m one basm of attIac topology to create ngorous theoretical models of cha tIOn may move the system to a new basm of attl actIOn otic dynamICS Meteorologist Edward Lorenz dIS and thus cause It to evolve toward a new attractor covered that a Simple system of equatIOns he had devI"ecl to Simulate the earth's weather on a pnmltlve ChaotIC beha vlOr wlthm a workmg model would be computer dIsplayed a surpnsmg type of senSItIvIty eaSIer to recognIze If all 01 bIts Imtlatmg neal an erra the shghtest change m the Imtlal condllIons,eventually tIC OJ b,t were also elratlc However, the potentIally would lead to weather patterns bearIng no resem fractal structure of the regIOn of senSItive dependence blance to those generated m the ongmalrun can allow ImtIaI pomts whose orbIts behave "sensIbly" and Imtlal pomts whose orb,t, are enallc to coeXIst mseparably m the state space hke two mtermmgled BIOlOgIst Robert May llsed the 10gIslic difference equa clouds of dust Thus, SImulatIOn of a model at a few tIOn xnT I = IXn (I-x,) to model populatIOn level, x, tI lal pOlnts cannot rule out the posslblhty of chaotIc over successive tIme penods He observed that for dynamICS Rather, we need a deeper understandmg of some chOIces of the growth rate parameter, r, the pop the mathematical propel ties of our models Nor can ulatIOn level would converge, for other chOices It chaotic dynamICS be d,sm,ssed as ansmg only m a few would cycle among a few values, and 'for slill others It qUIrky speCial cases As we shall see, It anses even would fluctuate seemmgly randomly, never achIevmg when the system's law of motIOn IS a Simple quadratIC either a steady state 01 any dlscermble repeatmg pat j tern When he attempted to graph the populatIOn level agambt the glowth 1ate parameter, he observed a The DIscovery of Chaos strangely patterned, fragmented set of pomts
) Recent years have Witnessed an explOSIOn of mterest and activity m the area of chaotiC dynamiCs What PhYSICISt MItchell FeIgenbaum Investigated the 1 behaVIOr of dynamical systems whose equatIOns of accounts for thiS new VISlblhty, WhICh extends even beyond the reseal ch commumty mto the pubhc medIa? molion anse from ummodal (hIll-shaped) functIOns He noticed that cel tam parameter values that sent the To plovlde an answer, we brIefly trace the historIcal system Into repeatIng cycles always displayed the de,elopment of the subject same numellcally precise pattern no matter which dynamical system was exammed, the ratIOs of succes The first recogrutIOn of chaotIC dynamICs IS attnbuted sive distances between these parameter values always to HenrI Pomcare, a FI ench mathematiCian whose converged to the same constant, 4 66920 Feigen work on celestial mechamcs around the turn of the baum had (hscovered a umversal property of a class of centUl'y helped found the study of dynamical systems, nonhnear dynamIcal systems HIS discovery ultImately systems m which some structure (perhaps a solar sys clanfied how systems can evolve toward chaos tem, perhaps-as now understood-an economy) changes over tIme accordmg to pI edetermmed rules Pomcare foresaw the potential for unpredlctablhty m Thus, as these and other examples demonstrate, while dynamical systems whose equatIOns of motIOn were mathematlClans were developmg the theory of non nonlmear However, neither the mathematical theory lInear and chaolic dynarmcs, SCIentists m dIVerse dlscI 3 plines were witnessing and discoverIng chaotic Henceforth, for brevity, we denote by phenomena for themselves Ultimately, researchers learned of one another's findmgs and recogruzed their fn, common ongm the nth Iterate of a functIOn f Thus, fI(x) = f(x), f'(x) = f(f(x», f3(x) = f(f(f(x»), and so on By conventIOn, The role of the computer m the emergence of the con fO(x) = x Of course, fn IS Itself a functIOn rfshould not temporary understandmg of dynamical systems IS dif be confused With the nth derIVatwe of f, which IS ficult to exaggerate As we now realize, even the customanly denoted simplest system_s can generate beWllderlllgly complI cated behavIOr The development of modern computer fen) power and graphics seems to have been necessary before researchers could put the full picture of non Orbit Diagrams hnear and chaotic dynarrucs, qUIte hterally, mto focus Fortunately for expository purposes, many of the The Mathematics of Chaos Important features of dynamical systems are present m one-chmenslOnal systems In fact, one of the Impor We now explam some of the basIc mathematical Ideas tant findmgs of chaos research has been tnat discrete mvolved m noniineal dynamics and chaos We also dynamical systems generated by IteratIOn of even the adopt a slIghtly different perspective In the above most elementary nonlmear scalal functIOns are capable discussIOn, we have ImplIcitly portrayed dynamical of chaotiC behaVIOr Thus, we shall concentrate on systems as bemg m motIOn m contmuous tIme How functIOns that operate on the number lme ever, the equatIOns of motIOn of such systems tYPically mvolve differential equatIOns, and a proper tI eatment For such functIOns, there IS a particularly COnVe!llent often reqUIres advanced mathematIcal machmel y It IS techmque for dmgramrrung orbits ConSider a functIOn generally much easlel to work With (and to under f and an Imtlal pomt x (fig 1) Begmmng at the pomt stand) discrete-time systems, m which time take; only (x, x) on the 45"lme, draw a dotted Ime vertIcally to mteger values representmg successive tIme perIOds the graph of f, the pomt of mtersectlOn WIll be (x, f(x» Let us shift OUI attentIOn to these systems From that pOInt, draw a dotted Ime honzontally to the 45' lIne, the pomt of mtersectlOn WIll be (f(x), f(x» From there, draw a dotted Ime vertically to the graph When the law of motIOn of a discrete dynamical system of f, the pomt of mtersectlOn Will be (f(x), f2(x» Con IS unchangmg over time, the movement of the system tmue thiS pattern of movmg vertically to the graph of f through time can be understood as a process of Iterat and then hQnzontally to the'45' line The resultmg diS mg a functIOn To estabhsh this pomt, conSider a typI play, called an orb,t dwgram, shows the behaVIOr of cal dynamic economic computer model, M, havmg k the orbit ongmatmg at x In particular, the orbit may endogenous vallables To start the model runmng, one enters an Imtlal conditIOn vector, vo, of k numbers The mode1 computes an output vector, M(vo), contam mg the new values of the k endogenous varIables at FIQl.lre 1 the end of the first time perIOd The model then acts Construclion of an orbit diagram on M(v ) and computes a new output vector, M(M(v », o o 