Unversal Behavior in Nonlinear Systems

8/3/2019 Unversal Behavior in Nonlinear Systems

1/24

Universaly Mitchel l J. Fe igenbau rnBehaviorNonlinear Systems

Universal numbers,5 = 4.6692016 ...

anda = 2.502907875 ...,

determine quantitativelythe transition from

smooth to turbulent orerratic behavior

for a large class ofnonlinear systems.

here exist in nature processes thatcan be described as complex orchaotic and processes that aresimple or orderly. Technology attemptsto create devices of the simple variety:an idea is to be implemented, and

various parts executing orderly motionsare assembled. For example, cars, air-planes, radios, and clocks are all con-structed from a variety of elementaryparts each of which, ideally, implementsone ordered aspect of the device .Technology also tries to control orminimize the impact of seemingly disor-dered processes, such as the complexweather patterns of the atmosphere, themyriad whorls of turmoil in a turbulentfluid, the erratic noise in an electronicsignal, and other such phenomena. It isthe complex phenomena that interest ushere.When a signal is noisy, its behaviorfrom moment to moment is irregular andhas no simple pattern of prediction.

given up asking for a precise cauprediction), s o tha t the goal of a descrt i on i s t o d e t e r m i n e w h a t tprobabilities are, and from this informtion to determine various behaviorsinterest-for examp le, how air tbulence modifies the drag on an a irplaW e know th at perfectly definite cauand simple rules can have statistical random) behaviors . Thus, modern cop u t e r s p o s s e ss " r a n d o m n u m bgenerators" that provide the statistiingredient in a simulation of an erraprocess. However, this generator dnothing more than shift the decimpoin t in a ra t ional number whrepeating block is suitably long. Accdingly, it is possible to predict what nth generated number will be. Yet, ilist of successive generated numbthere is such a seeming lack of order tall statistical tests will confer upon numbers a pedigree of randomneTechnically, the term "pseudorandoHow ever, if we ana lyze a sufficiently is used to indicate this natu re. O ne nlong record of the signal, we may find may ask whether the various comp

that signal amplitudes occu r within processes of natu re themselves might narrow ranges a definite fraction of thetime. Analysis of another record of thesignal may reveal the same fraction. Inthis case, the noise can be given astatistical description. This means thatwhile it is impossible to say what am-plitude will appear next in succession, itis possible to estimate the probability orlikelihood that the signal will attain somespecified range of values. Indeed, for the

be merely pseudorandom, with the import of randomness, which istestable, a historic but misleading ccept. Indeed our purpose here is to plore this possibility. What will praltogether remarkable is that some vsimple schemes to produce erratic nubers behave identically to some of erratic aspects of natural phenomeMore specifically, there is now coglast hundred years disorderly processes evidence that the problem of how a flhave been taken to be statistical (one has changes over from sm ooth to turbul

LOS ALAMOS SCIENCE/Summer


2/24

flow can be solved through its relation tothe simple scheme described in this arti-cle. Other natural problems that can betreated in the same way are the behaviorof a population from generation togeneration and the noisiness of a largevariety of mechanical, electrical, andchemical oscillators. Also, there is nowevidence that var ious Hamil toniansystems-those subscribing to classicalm e c h a n i c s , s u c h a s t h e s o l a rsystem-can com e unde r this discipline.T h e f e a t u r e c o m m o n t o t h e s ephenomena is that, as some externalparameter (temperature, for example) isvaried, the behavior of the systemchanges from simple to erratic. Moreprecisely, for some range of parametervalues, the system exhibits an orderlyperiodic behavior; that is, the system'sbehavior reproduces itself every periodof time T. Beyond this range, thebehavior fails to reproduce itself after Tseconds; it almost does so, but in fact itrequires tw o intervals of T to repeat it-self. That is, the period has doubled to2T. This new periodicity remains oversome range of parameter values untilanother critical parameter value isreached after which the behavior almostreproduces itself after 2T, but in fact, itnow requires 4T for reproduction. Thisprocess of successive period doublingrecurs continually (with the range ofparameter values for which the period is2"T becoming successively smaller as nincreases) until, at a certain value of theparameter, it has doubled ad infiniturn,so that the behavior is no longerperiodic. Period doubling is then acharacteristic route for a system tofollow as it changes over from simpleperiodic to complex aperiodic motion.All the phenom ena m entioned above ex-hibit period doubling. In the limit ofaperiodic behavior, there is a unique andhence universal solution common to allsystems undergoing period doubling.This fact implies remarkable conse-quences. For a given system, if we

denote by A the value of the parameterat which its period doubles for the nthtime, we find tha t the values A convergeto A (at which the motion is aperiodic)geometrically for large n. This meansthat

for a fixed value of 6 (the rate of onset ofcomplex behavior) as n becomes large.Put differently, if we define

(quickly) approaches the constantvalue 6. (Typically, 6 will agree with 6to several significant figures after just afew period doublings.) What is quiteremarkab le (beyond the fact that there isalways a geometric convergence) is that,for all systems undergoing this perioddoubling, the value of 5 is predeterminedat the universal value

Thus, this definite number must appearas a natural rate in oscillators, popula-tions, fluids, and all systems exhibiting aperiod-doubling route to turbulence! Infact, most measurable properties of an ysuch system in this aperiodic limit nowcan be determined, in a way that essen-tially bypasses the details of the equa-tions governing each specific systembecause the theory of this behavior isuniversal over such details. That is, solong as a system possesses certainqualitative properties that enable it toundergo this route to complexity, itsquantitative properties are determined.(This result is analogous t o the results oft h e m o d e r n t h e o r y o f c r i t i c a lphenomena, where a few qualitativeproperties of the system undergoing aphase transition, notably the dimen-sionality, determine universal critical ex-

ponents. Indeed at a formal level the theories are identical in that they fixed-point theories, and the numbefor example, can be viewed as a critexponent.) Accordingly, it is sufficienstudy the simplest system exhibiting phenomenon to comprehend the gencase.Functional Iteration

A random number generator is anample of a simple iteration schem e has complex behavior. Such a schgenerates the next pseudorandom nber by a definite transformation uponpresent pseudorandom number. In owords, a certain function is reevaluasuccessively to produce a sequencesuch numbers. Thus, if f is the functand x0 is a st artin g numb er (or "seedthen xo, xp ..., xn, ..., where

is the sequence of generated pseudord o m n u m b e r s . T h a t i s, t h e y generated by functional iteration.nth element in the sequence is

where n is the total number of appltions of f. [f"(x) is not the nth powef(x); it is the nth iterate of f.1 A propeof iterates worthy of mention isf"(frn(x))= frn(fn(x))= frn+"(x)since each expression is simply m +applications of f. It is understood th

LOS ALAMOS SCIENCEISummer 1980


3/24

It is also useful to have a symbol, o ,for functional iteration (or composition),so that

Now fn in Eq. (5) is itself a definite andcomputable function, so that x as afunction of x0 is known in principle.If the function f is linear as, for exam-ple,

for some cons tant a, it is easy to see that

so that, for this f,

is the solution of the recurrence relationdefined in Eq. (4),

Should I a I < 1, then x geometrica llyconverges to zero at the rate I /a. Thisexample is special in that the linearity off allows for the explicit computation offn .We must choose a nonlinear f togenerate a pseudorandom sequence ofnumbers. If we ch oose for our nonlinearf

then it turns o ut that fn is a polynominalin x of order 2". This polynomial rapidlybecomes unmanagea bly large; moreover,its coefficients are polynomials in a oforder up to 2 " ' and become equally dif-ficult to com pute . Th us even if x0= 0, xnis a polynomial in a of order 2"'. Thesepolynomials are nontrivial as can be sur-mised from the fact that for certain

values of a, the sequence of numbersgenerated for almo st all startin g points inthe range (a - a2,a) possess all themathematical properties of a randomsequence. T o illustrate this, the figure onthe cover depicts the iterates of a similarsystem in two dimensions:

Analogous to Eq. (4), a starting coor-dina te pair (xo,yo) is used in E q. (14) todetermine the next coordinate (x,,y,).Equation (14) is reapplied to determine(x2,y2)and so on. F or some initial points,all iterates lie along a definite ellipticcurve, whereas for others the iterates aredistributed "randomly" over a certainregion. It should be obvious that no ex-plicit formula will account for the vastlyrich behavior shown in the figure. Thatis, while the iteration scheme of Eq. (14)is trivial to specify, its nth iterate as afunction of (xo,yo) s unavailable. Pu t dif-ferently, applying the s implest ofnonlinear iteration schemes to itself suf-ficiently many times can create vastlycomplex behav ior. Yet, precisely becausethe sam e operation is reapplied, it is con-ceivable that only a select few self-consistent patterns might emerge wherethe consistency is determined by the keynotion of iteration and not by the par-ticular function performing the iterates.These self-consistent patterns do occurin the limit of infinite period doublingand in a well-defined intricate organiza-tion that can be determined a prioriamidst the immense complexity depictedin the cover figure.T h e F i x e d - P o i n t B e h a v i o r o fFunctional Iterations

Let us now make a direct onslaughtagainst Eq. (13) to see wha t it possesses.We want to know the behavior of thesystem after many iterations. As we

already know, high iterates of f rapidbecome very complicated. One way tgrowth can be prevented is to have tfirst itera te of x0 be precisely xO itsGenerally, this is impossible. Rather tcondition determines possib le x{s. Sua self-reproducing point is called a fixpoint of f. The sequence of iteratesthen x0, x0, xO, .. so that the behaviorstatic, or if viewed as periodic, it hperiod 1.

It is elementary t o determine the fixpoints of Eq. (13). For future covenience we shall use a modified formEq. (13) obtained by a translation inand some redefinitions:

s o tha t a s A, is varied, x = 0 is alwayfixed point. Indeed, the fixed-point codition for Eq. (15),

gives as the two fixed points

The maxim um value of f(x) in Eq. (is attained at x = 1/- and is equal toAlso, for 'k > 0 and x in the inter(0,1), f(x) is alwa ys positive. Thu s, if 7anywhere in the range [0,1], then aiterate of any x in (0,l) is also always(0,l). Accordingly , in all that follows shall consider only values of x and 'king between 0 and 1. By Eq. (16) for 07. < \/.,only x* = 0 is within ranwhereas for 1/


4/24

Where the lines intersect we have x = y= f(x), so that the intersections areprecisely the fixed points. Now, if wechoose an x0 and plot it on the x-axis, theordinate of f(x) at x0 is x,. T o obtain x,,we must transfer x, to the x-axis beforereapplying f. Reflection through thestraight l ine y = x accomplishesprecisely this operation. Altogether, toiterate an initial x0 successively,1. move vertically to the graph of fix),2. move horizontally to the graph of y =

x, and3. repeat steps 1, 2, etc.Figure 1 depicts this process for t = x.The two fixed points are circled, and thefirst several iterates of an arbitrarilychosen point x0. are shown. What shouldbe obvious is that if we star t from any x0in (0,l ) (x = 0 and x = 1 excluded), uponcontinued iteration x,, will converge tothe fixed point at x = '/,.N o matter howclose x0 is to the fixed point at x = 0, theiterates diverge away from it. Such afixed point is termed unstable. Alter-natively, for almost all x0 near enough tox = 1/ [in this case, all xO n (0,1)], theiterates converge towards x = l/,. Such afixed point is termed stable or is referredto as an attractor of period 1.Now, if we don't care about the tran-sient behavior of the iterates of xo, butonly about some regular behavior thatwill emerge eventually, then knowledgeof the stab le fixed point at x = '/ satisfiesour concern for the eventual behavior of'the iterates. In this restricted sense ofeventual behavior, the existence of an at-tractor determines the solution indepen-dently of the initial condition x0 providedth at x0 is within the basin of attraction ofthe attractor; that is, that it is attracted.The attractor satisfies Eq. (16), which isexplicitly independent of x0. This condi-tion is the basic theme of universalbehavior: if an a ttractor exists, the even-tual behavior is independent of thestarting point.

Fig. 1 . Iterates of x0at X = 0.5.

What makes x = 0 unstable, but x =stable? The reader should be able toconvince himself that x = 0 is unstablebecause the slope of f(x) at x = 0 isgreater than 1. Indeed, if x* is a fixedpoint of f and the derivative of f at x*,fr(x*), is smaller than 1 in ab solute value,then x* is stable. If l f'(x*)l is greaterthan 1, then x* is unstable. A lso, onlystable fixed points can account for theeventual behavior of the iterates of an ar-bitrary point.We now must ask, "For what valuesof 'k are the fixed points attracting?" ByEq . (15), fr(x)= 4X(1 -* 2x) so that

an d

For 0 < t. < -/, nly x* = 0 is stable. Att. = '/,x* = 0 and f'(x*)= 1. For < t.< 3 / , x* is unstable and x* is stable,while at >. = '/., f ' ( ~ * ~ )-1 and x"1sohas become unstable. Thus, for 0 <


5/24

Period 2 from the Fixed Point

What happens to the system when A sin the range % < A < 1, where there areno attracting fixed points? We will seethat as ?L increases slightly beyond A =%, f undergoes period doubling. That is,instead of having a stable cycle of period1 corresponding to one fixed point, thesystem has a stable cycle of period 2;that is, the cycle contains two points.Since these two points are fixed points ofthe function f 2 f applied twice) and sincestability is determined by the slope of afunction at itsfixed points, we must nowfocus on f2. First, we examin e a gr aph off 2 a t A just below x. Figures 2a and bshow f an d f2, respectively, at A = 0.7.T o understand Fig. 2b, observe firstthat, since f is symmetric about its max-imum at x = x, 2 is also symmetricabout x = x. lso, f 2 must have a fixedpoint whenever f does because thesecond iterate of a fixed point is still thatsame point. The main ingredient thatdetermines the period-doubling behaviorof f as A increases is the relationship ofthe slope of f2 to the slope of f. Thisrelationship is a consequence of thechain rule. By definition

where

W e leave it to the reader to verify by thechain rule that

and

an elementary result that determinesperiod doubling. If we start at a fixedpoint of f and apply Eq. (20) to xo= x*,so that x2 = x, = x*, then Fig. 2. = 0.7. x* is th e stable fmed point. The extrema o f f are located in (a

(22) constructing the inverse iterates of x =0.5.f2I(x*)= fl(x*)fl(x*)= [fl(x*)]8 LOS ALAMOS SCIENCE/Summer