45· descnbmg the economy at the end of the second tIme penod Successive output vectors are computed m the same manner Note that the model Itself, the law of motIOn, remams unchanged durmg this process In effect, M acts as a function, mappmg k-vectors to new k-vectors, applymg Itself Iteratively to the last computed functIOn value The state space of the econ omy IS the k-dlmenslOnal space Rk, and, for each Imtml conditIOn vector Vo, there_ IS a correspondlllg orbit , v0, M(vo), M(M(vo», M(M(M(vo»), ,descnbIng the future evolutIOn of the economy , (1'1,) 1'1,)) More generally, conSider any function f If f maps Its domalll (the set of all x for which f(x) IS defined) Into Itself, then, for each, Xo In the domaIn of f, the sequence xo, f(xo), f(f(xo», f(f(f(xo))), IS well defined and may be conSidered an orbit of a dynamical system determmed by f through IteratIOn be visualized from the mtersectlOn pomts marked on Let us fir"t dispense with the case a = I In this case, the 45" line, the dotted lines mdlcate the directIOn of If b = 0. then every h IS a fixed pomt of g (that IS, g(x) motIOn of the system Of course, the pomts (x,x), (f(x), = x), and (smce then, also gO(x) = x) the system f(x)), (f'(x), f'(x)), only look hke the orbit They always remamS at any Imtlal pomt In contrast, If b * reside m the plane, whereas the actual orbit, conslst 0, then no x IS a fixed pomt of g, mdeed, for any Imtlal mg of the numbers x, f(x), f'(x), , resides m the pomt x, go(x) diverges monotomcally as n--+ x to eIther state space, that IS, III the number Ime " 01 -x accordmg to whether b > 0 or b < 0
DynamIcs of Lmear Systems In dlscussmg the sn remallllng cases, that IS, the cases m which a * I, I take b to be an arbltralY num Though the basIc focus of this paper IS nonlinear ber In these cases, g has exactly one ft~ed pomt, dynamics, exammatlOn of linear systems provides b/(1-a), and any orbIt ongmatmg there remams there essential mtUltlOn about nonlinear ones Thus, we I next ehamme the behaVIOr of orbIts orlgmatmg at begm ,with an exhaustive treatment of the hnear case POll1ts other than b/(1-a) For this pm pose, I assume that the Imttal pomt x IS an arbltl ary number dlstmct Choose any numbers a, b, and consider the functIOn g from bl(1-a) defined by g(x) = ax + b To compute a typical orbit of g, observe that If a < -I, then gO(h) has no flmte 01 lI1hmte limit Rathel, It eventually altel nates between positive and g'(~) a(ax + b) + b negative numbers as ItS absolute value diverges monotomcally to oc a'x + b(1 +a), If a = -I, the fixed pomt b/(I-a) equals b/2, and
g"(x) = (-I)"(x-b/Z) + biZ
= {b-' If n IS odd g'(x) a'x + b(1+a+a'+a'), x If n IS even and. 111 general, go(x) = a"x + b(l+a+a'+a'+ Thus, g"(x) alternates endlessly between the (dlstmct) +ao-') If a = I, then numbers b-x and x
g"(x) = , + bn If -I < a < 0, gO (x) convelges to b/(l-a) while altel natmg above and belo .... It However, If a F 1, the formula fOl the sum of a geo If a = 0, then, for alln, go(x) = b Thus, the system metnc senes gIVes moves from the Initial pomt du ectly to band remams there go(x) = .lOX + b[l-aOl l-aJ If°< a < 1, go(~) converges monotomcally to b/(1-a) = ao rx - ...E.... ] + ...E.... The convel gence IS from above If x > b/(I-a) and from L I-a I-a below If x < bl(l-a)
Note that when a IS nonnegative, aOremams nonnega Fmally, If a > I, then go(,) dIverges monotolllcally, to tive, while when a IS negatIve, an alternates between '" If x > b/(1-a) and to -x If x < b/(1-a) negative and posItIve In particulal, when d. = -1, an alternates between -1 and I When a * ±I, the diS The pOSSible behaVIOrs of orbIts m the one-dImensIOnal tance between aO and °either converges monotomcally Imear system are Illustrated m figures 2(a)-Z(h) From to °or (liverges monotomcally to '" as n--+" accOl dmg these figures and the precedmg dISCUSSIOn, two lessons to whether lal < 1 or lal > 1 Usmg these facts, we emerge First, the fixed pomt IS of tel! at the "center of now analyze the behaVIOr of all the orbits generated by the actIOn" It IS to or from th,S pomt that orbits typ g, accol dmg to the varIOus pOSSibilities for the Stl uc Ically converge or dIverge Second, the slope param tural parametels a and b and the Imtlal POll1t h We etel, a, plays a pIvotal role 111 determmlllg orbIt shall find It convement to orgamze our analysIs alound dynamiCs These two prmclples hold as well for non the pOSSIble value of a We (hstmgUlsh seven cases (1) Imeat systems a < -1, (Z) a = -1, (3) -1 < a < 0, (4) a = 0, (5) °< a < 1, (6) a = 1, and (7) a > 1 Wlthm each of these cases, Fixed Pomts and PerIOdIC Points we conSider all pOSSible values of the I emammg stt uc tUl al pal ameter b and the mltlal pomt x. and we deter It IS not a comcldence that, m the hnear system, con mme the longrun behaVIOr of the orbit ongmatmg at x vergent orbits always converge to a fixed pomt of the when g has·structural parameters a and b underlymg functIOn In fact. thiS property holds In
5 FIQI.!fe 2(3) Figure 2(b) Orbit diagram for linear function (a<-1) Orbit diagram for linear function (a=-1)
g g
\ x b/1-a
Figure 2(c) Figure 2(d) Orbit diagram for linear function (-1 ______~b4-__~~------g x b/1-a general To estabhsh It, suppose a contmuous functIOn hmlt must be a fixed pOint Con espondl11gly, If an f has a convergent orbit x, f(x), f'(x), ,f"(x), economy converges to an eqUlhbllum, the eqUlhbllum Let L be the hmlt Then ' state must be a fixed pomt of the system function Closely related to fi>.ed pOints are pomts whose 01 bits may leave but later letu!'l1 ,(see fig 2(b)) A pomt x IS called a pel Wd1C pomt of f wIth penod n If f"(),.) ~ x The smallest positive n for which the latter equatIOn holds IS called the pm"e perIOd of x It can be shown that any perIOd of x IS a multiple of the pl'lme penod L, EvelY fi>.ed pomt of a functIOn f IS a periodiC pomt of f so that L IS a fixed pomt of f Thus, m partial answer of p"me perlOd 1 It IS also true that evel y perlO(hc to our gUldmg questIOn, "whele does the ,system go," pomt IS a fi>.ed pomt (though not of the same functlOn), we can reply If It convelges to any fimte hmlt, that smce the pellOdlclty COIl(htlOn f"(x) ~ "\ IS nothmg but 6 Figure 2(e) Figure 2(1) Orbit diagram for linear function lO(a(1) Orbit diagram for linear function (a=1, boO) 45' 45' FlglSe 2(g) Figure 2(h) Orbit diagram for linear function (a'1, b>, 0) Orbit diagram for linear function (a)1) g g 45' 45' the assertIOn that x IS a fixed pomt of the function [n cycles (Such an orbit IS called a cycle of length no ) As Thus, properties of fIXed pomts have counterparts for a consequence, whenever a system With a computa penodlc pomts, and vice versa tionally tractable law of motIOn IS mltlaitzed at a pomt known to have a small perIOd, the system's entire I f x IS a penodlc pomt of f havmg pnme penod l1o, then future evolutIOn can, as a practical matter, be necessanly, calculated f"O+ I(>.) = f(fno(X)) = f(x), In recent decades, there have been some remarkable dlScovenes concermng when the eXistence of a cycle of f"O+ 2(x) = f(f"O+'(x)) = f2(x), one length Impltes the eXistence of cycles of other lengths LI and Yorke (10) show that If f IS any contm and so on It follows that the orbIt of x reduces to a uous functIOn mappmg an mterval J mto Itself, and If fimte set conslstmg of the dlstmct pomts x, f(x), f2(x), some pomt m J IS penodic for f Wlth pnme perIOd 3, , f'w'(x), through which the system endlessly then, for every posItive mteger n, there IS a penodlc 7 pomt m J havmg pnme perIOd n I In bnef If thel e IS a and 2(e) that, If a curve (that IS, a nonlmeanty) were cycle of length 3, there mu~t be cycles of all lengths mtroduced mto the gI aph of the functIOn g at some dIS tance from b/(l-a), any orbIt Ol~gInatmg near enough The LI-Yorke Theorem IS actually only a part of a to b/(1-a) would stIll converge to b/(1-a) Such local more general result of Sarkovsku (see (6» that may be con vergence does not depend on the slope of the graph descllbed as follows LIst the entIre set of posItIve of the functIOn fal away flOm the fIxed pomt, what mtegers m the followmg manner matters IS only the slope, that IS, the denvatlve, m a neIghborhood of the fIxed pomt In fact, less 3,5,7, obvIOusly, but as we shall see momentanly, It IS really only the denvatJve at the fIxed pomt Itself that 2 3, 2 5, 2 7, matters 2-' D,- 2' 7, SimIlarly, all orbIts m the lInear system orIgmatmg elsewhere than b/(1-a) move away flOm b/(l-a) when 23 3, 2' 5, 2' 7, ever lal > 1 If a nonlmearlty wele mtlOduced mto the gI aph of g at a dIstance from b/(l-a), any orbIt ong matmg suffiCIently close to (but not preCIsely at) b/(l a) would stIll move away flom b/(l-a), at least ImtIally (the posslbLlIty of an eventual return IS another Issue) Agam, for such local "aversIOn" to b/(l-a), It tUl ns out 2n 3, 2n 5, 2n 7, that only the denvatIve at b/(l-a) Itself matters These observatIons lead to the followmg defimtlOns A fixed pomt p of a functIOn f IS called hyperboltc If If(p)1 ;t 1 When If'(p)1 < 1, p IS called attraetmg, whIle when If(p)1 > 1, p IS called repelhng These adjectIves are JustIfied by the followmg two propOSItIOns, whICh are readIly establIshed (1) If p IS an attractmg hyper This lIst IS called the "Sarkovsku ordermg" of the POSI bolIc fixed pomt, there IS an mterval contammg p such tIve mtegers Obselve that the odd mtegers exceedmg that any orbIt orIgInatmg therem converges to p (2) If I are hsted fIrst, followed by the varIOus posItIve p IS a repellmg hyperbolIc fIxed pomt, there IS an powel s of 2 tImes the odd mtegers exceedmg 1, fol mterval contammg p such that any orbIt ol~gmatmg lowed finally by the pure powel S of 2 m reverse order therem (but not at p Itself) eventually leaves the mter Now, suppose f IS a contmuous functIOn mappmg the val (at least temporarIly) For the functIOn shown m numbel Ime mto Itself Then, Sarkovskll'S Theorem figure 1, 0 IS attractmg hyperbolIc whIle the other two states that, If f has a penodIc pomt of pnme penod k fixed pOInts are repellmg hyperbolIc and k' IS any mteger appearmg latel m the lIst than k, then f also has a pelloclIc pomt of pllme pellod k' One In the lIterature, a perIodIC pomt x of f of pnme penod consequenc.e of this result Ib that If f hds any cycle n IS defined as hyperbolIc If l(fn),(x)1 ;t 1 The meamng who"le length IS not d pure powel of 2, then f must of thIS defmltlOn becomes transparent once It IS have cycles of Infimtely manv dlffel ent lengths Thus, I ecalled that x IS a fixed pomt of 1" for e)..ample, If an annual Iterative economIC model WIth one endogenous vanable exhIbIts a busmess cycle In hIgher dImensIOnal systems, the notIon of the derIV of length 5 years, then (fOl other Illltlal pomts) the atIve at a pOInt IS expressed m terms of a JacobIan model must be capable of generatmg busmess cycles of matnx, and a perIodIC pomt IS defined as hyperbolIc If mfimtely many othel lengths While Sarkovsku's The none of the eIgenvalues of thIS matrIx has complex orem pertams only to functIOns of one Val lable and modulus one (that IS, If none hes on the Ulllt CIrcle m thus would be clIrectly applIcable to, at most, a lImIted the complex plane) class of dynamIC economIc models, It does serve to Illustrate the theSIS that nonlInear dynamIcal systems When a fixed pomt p IS hyperbolic attractmg, the sys are lIable to Impose unobvlOus but empmcally I elevant tem can be conSIdered stable at p WIth respect to mathematical restnctlOns on economIc behavIOr changes m InItIal condItIOns Ifthe system IS ImtlalIzed at p, It WIll, of course, remam there More Important, Hyperbolic POInts though, the system WIll converge to p even If It IS not mltlalIzed there, as long as It IS ImtmlIzed suffiCIently ExammatlOn of the Imear system leveals that, when near p ever lal < 1, all 01 bIts converge to the fIxed pomt b/(l-a) However, It IS clear from figures 2(c), 2(d), In the same vem, a hyperbolIc repellIng fixed pomt p can be conSIdered a pomt of mstabllIty of the system lltallclzed numbel S III pcU entheses cIte SOUl ces Its ted III the Refel WIth respect to changes m mllIal condItIOns WhIle the ences sectIOn at the end of thiS aI tide system WIll remam at p If ImtIalIzed preCIsely there, It 8 will move away from p ,\,henever It IS Imtlahzed suf belong to the set C' of all such functlOns 2 Define the ficiently close to, but not at, p "C'-dlstance" between f and g as d,(f,g) = sup max {if(x)-g(x)l, If'(x)-g'(x)l, x Structural Stabihty The stabilIty property enjoyed by an attractmg hyper bohc pomt concerns the effect of a shght change m the where the supremum" IS taken over all x In J (For Imtlal condltlOn, the underlymg model, however, each x m J, there IS a correspondmg maximum of abso remams fixed We now discuss another notion of sta lute values as shown, the supremum IS over thiS set of blhty, structural stablhty, which concerns the effect of maxima) Then, f and g are conSidered "C'-close" when a shght change m the model Itself d,(f,g) IS small, that IS, when f IS pomtwlse close to g and the first r denvatives of fare pomtwlse close to those of g Figure 3 shows two functlOns that are CO In essence, a model IS structurally stable If small close but not CLclose changes m the model's structure leave dynamical