6/24

Since at = 0.7, lP(x*) I < 1, it follfrom Eq. (22) that

Also, if we start a t the extremum offthat xo = % and P(xo) = 0, it follfrom Eq. (21) that

for all n. In particular, f2 is extreme (a minimum) at x. Also, by Eq. (201will be extreme (an d a m aximum ) atxo that will iterate under f t o x = x , sthen xl = % and f'(xl) = 0. The se poith e inverses of x =%, are found by govertically down along x = x to yand then horizontally t o y =(Reverse the arrows in Fig. 1, and Fig, 2a.) Since f has a maximum, tha re tw o horizontal intersections hence, the two maxima of Fig. 2b.ability o f f to have complex behaviorprecisely the consequence of its douvalued inverse, which is in turn a reftion of its possession of an extremummonotone f , one that always increaalways has simple behaviors, whethenot the behaviors are easy to computelinear f is always monotone. The f'scare about always fold over and sostrongly nonlinear. This folding nlinearity gives rise to universality. Juslinearity in any system implies a defimethod of solution, folding nonlineain any system also implies a defimethod of solution. In fact folding nlinearity in the aperiodic limit of pedoubling in any system is solvable, many systems, such as various coupnonlinear differential equations, posthis nonlinearity.

To re turn to F ig . 2b , as A -+ %the maximum value of f increases toff(x*) -+- and f2'(x*) --+ +1. A s Acreases beyond %, I P(x*)l > 1 f2'(x*) > 1, so that f2 mus t developnew fixed points beyond those of is, f 2 will cros s y = x a t two more po~ i g .. A =0.75. (a) depicts the slow convergence to thefme dp0 int.f osc date s about This transition is depicted in Figsthe fa e d point. an d b for f and f2, respectively, at

LOS ALAMO S SCIENCEISummer 1980


7/24

0.75, and similarly in Figs. 4a and b at k= 0.785. (Observe the exceptionally slowconvergence to x* at X = 0.75, whereiterates approach the fixed point notgeometrically, but rather with deviationsfrom x* inversely proportional to thesquar e root of the nu mber o f iterations.)Since x; and x the new fixed points off2, are not fixed points of f, it must bethat f sends one into the other:

and

Such a pa i r of points, termed a 2-cycle, isdepicted by the limiting unwinding cir-culating square in Fig. 4a. Observe inFig. 4 b tha t the slope of f2 1s in excess of1 at the fixed point of f an d so is an un-stable fixed point of f2 ,while the two newfixed points have slopes smaller than 1,and so are stable; that is, every twoiterates of f will have a point attractedtoward x: if it is suficiently close to x;or towar d x; if it is suficien tly close tox;. This means that the sequence underf,

eventually becomes arbitrarily close tothe sequencex;, x;, x;, x;, ... ,

Fig. 4 . = 0.785. (a ) shows the outward spiralling to a stable 2-cycle. m e elementsso that this is a stable 2-cycle, or an at- the 2-cycle, x; and x;, are located as fmed points in 0).10 LOS ALAMOS SCIENCEiSummer 1


8/24

tractor of period 2 . Thus, we have obsved for Eq. (15) the first period d oublias the parameter k has increased.There is a point of paramount imptance to be observed; namely, f2 has tsam e slope at x; and a t x;. This pointa direct consequence of Eq. (201, sincexo=x then x l = x a nd vic e v er sa ,that the product of the slopes is tsam e. M ore gene rally, if x;, x ...,xan n-cycle so that

and

then each is a fixed point off" with idtical slopes:

an d

Fig. 5 . 1=1, .A superstable 2-cycle.xy and x; are at extremu off 2.LOS ALAMOS SCIENCEISummer 1980

From this observation will follow peridoubling ad infiniturn.A s k i s i n c r e a s e d f u r t h e r , tminimum a t x = will dro p as the sloof f 2 through the fixed point of f creases. At some value of k, denoted k,, x = will become a fixed point ofSimultaneously, the right-hand mimum will also become a fixed pointf2. [By Eq. (261, both elements of thecycle have slope 0.1 Figures 5a anddepict the situation that occur s at k =


9/24

Period Doubling Ad I@nitumWe are now close to the end of thisstory. As we increase k further, theminimum drops still lower, so that bothx: and x; have negative slopes. At someparameter value, denoted by A2, theslope at both x t and x; becomes equal to-1. Thus at A2 the same situation hasdeveloped for f 2 as developed for f at Al

= x . This transitional case is depicted inFigs. 6a an d b. Accordingly, just a s thefixed point of f at Al issued into being a2-cycle, so too does each fmed point of f2at A2 create a 2-cycle, which in turn is a4-cycle of f. That is, we have now en-countered the second period doubling.The manner in which we were able tofollow the creation of the 2-cycle at A,was to anticipate the presence of period2, and so to consider f2, which wouldresolve the cycle into a pair of fixedpoints. S imilarly, to resolve period 4 intofixed points we now should consider f".Beyond being the fourth iterate off, Eq.(8) tells us that f" can be computed fromf2 :

From this point, we can aband on f itself,and take f2 as the "fundamental" func-tion. Then, just as f 2 was constructed byiterating f with itself we now iterate f2with itself. The manner in which f2reveals itself a s being an iterate of f is theslope equality at the fixed points of f2,which we saw imposed by the chain rule.Since the operation of the chain rule is"automatic," we actually needed to con-sider only the fixed point of f2 neare st tox = x; he behavior of the other fixedpoint is slaved to it. Thus, at the level off", we again need to focus on only thefixed point of P nearest to x = K: heother three fried points are similarlyslaved to it. Thus, a recursive schemehas been unearthed. We now increase kto k2, so tha t the fixed point of f" nearest Fig*6. ?L = A2 .x; and x; in @) have the same slow convergence as the B e d p i nto x = x s again at x = with slope 0. Fig. 3a.12 LOS ALAMOS SCIENCEISummer


10/24

there is a definite operation that, acting on functions, creates functions;particular, the op eration acting on f2"A or better, f2" at A,,,) will determif2" at \ ,.Also, since we need to ke

2%track of f only in the interval includithe fixed point of f2" closest to x = '/,asince this interval becomes increasingsmall as A, increases, the part of f thgenerates this region is also the restrtion of f to an increasingly small intervabout x = '/. (Actually, slopes of f points farther away also matter, bthese m erely set a "scale," which will eliminated by a rescaling.) The behaviof f away from x = '/, is immaterial the period-doubling behavior, and in tlimit of large n only the nature offmaximum can matter. This means thin the infinite period-doubling limit, Fig. 7 . A, = iy A superst& 4-cycle. The region within the dashed square in (a) functions with a quadra tic extremum wshould be compared with all of Fig. 5a . have identical behavior. [? I ( ' / , ) # 0 is t

Figures 7a and b depict this situation ff2 and f4, respectively. Wh en A, increasfurther, the maximum of f" at x = /,nomoves up, developing a fixed point winegative slope. Finally, at A, when tslope of this fixed point (as well as tother three) is again - , each fixed poiwill split into a pair giving rise to ancycle, which is now stable. Again, f8=o f4, and f" can be viewed as fundametal. We define A,, so that x = '/, again ifixed point, this time of f8. The n at A4 tslopes are - , and another period douing occurs. Always,

Provided that a constraint on the ranof A, does not prevent it from decreasithe slope at the appropriate fixed popast -1, this doubling must recurinflnitum.Basically, the mechan ism th at f2" usto period doub le at An +, is the s ammechan ism that f2"+l will use to doubat An+;. Th e functio n f2"' is co nst ruc tfrom f2" by Eq . (27), and similarly f2will be constructe d fro m f2"'l. Th u