behavlOr qualltatlvely unchanged To understand why Our next step m makmg precise the notlOn of struc this property IS unportant for empmcal work, suppose tural stablhty IS to clarify what IS meant by the that, from a collectlOn of economic models shanng the dynamlCal "eqUlvalence" of two dynamlCal systems same functlOnal form and dlffenng only m their values Toward thiS end, suppose f and g are contmuous func of some structural parameter vector w, we were to tlOns mappmg an mterval J mto Itself By a home attempt to select the model Mw that truly described omorphIsm' of J we mean a contmuous, mvertlble reahty Suppose further, thoug'h, that there eXisted functlOn mappmg J onto Itself Thus, a homeomor parameter vectors w arbltranly close to w0 whose cor phism of J IS a one-to-one correspondence between J respondmg models M. had dynamical behavlOr dlffer and Itself such that nearby pomts are sent to nearby mg from that of M.o Then, as a practical matter, we pomts Smce the mverse functlOn of a homeomorphism could never confidently determme the true economlC of J IS Itself necessanly contmuous, thlS "preservatlOn dynamiCs of the sltuatlOn, for even the shghtest error of nearness" operates m both directlOns As an exam m econometncally estlmatmg Wo (such as due to com ple, the functlOn h defmed for all x m the mterval puter roundmg) would leave us vulnerable to havmg arnved at a dyn-amlcally meqUlvalent M. What we Figure 3 would prefer IS for our parametenzed collectlOn of Functions CO -close but not C' -close models to satisfy the condition that, whenever w IS sufficiently close to wo, M. must be dynamically eqmvalent to MWQ ThiS property, structural stability, IS probably Imphcltly assumed by most economists engaged m computer modehng of dynamiC economic systems However, as we shall soon see, even the sim plest nonhnear systems can be structurally unstable Thus, structural stablbty cannot be taken for granted To make these Ideas more concrete, let us conSider the meanmg of structural stablhty for discrete one dlmenslOnal dynamlCal systems We first need to clar Ify what could be meant m thiS context by "a small change m the model's structure" Smce these systems are entirely determmed by the functlOn hemg Iterated, It makes sense to mterpret a small change m the system as meamng a small change in'the underlymg functlOn But, to change a functlOn sbghtly really means to mtroduce a new functlOn that !By conventiOn, when r = 0, fl r) = f, that IS, the Oth denvatlve of a IS, m some sense, near the ongInal How, then, can we function IS the functIOn Itself Since CO IS defined as the set of all functIOns wIth conlmuo!ls Oth derivative, en IS Simply the set of all measure the "nearness" of two functlOns? In the the contmuous functIOns ory of structural stablhty, the followmg method has :fJ'he supremum of a set of numbers, denoted "sup," IS the smallest proved effective Suppose f and g are r-tlmes differen upper bound of the set Thus, for example, the supr_e!flum of the tiable functlOns defined on an mterval J Usually, one oven mtervtli (0,1) IS 1 Supremum plays the role of m~xlmum for sets that may not have a largest element The supremum of J. sel assumes also that ~,) and g") are contmuous, so that f with no tillite upper bound IS ::c and g are r-tlmes contmuously dlfferentiable and thus ~From the Gleek hmneo- (Similar) + morphtSHl (form) 9 [-I,ll by hex) X3 IS a homeomorphIsm of thIs A precIse defInItIOn of structural stabIlIty IS now mtel val Wlthm reach Let f be an r-tlmes contmuously dIfferen tiable functIon mappmg an mterval J mto Itself Then, We say f and g are topolpgtcally conjugate If there f IS called C'-6tructurally stable If there eXIsts an E > eXIsts a homeomorphIsm of J such that, for each x m J, osuch that any r-tlmes contInuously dIfferentIable functIon g that maps J mto Itself and satIsfies d,(f.g) < h(f(x)) = g(h(x)) E IS topolOgIcally conjugate to f What thIs condItIOn e"presses IS that, whenever f Whtle verIfYIng structural stablhty can be dIfficult, sends a pomt x to f(x), g sends the pomt correspondmg examples of structural mstablhty ale not hard to find to X (namely hex)) to the pomt correspondmg to f(x) Define f by (namely h(f(x))) Thus, the behaVIOr of g corresponds 10 a contmuous one-to-one manner to the behaVIOr of f5 f(x) = x - x', When two functIOns are topolOgIcally conjugate, each preCIsely rephcates the dynamIcal propertIes of the and, for each E > 0, define g, by other, and the dynamIcal systems they generate may be consldel ed eqUIvalent To Illustrate thIs pomt, sup g,(x) = x - x' + <12 pose f and g are topolOgIcally cOnjugate by means of a homeomorphIsm h Also, suppose f has an'orblt orlg matmg at x and convergIng to p Then Note that, for each r, the functIOns f and g, are r-tlmes contmuously dIfferentiable and map the number hne hlp) = h(I~~!(X~ (here playmg the role. of the mtel val J) mto Itself A SImple computatIOn shows that, for every rand E, d,(f,g,) < E However, exammatlOn of the equatIOns x hm h(fn(x)) - x2 = x and x - X' + ./2 = x ImmedIately estabhshes n-~ that f has only one fixed pomt whtle each g, has two Thus, for no E > 0 can g, be topolOgIcally conjugate to hm h(f(f'-l(x))) f It follows that f cannot be C'-structurally stable n-~ If there IS a moral for agricultural economIsts 10 thIs hm.g(h(f'-l(x))) dISCUSSIOn, It would seem to be that greater emphaSIS n-~ should be placed on confirmmg the structural stablhty of a dynamiC model pnOl to ItS econometnc estunatton In the absence of structural stabIlIty, estImatIOn of a model would only smgle out one of a,number of dynam ICally meqUlvalent approximatIOns It would therefore hm g"(h(x)), serve no clear purpose n_oo An Example of ChaotIc DynamICs that IS, the orbIt of g ongmatmg at hex) converges to hlp) SImIlarly, suppose y IS a perIOdIC pomt of f of To gam a quahtatlve understand109 of what IS mvolved penod m Then In chaotIc, dynamICs, we now examme m detatl the gm(h(y)) gm-l(g(h(y))) class of functIOns F. (fL > 1) defined by I gm-l(h(f(y))) F .(x) = fLx(l-X) Usmg functIOns from thIs class as the law of mQtlOn of \ a dIscrete dynamical system, I shall mvestlgate the I' longrun behaVIOr of-all orbIts, followmg the notatIOn and approach of (6) h(tm(y)) FIrst, ·some baSIC facts (fIg 4) Let p. = (fL-l)/fL hey), Then, 0 < p. < 1, and p. IS a fIxed pomt of F. so that hey) IS a penodlc pomt of g of penod m Another fixed pomt IS 0 SlIIce F .