LOS ALAMOS SCIENCE/Summer 1980


11/24

generic circumstance.] Therefore, theoperation on functions will have a stabhfixed point in the space of functionswhich will be the common universal limiiof high iterates of any specific functionTo determine this universal limit wtmust enlarge our scope vastly, so thaithe role of the star ting poin t, x0, will beplayed by an arbitrary function; the attra ding fixed point will become a universal function obeying an equation implicating only itself. The role of the function in the equation x0= f(xo) now musibe played by an operation that yields 2new function when it is performed upora function. In fact, the heart of thiioperation is the functional compositiorof Eq. (27). If we can determine the exact operator and actually can solve it:fixed-point problem, we shall understancwhy a special number, such as 6 of Eq(3), has emerged independently of thespecific syste m (the starting function) wchave considered.The Universal Limit of High Iterates

In this section we sketch the solutiorto the fixed-point problem. In Fig. 7a, 2dashed square encloses the part of f2 halwe must focus on for all further periocdoublings. This square should be cornpared with the unit square that cornprises all of Fig. 5a. If the Fig. 7a squarcis reflected through x = %, y = '/, ancthen magnified so that the circulatiorsquares of Figs. 4a and 5a are of equasize, we will have in each square a pieccof a function that has the same kind o;maximum at x = '/, and falls to zero aithe right-hand lower corner of the circulation square. Just as f produced thiisecond curve of f 2 in the square as I. increased from I., to K2, so too will f 'produce another curve, which will besimilar to the other two when it has beermagnified suitably and reflected twiceFigure 8 shows this superposition for thtfirstfive such functions; at the resolutiorof the figure, observe that the last three

A DISCOVERY

The inspiration for the universality theory came from two sources. Fi19 7 1 N. Metropolis, M. Stein, and P. Stein (all in the L A SL T heoreticasion) discovered a curious property of iterations: as a pa rame ter is variebehavior of iterates varies in a fashion independent of the particular fuiterated. In particular for a large class of functions, if at some value param eter a certain cycle is stable, then a s the parameter increases, th e creplaced successively by cycles of doubled periods. This period doublintinues until an infinite period, and hence erratic behavior, is attained.Second, during the early 1970s' a scheme of mathematics called dynasystem theory was popularized, largely by D. Ruelle, with the notion"strange attractor." The underlying questions addressed were (1) how cpurely ca usal equation (for example, the Navier-Stokes equ ations th at defluid flow) come to demonstrate highly erratic or statistical properties ahow cou ld these statistical properties be com puted. This line of thou ght mwith the iteration ideas, and the limiting infinite "cycles" of iteration s ycame to be viewed as a possible means to comprehend turbulence. Indbecame inspired to study the iterates of functions by a talk on s uch m attS. Smale, one of the creators of dynamical system theo ry, at Aspe n in themer of 1975.My first effort at understanding this problem was through the coanalytic properties of the generating function of the iterates of the quam ap

This study clarified the m echanism of period doubling and led to a ra thferent kind of equation to determine the values of X at which the periodbling occurs. The new equations were intractable, although approximatetions seemed possible. Accordingly, when I returned from Aspnumerically determined some parameter values with an eye toward discesome patterns. A t this time I had never used a large computer-in fac t mcomputing power resided in a programmable pocket calculator. Now,machines are very slow. A particular parameter value is obtained itera(by Newton's meth od) with each step of iteration requiring 2" iterates map. F or a 6 4-cycle, this means 1 minute per step of Newton's metho d. same time as n increased, it became an increasingly more delicate matlocate the desired solution. However, I immediately perceived the &'sconverging geometrically. This enabled me t o predict the next value wcreasing accuracy as n increased, and so required just one step of Newmethod to ob tain the desired value. T o the best of my knowledge, this obtion of geometric convergence has never been made independently, for th

LOS ALAMOS SCIENCE/Summer


12/24

pie reason that the solutions have always been performed automatically onlarge and fast computers!That a geometric convergence occurred was already a surprise. I was in-terested in this for two rea sons: first, to gain insight into my theo retical work,as already mentioned, and second, because a convergence rate is a number in-variant under all smooth transformations, and so of mathematical interest. Ac-cordingly, I spent a part of a day trying to fit the convergence rate value,4.669. to the mathematical constants I knew. The task was fruitless, save forthe fact that i t made the number memorable.At this point I was reminded by Paul Stein that period doubling isn't a u-nique property of the quadratic map, but also occurs, for example, in

Xn+ = ?L sin TTX- .However, my generating function theory rested heavily on the fact that thenonlinearity was simply quadratic and not transcendental. Accordingly, my in-terest in the problem waned.

Perhaps a month later I decided to determ ine the A's in the trans cend entalcase numerically. This problem w as even slower to compute than the quad raticone. Again, i t became apparent that the h's converged geom etrically, andaltogether amazingly, the convergence rate was the sa me 4.669 t ha t I remem-bered by virtue of my efforts to fit it.Recall that the work of Metropolis, Stein, and Stein showed that precise

qualitative features are independent of the specific iterative scheme. Now Ilearned that precise quantitative features also are independent of the specificfunction. This discovery represents a complete inversion of accustomed ritual.Usually one relies on the fact that similar equations will have qualitativelysimilar behavior, but quan titative predictions depend on the details of the equ a-tions. The universality theory shows that qualitatively similar equations havethe identical quantitative behavior. F or e xample, a system of differential equ a-tions naturally determines certain maps. The computation of the actualanalytic form of the map is generally well beyond present mathematicalmethods. However, should the map exhibit period doubling, then precise quan-titative results are available from the universality th eory because th e theory a p-plies independently of which map it happens to be. In particular, certain fluidflows have now been experimentally observed to become turbulent throughperiod doubling (subharmonic bifurcations). F rom this one fact we kno w th atthe universality theory applies-and indeed correc tly determ ines the preciseway in which the flow becomes turbulent, without any reference to the under-lying Navier-Stokes equations.LOS ALAMOS SCIENCE/Summer 1980

curves are coincident. Moreover, scale reduction that f2 will determinef* is based solely on the functional cposition, so that if these curves forf 2 " , converge (as they obviously dFig. 8), the scale reduction from levelevel will converge to a definite consBut the width of each circulation sqis just the distance between x = % wit is a fixed point of f2" and the fpoint of f2"next nearest to x = '/, (F7a and b). That is, asymptotically,separation of adjacent elementsperiod-doubled attractors is reduceda constant value from one doublinthe next. Also from one doubling tonext, this next nearest elem ent alternfrom one side of x = % to the other.dn denote the algebraic distance fro= to the nearest element of the atttor cycle of period 2", in the 2"-cyc?in. A positive number a scales thistance dow n in the 2""-cycle at ?in

But since rescaling is determined onlfunctional composition, there is sfunction that composed with itself reproduce itself reduced in scale byThe function has a quadratic maximat x = \{^ is symmetric about x =can be scaled by hand to equal 1 at. Shifting coordinates so tha t x = '/x = 0, we have

Substituting g(0) = 1, we have

Accordingly, Eq. (29) is a definite etion for a function g depending othrough x2 and having a maximum at x = 0. There is a unique smooth stion to Eq. (29), which determines


13/24

Knowing a, we can predict through Eq.(28) a definite scaling law binding on theiterates of any scheme possessing perioddoubling. The law has, indeed, been am-ply verified experimentally. By Eq. (29),we see that the relevant operation uponfunctions that underlies period doublingis functional composition followed bymagnification, where the magnificationis determined by the fixed-point condi-tion of Eq. (29) with the function g thefixed point in this space of functions.However, Eq. (29) does not describe astable fixed point because we have notincorporated in it the parameter increasefrom L,, o Thus, g is not thelimiting function of the curves in the cir-culation squares, a lthough it is intimatelyrelated to tha t function. T he full theory isdescribed in the next section. Here wemerely state that we can determine thelimiting function and thereby can deter-mine the location of the actual elementsof limiting 2"-cycles. We also have es-tablished that g is an unstable fixed pointof functional composition, where the rateof divergence away from g is precisely 6of Eq. (3) and so is computable. Accor-dingly, there is a full theory that deter-mines, in a precise quantitative way, theaperiodic limit of functional iterationswith an unspecified function f.