(1) = 0, the orbIt ongmatmg at 1 goes ImmedIately to 0 and .remams "Topology IS the study of those propertleS,a mathematical object there Fmally, It IS easy to show that 'any orbIt of F. retams when It IS contmuously transfonned The tenn "conjugate" ongmatmg at a pomt less than 0 (such as the pOInt x. orlgmates In the Latin com· (together) + Jugum (yoke) and bterally of fig 4) or greater than 1 (such as the pomt Xl of fig med.IlS "JOined or yoked togetner" Here. It IS fand g that are "yoked togelhel" by h 4) dIverges to _00 10 Figure ./I assume, therefore, that I'- > 2 + j'D, and WIll eventually Orbits of FII find that on the set of Imtlal pomts In [O,IJ whose orbIts nevel leave [0,1], F. behaves chaotlcally Let II be th,s bet That IS, let >\ be the set of all x m [0,1] for whICh each term of the 01 bIt of x, IS III [0,1] The first task IS determining the structure of II, whICh WIll be done by ascertammg the structure of the complement of II, the set of those pomts of [0,1] that are not m II For each n = 0, 1, 2, 3, , let A" be the set of all x III [0,1] whose tilSt n "; 1 orbIt terms, x, are m [0,1] but whose next orbIt term, F""'(X), IS not Observe that II consIsts preCIsely of those• pOints of [O,IJ that he In none of the A;s Moreover, the A;s are pall"Wlse dlsjomt Thus, one can Imagine construct mg II through the followmg recursIve process from the mterval [O,IJ, til st remove the subset Ao, next, Next, suppose 1 < I'- < 3 Smce F :(0) = I'- > 1, °IS from what lemams, remove A" and so on In genelal, hyperbohc repelhng On the other hand, smce F:(p.) when Ao, A" ,A" have been removed from [0,1], = 2-1'- and -1 < 2-1'- < 1, p. IS hypel bohc attlactmg A"+I must stIll (by d,sjomtness) he mtact In the One can show that the basm of attractIOn of p. IS pre remammg subset of [0,1] Remove An' and contmue cIsely the open Interval (0,1), any orb,t onglnatlng m thIS process ad mjlntturn When all of the A;s have this Interval (such as at the pOint X 2 of fIg 4) con been removed from [0,1], the subset of [0,1] that verges to p. We have thus determined the longrun remains will be pI eClsely II behaVIOr of all orbIts of F. for all values of I'- In the range 1 < I'- < 3, and we have found nothmg unusual To pIcture what thIS process actually looks hke, we In the dynanucs ansmg m th,s parameter range rely on the fact that a pOint x hes m An + I If and only If F.(x) hes m A" (ThIS property follows from the detim However, as I'- Increases beyond 3, F. undelgoes tlOn of the A;s In mathematICal parlance, A,,+ I IS the vanous quahtatlve changes Among these IS a change pre-Image of A" relative to the functIOn F. ) Now, Ao that occurs as I'- passes 4 the maxImum value of F. IS clearly an open mterval of length less than 1 cen (namely F /112), whIch equals 1'-/4) Increases beyond 1, tered at 112 Thus, removmg Ao flOm [0,1] leaves two and some POints m [0,1] are thus mapped outside of dlsjomt closed mtervals, BI and B' (fIg -5) To con o 0 [O,IJ by F. FOI any such pomt x, we have F .(x) > 1, strllct AI, vlsuahze a copy of Ao on the. y-axIs by and It follows flam a prevIOus I emark that the orb,t of reflectmg Ao around the 45' Ime (fig 6) Then, deter F.(x), mille from the graph of F. what pomts on the x-axIs are mapped by F. Into Ao The set of all such pOints will be Al (fig 7) Note that, smce the graph of F. nses continuously flOm °beyond 1 and then (fruther to that IS, the lIght) descends contmuously from beyond 1 back down to 0, Al conSIsts of two dISJOint open mtervals. F .. (x), F'(x), FJ(x), F"+'(x) each lYing inSIde of (and at a pOSItive dIstance from the r • • '. ' endpOints of) one of the closed mtelVdls B', BZ Thus, must d,verge to _00 Hence, the orbIt of x Itself must removmg both A" and A, from [0,1] leaves 'heh1ndfour dIverge to _00 More generally, any orbIt that ongl dlsjomt closed lIItel vals nates m [0,1] but does not remain In [0,1] must dIverge to-oo A, IS constructed SImIlarly Vlsuahze a copy of Al on the y-axIs and determine the set of all pomtb on the Of partICular mterest IS the parameter I ange I'- > 2 + x-axIs that are mapped by F. mto AI, that set WIll be /5 Although there ale smaller values of I'- fOl whIch A" and It wJ!1 consIst of fOUl dlsjomt open mtervals, chaotIC dynamICs appeal s, Devaney (6) has sho"n each IYlllg inSIde of (and at it posltlve dIstance from the that, when I'- > 2 + j!l, the clemonstratlOn of chaotIC endpolllts of) one of the four closed mtervals left dynamICs can be accomphshed relatIvely sImply We behmd aftel the removals of Ao and Al from [0,1] 11 Figure 5 FIgure 6 Removing the interval Ao from [0,1] Copymg the interval Ao onto the y-axis 45' ~1 BI 112 B' o ~ 0 AO constructIOn of a classIc mathematIcal obJect.called the FIgure 7 Constructing the set A1 Cantor set, a set defined by remOVIng from [0,1] the open "mIddle third" Interval (113, 213), then removIng the open mIddle third Intervals (1/9, 2/9), (7/9, 819) from the two closed Intervals remaInIng, and so on ad "'i,mturn, always. removIng the open mIddle thn'd Interval of each closed Interval remaInIng after the prevIOus removals The Cantor set has long been cele brated In mathematIcs for satIsfyIng the folloWIng two conditIOns (1) Its "length" IS ° (Indeed, by the for mula for the·sum of a geometnc senes, the total length of the dlsJomt mtervals removed from [0,1] In con structmg the Cantor set IS r 1 (113) + 2 (119) + 4 (1127) + ~ 2n-I/3n n=1 x (1/2)L(2I3)' n=l ~ ~ B' o B' o = 1 ModIfIed forms of the Cantor set havmg posItIve length can be constructed by remOVIng shorter Inter vals ) Yet, (2) the Cantor set contaIns as many POInts as all of [0,1] (SpecIfically, It can,be.shown that there· ThIS process can be contInued In general, A, WIll con eXIsts a one· to-one correspondence between the Can SISt of 2' dIsJoInt open Intervals, each IJ:Ing InsIde of, tor set and [0,1] Smce the elements of both sets can and at a posItIve dIstance from the endpoInts.of, one of thus be paIred off, the total number of pomts In each the 2' closed Intervals left behInd after the removals of set must be the same The fact that thIs number hap· ~, ,An_I pens to be Inflmte should not be held agaInst It Infimte sets have sIzes too) Thus, In brIef, I\. IS constructed by remOVIng an open Interval from the mIddle of the closed Interval [0,1], I\. IS known to share property (2) WIth the Cantor set then remOVIng open Intervals from the 'mlddles of the However, It also shares two further propertIes havmg remaInIng closed Intel vals, and so on, ad tnjuutmn more dIrect empmcallmphcatlOns first, I\. IS p.e1ject, This constructIOn bears a stl1kIng resemblance to the whose slgmficance here IS that, as near as deSIred to 12 any pomt of A, one can always find another pomt of A attractor IS called a strange attractor) and m the form That IS, no pomt of A IS Isolated Second, /I. IS