Some Details of the Full TheoryReturning to Eq. (28), we are in aposition to describe theoretically the uni-versal scaling of high-order cycles and

the convergence to a universal limit.Since dn is the distanc e between x = '/,and the element of the 2"-cycle at ?innearest to x = '/, and since this nearestelement is the 2 " ' iterate of x = 1/,(which is true because these two pointswere c oincident before the nth perioddoubling began to split them apart), wehave

Fig. 8. The superposition o f the suitably magnified dotted squares o f f " ' a t \ (aFigs. 5a , 7a , ...).

For future work it is expedient to per-form a coordinate translation that movesx = '/, to x = 0. Thus, Eq. (32) becomes

Equation (28) now determines that therescaled distances,

will converge to a definite finite value asn --+ oo. That is ,

must exist if Eq. (28) holds.However, from Fig. 8 we ksomething stronger than Eq. (34). Wthe nth iterated function is magnified(-a)", it converges t o a definite functEquation (34) is the value of this ftion at x = 0. After the magnificatthe convergent functions are given

Thus ,

LOS ALAMO S SCIENCEISummer


14/24

Fig. 9. The function g,. The squares locate cycle elements.

is the limiting function inscribed in thesqua re of Fig. 8 . Th e function g,(x) is, bythe argument of the restriction o f f to in-creasingly small intervals ab out its max-imum, the unive rsal limit of all iterates ofall f s with a quadra tic extremum. In-deed, it is numerically easy to ascertainthat gl of Eq. (35) is always the samefunction independent of the fi n Eq. (32).What is this universal function goodfor? Figure 5a shows a crude approx-imation of g, [n = 0 in the limit of Eq.(35)], while Fig. 7a shows a better ap-proximation (n = 1). In fact, the extremaof g, near the fixed points of g, supportcirculation squares each of which con-tains two points of the cycle. (The twosquares shown in Fig. 7a locate the fourelements of the cycle.) T hat is, gl deter-mines the location of elements of high-

order 2"-cycles near x = 0. Since g, isuniversal, we now have the amazingresult that the location of the actual ele-ments of highly doubled cycles is univer-sal! The reader might guess this is a verypowerful result. Figure 9 shows g, o ut tox sufficiently large to have 8 circulationsquares, and hence locates the 15 ele-ments of a 2"-cycle nearest to x = 0.Also, the universal value of the scalingparameter a, obtained numerically, is

Like 6, a is a number that can bemeasured [through an experiment thatobserves the d n of Eq. (28)] in anyphenomenon exhibiting period doubling.If g, is universal, then of course itsiterate gi also is universal. Figure 7b

depicts an early approximation to iterate. In fact, let us define a new versal function go, obtained by scagi:

(Because g, is universal and the iterof our quadratic function are all smetric in x, both g, an d go are sy mm efunctions. Accordingly, the minus within g i can be dropped with impunFrom Eq. (35), we now can write

[W e introduced the scaling of Eq . (37provide one power of a per period dbling, since eac h successive iterate of

LOS ALAMOS SCIENCE/Summer 1980


15/24

reduces the scale by a].In fact, we can generalize Eqs. (35)and (38) to a family of universal func-tions g,:

To understand this, observe that golocates the cycle elements as the fixedpoints of go at extrema; g, locates thesame elements by determining two ele-ments per extrem um. Similarly, gr deter-mines 2, elements about each extremumnear a fixed point of g,. Since each f 2 salw ays magnified by (-a)" for eac h r,the scales of all g are the same. Indeed,g, for r > 1 looks like g, of Fig. 9, exceptthat each extremum is slightly higher, toaccommodate a 2r-cycle. Since each ex-tremum must grow by convergentlysmall amounts to accommodate higherand higher 2r-cycles, we are led to con-clude that

must exist. By Eq. (39),

Unlike the functions g,, g(x) is obtainedas a limit of f2" t a fixed value of X,. In -deed, this is the special significance ofit is an isolated value of X, at whichrepeated iteration and magnification leadto a convergent function.We now can write the equation that gsatisfies. Analogously to Eq. (37), it iseasy to verify that all g are related by

By Eq. (40), it follows that g satisfies

The reader can verify t hat Eq. (43) is in-

variant under a magnification of g. Thus,the theory has nothing to say about ab-solute scales. Accordingly, we must fixthis by hand by setting

Also, we must specify the nature of themaximum of g at x = 0 (for example,quadratic). Finally, since g is to be builtby iterating a - x2, it must be bothsmooth and a function of x through x2.With these specifications, Eq. (43) has aunique solution. By Eqs. (44) and (43),

so that

Accordingly, Eq. (43) determines atogether with g.Let us comment on the nature of Eq.(43), a so-called functional equation.Because g is smooth, if we know itsvalue at a finite number of points, weknow its value to some approximationon the interval containing these points byany sufficiently smooth interpolation.Thus, to some degree of accuracy, Eq.(43) can be replaced by a finite coupledsystem of nonlinear equations. Exactlythen, Eq. (43) is an infinite-dimensional,nonlinear vector equation. Accordingly,we have obtained the solution to one-dimensional period doubling through ou rinfinite-dimensional, explicitly universalproblem. Equation (43) must be infinite-dimensional because it must keep trackof the infinite number of cycle elementsdemanded of any attempt to solve theper iod-doubl ing problem. Rigorousmathematics for equations like Eq. (43)is just beyond the boundary of presentmathematical knowledge.At this point, we must determine twoitems. First, where is 6? Second, how d owe obtain g,, the real function of interestfor locating cycle elements? The two

problems are part of one question. Eqtion (42) is itself an iteration scheHowever, unlike the elements in Eq. the elements acted on in Eq. (42) functions. The analogue of the functo f f in Eq. (4 ) is the operation in functspace of functional comp osition followby a magnification. If we call this option T, and an element of the functspace v , Eq. (42) gives