totally of the boundary between competIng baSinS of attrac dtsconnected (It containS no open Intervals),b from tIOn (ConSIder an economIc model that allows (lIf whIch It follows that, as near as deSIred to any pOint of ferent l111tlal condItIOns to generate dIfferent A, one can always find a pomt of [0,1] that IS not m /I. eqUIlIbrIa Here, the boundary between baSInS of As a consequence, whenever the dynamIcal system attractIOn correspondmg to dlstmct eqUlhbrIa may be a generated by F. IS ImtIalIzed on /I., ItS longrun fractal exhIbItIng a type of sensItIvIty to the inItIal behaVIOr IS "mfimtely sensItIve" to errors m the mltIal condItIOn noted earher the shghtest movement away condItIOn, Since wlthm any Interval (no matter how from an ImtIaI pomt lYing m one basm of attractIOn small) around an Intended startmg pOint In A, there may move the system to a new basm of attractIOn and eXIst both pOints 111 /I. (whose orbIts, by defImtlOn, thus cause It to evolve toward a new attractor The remaIn In [0,1]) and POInts not In A (whose Olblts eqUlhbrlUm generated by an Imtlal condItIOn lymg on dIverge to _00) Thus, If one attempted to study thIS thIS boundalY would be unpredICtable) In addItIOn, dynamIcal system on a computer, ineVItable rounding fractals can appear m the form of the state space errors In determining the pOints of /I. would make regIOn on whIch chaotIC behaVIOr IS mamfested (see the accurate sImulatIOn over A ImpOSSIble next sectIon) ThIS sensItIvIty of orbIts to the Imtlal condItIOn, whIle suggestwe of true "sensItIve dependence on ImtIaI con SymbolIc DynamICS dItIOns," must be carefully dIstingUIshed from It The sensItIvIty Just descnbed compares orbIts ong111atIng Havmg determmed the structure of the set A of all m /I. WIth orbIts orIgmatIng outSIde A In contrast, pOints whose F.-orbIts IemaIn In [0,1], I now demon true sensItIve dependence refers to a kmd of separat strate the chaotIC behaVIOr of F. on thIS set mg behaVIOr between orbIts ongmatIng nearby wlthm the same set A more formal deflmtlOn wIll be It turns out that If one attempts to analyze the orbIts prOVIded later m thIS artIcle of F. by d11 ect computatIOn the problem soon becomes prohIbitively complex Therefore, one constructs a Irregular sets such as A and the Cantor set have model that abstracts f,om the phenomenon under recently come to be referred to as "fractals" (from the study only ItS essentIal featUl es More specIfically, I LatIn fractus, meanmg "broken" and reflect111g the wIll construct a new dynamICal system that IS dynam dIsconnected character of such sets) Though the sCIen ICally eqUIvalent to the system dete! mined by F. on A tIfIC commumty has not yet arrIved at a consIstent but fal easIer to analyze ThIS apPIoach, used com usage of thIS term, one often sees the follOWing mdl monly m the theory of dynamIcal systems, IS called the VIdual or Jomt cntena exhlbltmg a hIgh degree of Jag method of symbohc d1jlw?n1cS gedness, self-SImIlar (that IS, defined by a recurSIVe process In such a way that any part of the set, when The state space of our model dynamIcal system WIll be magnIfied, looks the same as the entIre set), and hav the set L2 of all sequences of o's and l's We can repre Ing nomnteger dImenSIOn (There are many ways to sent a tYPIcal element of L, as an mfimte vector, extend the usual concept of dImenSIOn (0 for a pOint, 1 for a curve, 2 for a surface, and so on) to more com phcated sets Hausdorff dImenSIOn (13), perhaps the most WIdely used, assIgns to the Cantor set a dImen where St' eIther 0 or 1, IS the lth term of the c;;equence sIOn of In 211n 3, or approxImately 0 63 Some other Note that we are startmg our sequences WIth a "Oth" notIOns of dImenSIOn suggested for applIcatIOn to term rather than a "1st" term That IS, our sequences fractal sets are tnformatton dImenSIOn, correlatwn al e defined on the set of nonnegatIve mtegers rather dImenSIOn, and Lyapunov dImenSIOn (see (J 7)) ) than the set of posItIve mtegers Later, thIS arrange ment Will enable us to assocIate the terms x, F .(x), UntIl relatIvely recently, the Cantor-lIke sets now F2(X), Fol(X), of an orbIt of F. WIth the terms so, s" called fractals were conSIdered exotIc structures S;!.• S3, •of a certam sequence m L2 belongmg solely to the world of pure mathematICs The dIscovery of theIr mtImate connectIOn WIth non An example of an element of L2 IS the vector lInear dynamICs has been stnkmg However, they are now understood to be a typIcal concomItant of non (0, 1, 0, 1, 0, 1, ), hnear dynamIcal systems (See, for example, (9, 12, 17) and the references contamed therem ) They have (that IS, (S0' s" s" S3, ), where s, = 0 If lIS even and been detected In the form of attractors (a fractal s, = 1 If I IS odd) Agam, any sequence of O's and 1's IS allowed as an element of L2 6'fhls c property should seem at least plaUSible m VIew of the method of construcLlOn of!\ It IS m provmg thIS property thdt the The system functIon m our model-the functIon whose assumption that 1.1 > 2 + fi) IS first put to use See (6) Interestmgly. the property Implies that every pOint of A IS on the boundary of A dynamICs on Ll WIll parallel that of F" on A-WIll be the 13 shift aperator, (J, defined at any sequence (so, Sl> S2, S3, topologically conlugate If there eXIsts a homeomor ) In~, by phIsm h between X and Y such that, for each x In X, h(f(x)) = g(h(x)) Observe that (J maps a sequence to a new one whobe As WIth our earllel defimtlOn, topologically conjugate Ith term IS the (I + l)st tel m of the orIginal (Note also functIOns map correspondmg POints to corresponding that the inItial term of the onglnal sequence IS Ignored pOints, exhIbIt the same dynamical propertIes, and In formIng the new one Thus (J, like Fe' IS not invert may be consIdered dynamIcally eqUIvalent Ible ) Symbolically, we may wrIte We now define a homeomorphIsm bet"een A and ~, by ((J(S)), = S,+I, means of whIch F. and (J can be shown to be conJu gate Recall that, under our continUIng assumptIOn for any sequence s In ~, that f.L > 2 + /5, we eal lIer defined Ao as the set of POints In [0,1] whose F.-values lie outsIde [0,11, and Next, we make precIse the meamng of the dynamIcal we define,]' B' and B2 as the ,dISJOint closed Intel vals eqUl valence of (J and F" To do so, we generalize our remaining wh~n A,l I~ removed from [0,1] (see fig 5) earlier notIOn of dynamIcal eqUIvalence, whIch relied To facIlItate our defllllllg a homeomorphIsm, we on the concepts of homeomol phlsm and topologICal rename the Intervals B' and B..! as "I " and "I " o 0 0 " conjugacy First, howe,er, we must Introduce a gen respectIvely (fig 8) eral defimtIOn of "contmuous functIOn" We define a functIOn, S, from A to ~2 as follows Let ... Suppose f IS a functIOn mappIng a set X Into a set Y be an arbItrary pomt In A By defimtlOn, the 01 bIt of x Recall the IntUItive meanIng of contlllUlty as x nevel leaves [0,1], thus, the 01 bIt must lemaIn WIthin approaches x', f(x) approaches f(x ') Suppose each of X the sets 10, I, We assocIate WIth x a sequence, S(:\), of and Y has been assIgned some measure of the "dIs o's and l's whose'lth term (I = 0, 1, 2, 3, ) IS defined tance" between ItS POInts (Just as the sets X and Y by can be qUIte dIfferent, these measures of distance can o If F'(>.) hes In I" be qUIte dIfferent too) Then, the'IntUltlve meanIng of ( Sex) = { • contInUIty becomes as the dIstance (In X) between" ~ , 1 If F'(x) hes In I, and x' applOaches 0, the distance (In Y) between f(x) • and f(x') approaches ° More formally, f IS continuous sex), called the "tmera", of x, IS obVIOusly an element at x' If, for any. > 0, there eXIsts a 5 > °such that, of ~2 Thus, S IS a functIOn mapping A to~, Rather whenever the distance In X between X and x' IS less amazingly, It turns out that S IS, In fact, a homeomOl than 5, the dIstance In Y between f(x) and f(x') IS less phlsm between A and~, MOIeovel, by means of S, F" than. and (J can be shown to be topologically conjugate The The set A comes eqUIpped wIth a natural measure of distance the absolute value of the dIfference between FI{jUfe 8 two POInts As for ~2' we now defIne the dIstance The Intervals 10 and 11 between any of ItS sequences sAnd t to be des, t) = }: Is,-t,j 1=0 21 It IS not dIfficult to prove that, wIth thIs dIstance measure, (J IS contInUOUS on ~2 We already know, of course, that F" IS contmuous on A We are now able to generalize our earlier notIOns of homeomorphIsm, topologIcal conjugacy, and (thus) dynamIcal eqUIvalence to apply to (J and F" Suppose X and Yare any sets each of whICh has been assIgned a dIstance measUle Then, by a hO?neomo'p/u.m between X and Y we mean an Invertible functIOn map 1/2 ~1 ping X onto Y such that both the functIon and Its " mverse are continUOUS Suppose f IS a contmuous func tIOn mapping X Into Itself and g IS a continUOUS func tIOn mapping Y Into Itself Then, we say f and g al e proofs of these facts go beyond the scope of this artICle "0,0", "0,1", "1,0", and "1,1"), then all stnngs of' length (see (6)) 3, and bO on In general, aftel all of the 2" stllngs of length n have been listed, contmue With all strmgb of The conjugacy between F" and (J estabhshes their length n + 1 and beyond Thus dynamical eqUivalence and permits Information gleaned from (J to be apphed to F" To begm to explOIt s* = (0, 1,0, 0, 0, 1, 1,0, 1, 1,0,0 0,0,0, 1, thiS feature, let us determIne how many POInts of perIOd n F" has Now, by dynamical eqUIvalence, " We shall show that the orbit of s' With lespect ·to (T Ib must have exactly the same number of pomts of penod dense In L2 In fact, given any sequence S 111 Ll and dny n as F" However, for a sequence s m L2 to be of penod £ > 0, choose n so that 1/2" < <, and obsel ve that the n With respect to " means that shIftmg s n times pro stnng So, , s" conslstmg of the fil,t n + 1 termb of s duces s agam, that IS, s, co = s, for each I It follows must appear somewhele m s' By the defimtlOn of (J, that·s must be a repeatmg sequence of the form there must therefore eXist a k such that sn· ), There are precisely 2" ways of arrangIng o's and I's to that IS, such that the filst n + 1 telms of (Jk(S') and s form a finite stl mg so, , Sn_lJ hence, (J' must have agree As before, the formula defilllng OUI chstance exactly Z" penodlC pomts of penod n The same holds, meaSlll e then Implies that then, for F" Usmg symbohc dynamiCs, we have thus established that, for example, F" has 64 penodic d«Jk(S'),S) ::::. 112" < < pomts of penod 6, or ZII7 penodlc pomts of penod 117 It follows that the orbit of s· IS dense III L2 By the A subset X' of a set X endowed With a distance meas dynanllcal eqUIvalence between F" dnd (T, we tan thus Ule IS called dense m X If, for any pomt x m X, one can conclude that S-I(~*), the pomt III 1\ COI'l espol1(hng to find some pOInt fi om X' as close to x as deSired The s· undel the mverse of the Itmeral y homeomOJ phlsm set of all periodiC pomts of (J IS dense In L, In fact, S, has a dense F .-01 bit m 1\ FOi breVity, put '(i = gIven any sequence s m L2 and any £ > °(no matter S-I(S*) how'small), choose n so that liZ" < £ Define a repeat Ing, hence penodlC, sequence s' m L2 by The fact that the OJ bit of ,,* IS dense m A Imphes that, for any" m 1\ and any £ > 0, thel e IS an orbit pomt Fm(x*) lymg In the mtelVal ("-£,'+£) Howevel, a Simple• argument shows that the orbit of Fm(x*), (Note that s' merely repeats the first n + 1 terms of • s ) Flom the definitIOn of our distance measure d( , ) on L" It follOlvs that , , must also be dense III A Thus, al bltl allly neal any des,s') = L 0 + L 1=0 l=n.J..] pomt of 1\ there 0l1gInates a dense orbit Such an 01 bit , would appeal ell'atlc and essentIally random, for It < L liZ' would endlessly "dance" alOund A, VISltlllg and I=n+ 1 revlsltmg the vlclmty of each pomt of A mflmtely many times 112" Recent findmgs by both economists (4) and mathemati <£ CIans (14) have shed adclitlOnal hght on tlus seemmgly landom character of chaotiC 01 bits Thele IS now e'l Thus, the set of perIOdiC POInts of (J IS dense In L2 dence that chaotiC behavlOl IS llldeed often legit Smce dynamical eqUIvalence I~ known to encompass Imately stochastiC m the sense that chaotiC OJ bits may denseness propertIes \\ e can conclude that the set of be realizatIOns of a stochastiC process defined on the pellodlc pomts of F. IS dense m A state space In thiS SituatIOn, the IOl1gI'Un behavlOl of an econOn1lC variable may be deSCrIbed by an endoge The precedmg lesult tells us that cyclic 01 bits can be nously generated 10l1gIun probablhty distributIOn Day found ongmatlllg aI bltranly near any pomt of A We and Shafer (4) proved the e"lstence of such 15 Fmally, we defme the hallmark of chaos-sensItIve economic theory to beheve that a given dynamiC dependence on InItial conditIOns-and verify that F" economic process IS lInear, the process must be Viewed exhibits this property Let X be any set endowed WIth as at least potentially lIable to the type of chaotiC a distance measure and f a functIOn mappmg X to behaVIOr descnbed here Itself We say f exhibits senSttwe dependence on tm ttal cond,tIOns If there eXIsts a B > °with the follow Chaos theory suggests that the long-range predICtIOn mg property for any x m X and any. > 0, there IS a