In terms of T, Eq. (42) now reads

and Eq. (43) readsg =Tlgl (Thus, g is precisely the fixed point oSince g is the limit of the seq uenc e g,,can o btain g, for large r by linearizinabout its fixed point g. Once we havein the linear regime, the exact repeaapplication of T by Eq . (47) will provg, . Thus , we mus t inves t igate stability of T at the fixed pointHowever, it is obvious that T is unstaat g: for a large enough r, g is a pointbitrarily close to the fixed point g; by (479, successive iterates of g undemove away from g. How unstable isConsider a one-parameter family functions flw,which means a "line" in function space. For each f, there isisolated parameter value A , or whrepeated applications of T lead to cvergence towards g 1 Eq . (4 I ) ] . Now, function spac e can be "packed" withthe lines corresp ond ing t o the var ious The set of all the points on these lispecified by the respective A ' s detmines a "surface" having the prop ethat repeated applications of T t o apoint on it will converge to g. This is surface of stability of T (the "stamanifold" of T through g). But throueach point of this surface issues out cor responding l ine , which is o

18 LOS ALAMO S SClENCEISummer 1


16/24

dimensional since it is parametrized by asingle parameter, A. Accordingly, T isunstable in only one direction in functionspace. Linearized about g, this line of in-stability can be written as the one-parameter family

which passes through g (at 'k = 0) anddeviates from g along the unique direc-tion h. But f, is just one of our transf or-mations 1 Eq. (4)1! Thus, as we vary 'k, f,bwill undergo period doubling, doublingto a 2"-cycle a t A n. By E q. (41), ' k orthe family of func tions f, in Eq. (49) is

Thus, by Eq. (1)

Since applications of T by Eq. (47)iterate in the opposite direction (divergeaway from g), it now follows that therate of instability of T along h must beprecisely 6.

Accordingly, we find 6 and g, in thefollowing way. First, we must linearizethe operation T about its fixed point g.Next, we must determine the stabilitydirections of the linearized operator.Moreover , we expect there to beprecisely one direction of instability. In-deed, it turns out that infinitesimal defor-mations (conjugacies) of g determinestable directions, while a unique unstabledirection, h, emerges with a stability rate(eigenvalue) precisely the 6 of Eq. (3).Equ atio n (4 9) at 'kr is precisely gr forasymp totically large r. Thus gr is knownasymptotically, so that we have enteredthe sequence gr and can now, byrepeated use of Eq. (47), step down t o g,.All the ingredie nts of a full description ofhigh-order 2"-cycles now are at handand evidently are universal.

Although we have said that the func-tion g, universally locates cycle elements

near x = 0, we must understand that itdoesn't locate all cycle elements. This ispossible because a finite distance of thescale of g, (for example, the location ofthe element nearest to x = 0) has beenmagnified by an or n diverging. Indeed,the distances from x = 0 of all elementsof a 2"-cycle, "accurately" located by gl ,are reduced by -a in the 2""-cycle.However, it is obvious that some ele-ments have n o such scaling: because f(0)= a in Eq. (13), and an-+ , hich isa definite nonzero number, the distancefrom the origin of the element of the 2"-cycle farthest to the right certainly hasnot been reduced by -a at each perioddoubling. This suggests that we mustmeasure loc ations of elements on the farright with respect to the farthest rightpoint. If we do this, we can see that thesedistances scale by a2, since they are theimages through the quadractic max-imum o f f a t x = 0 of elements close to x= 0 scaling with -a. In fact, if we imageg, through the maximum o f f ( through aquadratic conjugacy), then we shall in-deed obtain a new universal functionthat locates cycle elements near theright-most element. The correct descrip-tion of a highly doubled cycle nowemerges as one of universa l localclusters.

We ca n state the s cope of universalityfor the locat ion of cycle e lementsprecisely. Since f('kl, x) exactly locatesthe two elements of the 2'-cycle, andsince f('k,, x) is an approxim ation to g , In= 0 in Eq. (35 )], we evidently can loc ateboth points exactly by appropriatelysealing g,. Next, near x = 0, f2('k2,x) is abetter approximation to g, (suitablyscaled). However, in general, the moreaccurately we scale g, to determine thesmallest 2-cycle elements, the greater isthe error in its determination of the right-most elements. Again, near x = 0, ?(A3,x) is a still better a ppro xim ation to g,. In-deed, the suitably scaled g, now candetermine several points about x = 0 ac-curately, but determination of the right-

most elements is still worse. Infashion, it follows that g l, suscaled, can d etermine 2^ points of tcycle near x =0 for r


17/24

elements not by their location on the x-axis, but rather by their ord er a s iteratesof x = 0. Because the time sequence inwhich a process evolves is precisely thisordering, the result will be of immediateand powerful predictive value. It isprecisely this scaling law that allows usto compute the spectrum of the onset ofturbulence in period-doubling systems.What must we compute? Firs t, just asthe element in the 2"cycle nearest to x =0 is the element halfway around the cy-cle from x = 0, the element nearest to anarbitrarily chosen element is preciselythe one halfway aro und the cycle from it.Let us denote by dn(m ) the distance bet-ween th e mth cycle element (xm) and theelement nearest to it in a 2"-cycle. [Thedn of Eq. (28) is dn(0)].As just explained,

How ever, xrn is the m th iterate o f x, = 0.Recalling from Eq. (6) that powers com-mute, we finddn(m) = frn(^O)

- rn(L n, 2"-l(Ln,0)) . (53)Let us, for the moment, specialize to mof the form 2 " r , in which case

Fo r r > I forn large), we have, by Eq. (39),

The object we want to determine is thelocal scaling a t the m th element, that is,the ratio of nearest separations at the mthiterate of x = 0, a t successive values of n.That is, if the scaling is called o,

IObserve by Eq. (28), the definition of a,that on(0)- - a ) . ) Specializing againto m = 2 " r , wh ere r


18/24

and so forth. [ It is easy to verify fEq. (52) that o is periodic in t of pe1, and has the symmetry

Fig. 10 . The trajectory scaling function. Observe that o (x + 1 / 2 ) = - o (x). and

Accordingly, we have paid attentioits first half 0 < t < '/,.I With o we alast f inished with one-dimensiiterates per se.Universal Behavior in H igher Dimsional Systems

So far we have discussed iteratioone variable; Eq. (15) is the prototEquation (14), an exam ple of iteratitwo dimensions, has the special propof preserving areas. A generalizatioEq. (14),

with Ib l < 1, contracts areas . Equa(61) is interesting because it possesso-called strange attractor. This man attractor (as before) constructefolding a curve repeatedly upon (Fig. 11) with the consequent propthat two initial points very near toanother are, in fact, very far from other when the distance is measalong the folded attractor, which ipath they follow upon iteration. means that after some iteration, theysoon be far apart in actual dis tancwell as when measured along the attor. This general mechanism givsystem highly sensitive depende nce i ts in i t ia l condi t ions and a statistical chara cter: since very smalf e r e n c e s i n i n i t i a l c o n d i t i o n smagnified quickly, unless the initialditions are known to infinite preciall known knowledge is eroded rapidFig. 1 1 . The plotted points lie on the " strange attractor" of Dufflng's equa tion. future ignorance. Now, Eq. (61) e