of nonlInear economic processes may be subject to the pomt x' m X ",Ithm a distance of. from x such that, same baSIC mathematical hmltatlOns as long-range for some n, the distance between fn(x) and fn(,,') weather predictIOn In both cases, future behaVIOr exceeds B HeunstlcaUy, sensItIve dependence means may appear mdependent of the mltlal conditIOns that that there IS a constant B > °such that, arbitrarIly produced It close to any pomt of X, one can find another pomt of X whose orbit eventually diverges (even If only tem It IS difficult to study thiS subject WIthout expenenc poranly) from that of the given pomt by more than 8 mg a certam humilIty concernIng our abilIty to control (Lyapunov exponents are sometimes used as a prag nonlmear econormc processes through polIcy mterven matic measure of this divergence (17) ) Although this tlOn Nonlmear systems can behave m a countenntUl discrepancy m orblts'ls reqUIred only to exceed 8, not tlve manner The conditIOns under which we can to be arbitrarIly large m absolute terms, It should be properly use mathematICal models to predict the long noted that the ratIO of this discrepancy to the distance run ImplIcatIOns of polIcy actIOns need to be clanfied between x and x' wIll become arbltranly large when x' IS chosen arbltranly close to x It IS m thiS sense that The dlscovelles of recent years might seem to have sensitive dependence Imphes unpredictable longrun revealed mtnnslc mathematIcal lImits to economic pre behaVIOr for orbits ongmatmg arbitrarIly near one dICtIOn Yet, a deepe! understandmg of the lImitatIOns another of longrun pomt predICtIOn should ultimately enhance, not diminIsh, the accuracy and credibilIty of the mfor To show that F" exhibits senSitIVe dependence on InI matlOn we prOVide A further enhancement may tIal conditIOns, let B be any positive number less than derIve from the replacement, m certam cases, of long the distance between the mtervals 10 and I" that IS, run pomt forecasts by forecasts based, m part, on less than the length of Ao (see fig 8) Choose any x m endogenous longrun probabilIty clIstnbutlOns A and any. > ° Smce, as noted earher, no pOint of A IS Isolated, there must eXist a pomt x' m A dlstmct We have attempted m thiS article to sketch some of from x whose distance from x IS less than E However, the major themes of contemporary nonhnear smce the Itmerary mappmg, S, IS mvertIble, dlstmct dynamiCs Many tOPICS, however, had to be omitted pomts m !I. must have distInct Itmerary sequences (For example, we have not discussed how the non Thus, for some n, hnear dynamiCs lIterature might con trIbute to unprovements m shortrun forecastmg models such as models of stock mal ket behaVIOr An anonymous ref eree suggests that. a small pel centage reductIon of that IS, either Fn(x) IS m 10 and Fn(x') IS m I I or vice shortrun forecast errors may be pOSSible Such a versa It follows at once that thl distance between reductIOn would be of particular mterest to arbitrage Fn(x) and Fn(x') exceeds B, which proves the result tJ aders ) Recommended further background readmg " " would mclude (m order, 9, 6, 17, 12, 7, 15) For a sam WhIle the exammatlOn of chaotiC behaVIOr over a plIng of the nascent economics lIterature on chaos, see fractal set such as !I. can reveal Important aspects of (1, 2, 3,4, 5, 8, 11, 16, 18) the subject, chaos should not be Viewed as a phe nomenon that appears only on unusual sets One can References show, for example, that the tiInctlOn F, given by F,(x) 1 Benhablb, Jess, and Rlcha!d H Day "A Charac = 4x(l-x) IS chaotiC on the entire mterval [0,1] Nor IS tenzatlOn of ErratiC Dynarmcs m the Overlappmg the e"lstence of chaos overly sensItIve to functIOnal GeneratIOns Model," Journal of Economtc form The chaotIc dynamiCS we have described for F" Dynanl1cs and Control Vol 4, 1982, pp 37-55 wIll also be exhibited by essentially any hill-shaped functIOn With suffiCiently large slope 2 Brock, W A , and A G MallIans DIfferentIal Conclusions Equatwns, Stabtltty and Chaos 111 DynamIC EconomICs Amsterdam ElseVier SCience Pub Recent findmgs m the field of nonlmear dynamical sys hshers B V , 1989 tems warrant a rethInkmg of traditIOnal attitudes toward economiC dynamiCs It IS now known that en a 3 Brock, W A , and C L Sayers "Is the Busmess tIc longrun behaVIOr and varIOus fOrIns of sensitivity to Cycle Characterized by Determmlstlc Chaos?" mltlal conditIOns can arise m even the Simplest non Journal oj Monetarlf Econormcs Vol 22, 1988, hnear models Unless there are sound reasons In pp 71-90 16 4 Day, Richard H , and Wayne Shafer "Ergodic 13 Morgan, Frank Geometnc Measure Theory San FluctuatIOns m DeternumstlC EconoJnlc Models," Diego Acaderrnc Press, 1988 Journal oj Economw Behamor and Orgamzatwn Vol 8, 1987, pp 339-61 14 Ornstem, D S , and B Weiss "StatistICal Proper ties of ChaotiC Systems," Bullettn oJ the Amencan 5 Dendrmos, Dlmltnos S , and MIChael Soms Chaos Mathematical Society Vol 24, No 1, Jan 1991, and Socw-Spattal Dynarmcs New York pp 11-116 Spnnger-Verlag, 1990 15 Parker, Thomas S , and Leon 0 Chua Practical 6 Devaney, Robert L An Introductwn to Chaotic Numertcal Algortthms Jar ChaotiC Systems New Dynamwal Systems New York Addison-Wesley York Spnnger-Verlag, 1989 Publishmg Co , 1987 16 Ramsey, James B , Chera L Sayers, and Philip 7 Edgar, Gerald A Measure, Topology, and Fractal Rothman "The Statistical Properties of Dimen Geometry New York Spnnger-Verlag, 1990 sIOn CalculatIOns U smg Small Data Sets Some EconoJnlc ApplicatIOns," Internatwnal Economic 8 Frank, Murray, and Thanasls Stengos "Chaotic Remew Vol 31, No 4, Nov 1990, pp 991-1,020 Dynamics m Economic Time-Series," Journal of EconomtC Surveys Vol 2, No 2, 1988, pp 17 Rasband, SNell Chaot!c Dynamlcs oJNonhnear 103-33 Systems New York John Wiley and Sons, 1990 9 GleICk, James Chaos New York Pengum Books, 18 Savlt, Robert ''When Random IS Not Random An 1987 IntroductIOn to Chaos m Market Pnces," The Journal oj Futures Markets Vol 8, No 3, 1988, 10 LI, Tlen-Ylen, and James A Yorke "Period Three pp 271-89 ImpiJes Chaos," Amertcan Mathematteal Monthly Vol 82, Dec 1975, pp 985-92 19 WeiSS, Michael D "Chaos, EconomICs, and Risk," QuantlJytng Long Run Agncultural Risks and 11 Mlrowski, Philip "From Mandelbrot to Chaos m Evaluattng Farmer Responses to Risk Proceed EconoJnlc Theory," Southern Economic Journal mgs of a senunar sponsored by Southern RegIOnal Vol 57, No 2, Oct 1990, pp 289-307 Project S-232, San AntOniO, Mar 17-21, 1991 Department of Agncultural EconoJnlcs and Rural 12 Moon, FrancIs C Chaotle VibratIOns New York SOCIOlogy, Umv of Arkansas Forthcommg John Wiley and Sons, 1987 17