LOS ALAMOS SCIENCEISummer 1980


19/24

in to the ear ly s tages of s ta t i s t ica lbehavior through per iod doubl ing.Moreover , 5 of Eq. (3) is again the rateof onset of complexity, and a of Eq. (3 1)is again the rate at which the spacing ofadjacent attractor points is vanishing.Indeed, the one-dimensiona l theorydetermines all behavior of Eq . (61) in theonset regime.In fact, dimensionality is irrelevant.The same theory, the sam e numbers, etc.also work for iterations in N dimensions,provided that the system goes throughperiod doubling. The basic process,wherever period doubling occurs a dinfiniturn, is functional com position fromone level to the next. Accordingly, amodification of Eq. (29) is at the heart ofthe process, with composition on func-tions from N dimensions to N dimen-sions. Should the specific iteration func-tion contract N-dimensional volumes (adissipative process), then in general thereis one direction of slowest contraction,so that after a number of iterations the Fig. 12a. The most stable 1-cycle o f Dll/fing's equation in phase space (e)process is effectively one-dimensional.Put differently, the one-dimensional solu-tion to Eq. (29) is always a solution to itsN-dimensional analogue. It is the rele-vant fixed point of the analogue if theiteration function is contractive.Universal Behavior in DifferentialSystems

The next step of generalization is toinclude systems of differential equations.A prototypic equation is Duffing's os-cillator, a driven damped anharmonicoscillator,i + k i + x3= bsin 2 n t . (62)Th e periodic drive of period 1 determinesa natural time step. Figure 12a depicts aperiod 1 attracto r, usually referred to a sa limit cycle. It is an attractor because,for a range of initial conditions, the solu-tion to Eq. (62) settles down to the cycle.It is period 1 because it repeats the sam e Fig. 12b. The most stable 2-cycle of Dufflng's equation. Observe that it is two dcurve in every period of the drive. placed copies of Fig. 12a.2 2 LOS ALAMOS SCIENCEISummer 19


20/24

Fig. 12c. The most stable 4-cycle of Duffing's equation. Observecopies of Fig, 12b have either a broad or a narrow separation.Figures 12b and c depict attractors ofperiods 2 and 4 as the friction or da mp-ing constant k in Eq. (62) is reducedsystematically. The parameter values k= L),Xi , Xi, ..., are the damping cons-tants corresponding to the most s table2"-cycle in analogy to the \. of the one-dimensional functional iteration. Indeed,this oscillator's period doubles (at leastnumerically!) ad infinitum. In fact, by k= L5, he 61 of Eq . (2) has converged to4.69. W hy is this? Instead of consideringthe entire trajectories as shown in Fig.12, let us consider only where the trajec-tory point is located every 1 period ofthe drive. The 1-cycle then producesonly one poin t , whi le the 2-cycleproduces a pair of points, and so forth.This time-one map [if the trajectorypoin t is (x,;) now , where is it on e periodlater?] is by virtue of the differentialequation a smooth and invertible func-

that the displacedtion in two dimensions. Qualitatively, itlooks like the map of Eq. (61). In thepresent state of mathematics, little canbe said about the analytic behavior oftime-one maps; however, since ourtheory is universal, it makes no dif-ference that we don't know the explicitform. We still can determine the com-plete quantitative behavior of Eq. (62) inthe onset regime where the motion tendsto aperiodicity. If we already know, bymeasurement, the precise form of thetrajectory after a few period doublings,we can compute the form of the trajec-t o r y a s t h e f r i c t i o n i s r e d u c e dthroughout the region of onset of com-plexity by carefully using the full powerof the universality theory to determinethe spacings of elements of a cycle.

Let us see how this works in somedetail. Consider the time-one map of the

D u f i g ' s oscillator in the superstablcycle. In particular, let us focus onelement at which the scaling functio(Fig. 10) has the value q,, nd for wthe next iterate of this element alsothe scaling an. (The element is not big disco ntinuity of 0.) It is then intuthat if we had taken our time-oneamination of the trajectory at valuetime displaced from our first choicewould have seen the same scaling athis part of the trajectory. That is,differential equations will extend map-scaling function continuously function along the entire trajectorythat, if two successive time-one elemhave scaling on, hen the en tire stretctrajectory over this unit time intervalscaling oo. In the last section, we wmotivated to con struct CT a s a functiot along an interval precisely towardsend.T o implement this idea, the first stto define the analogue of dn. We reqthe spacing between the trajectorytime t and at t ime T n/2 where the peof the system in the 2"-cycle is

That is, we define

(There is a d for each of the N variafor a system of N differential equatioSince o wa s defined as periodic of pe1, we now have

The c ontent of Eq. (65), based on thdependence arising solely through thin a, and not on the detailed form oalready implies a strong scaling pretion, in that the ratio

when plotted with t scaled so that TLOS ALAM OS SCIENCEISummer 1980


21/24

1, is a function independent of n. Thu s ifEq. (65) is true for some o, whatever itmight be, then kno wing xn(t), we cancompute dn(t) and from Eq. (65) dn+i(t) .As a consequence of periodicity, Eq.(64) for n -+ + 1 can be solved forx,,+,(t) (through a Fourier transform).Th at is, if we have measured any chosencoordinate of the system in its 2"-cycle,we can compute its time dependence inthe 2""-cycle. Because this proce dure isrecursive, we can compute the coor-dinate's evolution for all higher cyclesthrough the infinite period-doublinglimit. If Eq. (65) is true and o not known,then by measurement at a 2"-cycle andat a 2"+1-cycle, o could be constructedfrom Eq. (65), and hence all higher orderdoublings would again be determined.Accordingly, Eq. (65) is a very powerfulresult. However, we know m ~ c ~ Fig. 13 a. The ratio of nearest copy sep arations in the 8-cycle and 16-cy cle for DThe universality theory tells US that ing3 equation.period doubling is universal and thatthere is a unique function o which, in-deed, we have computed in the previoussection. Accordingly, by measuring x(t)in some chosen 2"-cycle (the higher then, the more the number of effectiveparameters to be determined empirically,and the more precise are the predic-tions), we now can compute the entireevolution of the system on its route toturbulence.How well does this work? The em-pirically determined o [for Eq. (62)] ofEq. (65) is shown for n = 3 in Fig. 13aa nd n = 4 in Fig. 13b. The figures wereconstructed by plotting the ratios of dn +,and dn scaled respective to T = 16 in

Fig. 1 3a and T = 32 in Fig. 13b.Evidently the scaling law Eq. (65) is be-ing obeyed. Moreover, on the samegraph Fig. 14 shows the empirical o forn = 4 and the recursion theoretical o ofFig. 10. The reader should observe thedetail-by-detail agreement of the two. Infact , i f we use Eq. (65) and thetheoretical o with n = 2 as empirical in-put, the n = 5 frequency spectrum agrees Fig. 13b . The same quantity as in Fig. 13a, but for the 16-cycle and32 -cycle . Here, twith the empirical n = 5 spectrum to time axis is twice as compressed.24 LOS ALAMOS SCIENCE/Summer 1


22/24

Fig. 14 . Figure 13b overlayed with Fig. 10 compares the universal scaling function o with the empirically determined scalingnearest copy separa tions rom the 16-cycle to the 32-cycle o r Dnfflng's equation.

within 10%. The n = 4 determines n = 5to within I%.) Thus the asymptotic un-iversality theory is correct and is alreadywell obeyed, even by n = 2!Equations (64) and ( 65 ) are solved, asment ioned above, th rough Four iertransforming. The result is a recursivescheme th at determines the Fou rier coef-f ic ien ts of ~ , ,+~( t )n terms of those ofx(t) and the Fourier transform of the(known) function o(t). T o employ theformula accurately requires knowledgeof the entire spectrum of xn (amplitudeand phase) to determine e ach coefficientof However, the formula enjoys an

approximate local prediction, whichroughly determines the amplitude of acoefficient of in term s of the am -plitudes (alone) of x near the desired fre-quency of x + ~ .What does the spectrum of a period-doubling system look like? E ach time theperiod doubles, the fundamental fre-quency halves; period doubling in thecont inuum vers ion is termed half -subharmonic b i furcat ion , a typ icalbehavior of coupled nonlinear differen-tial equations. Since the motion almostreproduces itself every period of thedrive, the amplitude at this original fre-

quency is high. At the first subharmhalving, spectral components of thehalves of the drive frequency comeOn the rou te to ape r iod ic i ty tsatura te at a certain amplitude. Sincmotion more nearly reproduces ievery two periods of drive, the saturated subharmonics , a t th e fourths of the original frequency,smaller still than the first ones, anon, as each set of odd 2"ths com esbeing. A crude approximate predicof the theory is that whatever the systhe saturated amplitudes of each sesuccessively lower half-frequen

LOS ALA MO S SCIENCE/Summer 1980


23/24

define a smooth interpolation located 8.2dB below the smooth interpolation of theprevious half-frequencies. [ This is shownin Fig. 15 for Eq. (62).] After subhar-monic bifurcations ad infiniturn, thesystem is now no longer periodic; it hasdeveloped a continuous broad spectrumdown to zero frequency with a definiteinternal distribution of the energy. Thatis, the system emerges from this processhaving developed the beginnings ofbroad-band noise of a determinednature. This process also occurs in theonset of turbulence in a fluid.The Onset of Turbulence

The existing idea of the route to tur-bulence is Landau's 1941 theory. Theidea is that a system becomes turbulentthrough a succession of instabilities,where each instability creates a newdegree of freedom (through an indeter-minate phase) of a time-periodic naturewith the frequencies successively higherand incommensurate (not harmonics);because the resulting motion is thesuperposition of these modes, it is quasi-periodic.In fact, it is experimentally clear thatquasi-periodicity is incorrect. Rather, toproduce the observed noise of rapidlydecaying correlation the spectrum mustbecome continuous (broad-band noise)down to zero frequency. The defect canbe eliminated through the production ofsuccessive half-subharmonics, whichthen emerge as an allowable route to tur-bulence. If the general idea of a succes-sion of instabilities is maintained, thenew modes do not have indeterminatephases. However, only a small numberof modes need be excited to produce therequired spectrum. (The number ofmodes participating in the transition is,as of now, an open experimental ques-tion.) Indeed, knowledge of the phases ofa small number of amplitudes at an earlystage of period doubling suffices todetermine the phases of the transition

Fig. 15. The subharmonic spectrum of Dufflng's equation in the 32-cycle. The docurve is an interpolation of the odd 32nd subharmonics. The shorter dashed curvconstructed similarly for the odd 16th subharmonics, but lowered by 8.2 dB. longer dashed curve of the 8th subharmonics has been dropped by 16.4 dB, andsolid curve of the 4th subharmonics by 24.6 dB.

Fig. 16. The experimental spectrum (redrawn from Libchaber and Maurer) of a vecting fluid at its transition to turbulence. The dashed lines result from droppihorizontal line down through the odd 4th subharmonics (labelled 2) by 8.2 and dB.LOS ALAMOS SCIENCE/Summe


24/24

THEMitchell J. Feigenbaum is a mathematical physicist andinventor of the universality theory. He earned hisbachelor of electrical engineering degree from the CityCollege of New York in 1964 and his Ph.D. in elemen-tary particle physics from Massachusetts Institute ofTechnology in 1970. He was a Research Associate atCornell University and Virginia Polytechnic Institutebefore becoming a Staff Member in the Theoretical Divi-sion at LASL in 1974. In M ay, 1980, he received aLASL Distinguished Performance Award for his univer-sality theory and development of the first quantitative un-derstanding of the onset of turbulence. He is presentlywriting a monograph on this work for the series Progressin Physics published by Birkhauser Boston, Inc. Hisother technical interests include field theories andfunctional integrals, the renormalization group and phasetransitions, and nonlinear dynamics generally. His peersat LASL often turn to him as the resident consultant onrelated technical matters.

spectrum. What is important is that apurely causal system can and doespossess essentially statistical properties.Invoking ad hoc statistics is unnecessaryand generally incompatible with the truedynamics.A full theoretical computation of theonset demands the calculation of suc-cessive instabilities. The method usedtraditionally is perturbative. We start atthe static solution and add a small time-depend ent piece. Th e fluid equations arelinearized about the static solution, andthe stability of the perturbation isstudied. To date, only the first instabilityhas been computed analytically. Oncewe know the parameter value (for exam-ple, the Rayleigh number) for the onsetof this first time-varying instability, wemust determine the correct form of thesolution after the perturbation has grownlarge beyond the linear regime. To thissolution we add a new time-dependentperturbative mode, again linearized (nowabout a time-varying, nonanalyticallyavailable solution) to discover the newinstability. To date, the second step ofthe analysis has been performed onlynumerically. This process, in principle,can be repeated again and again until asuitably turbulent flow has been ob-tained. At each successive stage, thecomputation grows successively moreintractable.However, it is just at this point that

stabilities have entered to reach theasymptotic regime. Since just two suchinstabilities already serve as a good ap-proximate starting point, we need only afew parameters for each flow to em-power the theory to complete the hardpart of the infinite cascade of more co m-plex instabilities.

Why should the theory apply? Thefluid equations make up a set of coupledfield equations. They can be spatiallyFourier-decomposed to an infinite set ofcoupled ordinary differential equations.Since a flow is viscous, there is somesmallest spatial scale below which nosignificant excitation exists. Thus, theequations are effectively a finite coupledset of nonlinear differential equations.The number of equations in the set iscompletely irrelevant. The universalitytheory is generic for such a dissipativesystem of equations. Thus it is possiblethat the flow exhibits period doubling. Ifi t does , then our theory appl ies .However, to prove that a given flow (orany flow) actually should exhibit dou-bling is well beyond present un-derstanding . All we can d o is experi-ment.Figure 16 depicts the experimentallymeasured spectrum of a convecting li -quid helium cell at the onset of tur-bulence. The system displays measurab leperiod doubling through four or fivelevels; the spectral components at each

asymptotic , the dotted l ines show crudest interpo lations implied for the n3, n = 4 component. Given the smamount of amplitude data, th e interpotions are perforce poor, while ignoraof higher odd multiples prevents cstruction of any significant interpolatat the right-hand side. Accordingly,do the crudest test, the farthest righand amplitude was dropped, and oscillations were smoothed away averaging. The experimental resu-8.3 dB an d -8.4 dB, are in surprisingood agreement with the theoretical 8

From this good experimental agrment and the many per iod doublingsthe clincher, we can be confident that measured flow has made its transitaccording to ou r theory. A measuremof 5 from its fundamental definitwould, of course, be altogether conviing. (Experimental resolution is insficient at present.) However, if we wobackwards, we find that the several pcent agreement in 8.2 dB is anperimental observation of a in the systto the same accuracy. Thus, the presmethod has provided a theore t icalculation of the actual dynamics infield where su ch a feat has been imposble since the construction of the NaviStokes equa tions. In fact, the scaling Eq. (65) transcends these equations, aapplies to the true equations, whatethey may be.

Unversal Behavior in Nonlinear Systems

Documents

Transcript of Unversal Behavior in Nonlinear Systems