Chapter 5: Fractals II In this chapter we address the question of why and have the values that they do and why they are the same for many di erent mapping functions. We will write an equation that embodies the observation of dilation symmetry, and Goals: we will nd that this equation determines the values of the exponents and . To understand why the scaling exhibited by period-doubling bi- In other words, knowing that dilation symmetry exists is enough information furcation sequences of one-dimensional maps is universal, and to to determine what the magni cation factors must be. This result explains the calculate the scaling exponents. observation of universality (di erent maps having the same exponents). The To examine other natural processes which give rise to fractal book by Cvitanovic in the reference list is a good resource for further reading structures. about this universality. Several arguments in this chapter are based on this source. This chapter continues our study of dilation-symmetric structures, or frac- We rst review how the iteration of a smooth function leads to n-cycles of tals. These structures are sometimes called “self-similar”. First, we will pursue ever longer length. Any smooth, positive function f(x)=rfo(x) that vanishes our investigation of the period-doubling bifurcation sequence of the logistic map for x = 0 and x = 1 must cross the line f = x when the “amplitude” r exceeds and get some understanding of why the scaling exponents and which char- some positive value. When r is just at this threshold value, f 0 = 1, and as r acterize it are universal. We hope that this will give you some sense of how increases from there, the derivitive at one of the crossing points becomes smaller understanding fractals can give new insight into natural phenomena. Then we than 1, so that the xed point is stable. We follow this point as r continues to will survey a few other systems in nature in which fractals arise. Some of these increases, and see that f 0 becomes more and more negative, eventually reaching other systems are not nearly so well understood, so perhaps you will get some f 0 = 1. At this point the xed point is marginally stable. Any further increase idea of the scope of the question of why fractals are seen in many di erent in r will drive the xed point unstable. To see what happens then, we look at situations. the second iterate f (2)(x). Up to this value of r it has had a stable xed point at the same x value as for f(x). When f 0 becomes -1 at the xed point, f (2)0 A. The universality of the period-doubling bifurcation sequence. 0 0 becomes +1, i.e., f f . It becomes tangent to the f = x line, making the f (2) In the previous chapter and in Required Project 3 we investigate the scaling xed point marginally stable. As r increases from here, this marginally stable behavior of the period-doubling sequence of the logistic map. We saw that the point moves and divides into two, one stable and one unstable. This is the same k sequence of r-values for the orbits of period 2 obeys behavior that f did for smaller r. It must be the same since f (2) is just another smooth function, as f was. The stable xed point represents a two-cycle of f,so r r lim k 1 k 2 = , (2) →∞ that there must be a second stable xed point of f nearby. As r continues to k rk rk 1 (2)0 increase, the derivitive f becomes -1 and this xed point becomes unstable, (4) (2) (2) 2k 1 by the same process that happened with f. But at this point f = f (f ) where =4.7...., and that the sequence of values yk f (x) satis es (4) (2) 1/2 is just tangent to the f = x line, and f repeats the process that f and f did before it. yk 1 yk 2 lim , = , Each time the stable xed points switch to a higher iterate of f, it means k→∞ y y k k 1 that f has an n-cycle or period of twice the length as before. For each pe- with =2.5..... These results mean that the bifurcation diagram looks the riod doubling there is a corresponding r value, that we denote rk for the kth (2k) same when it is magni ed about the point (x =1/2, r = r∞ =3.57....)bya doubling. Thus between rk and rk+1 f has a stable xed point that aren’t factor in the x-direction and a factor in the r-direction. This mapping stable for lower iterates. By continuing to increase r it appears that we may of the bifurcation diagram onto itself under certain rescaling factors is known increase the length of these cycles as much as we wish, leading to behavior of as dilation symmetry (because the diagram is invariant under dilation of co- arbitrarily great complexity. For the logistic map and many other f functions ordinate scales). In Required Project 3 you show that and are not noticeably the limit cycles become in nite for some nite r∞. This complexity as r → r∞ di erent for some di erent choices of the map function. is what we want to probe.
Physics 251/CS 279/Math 292 Chapter 5 typeset February 11, 2001 page 1 Physics 251/CS 279/Math 292 Chapter 5 typeset February 11, 2001 page 2 (2k) When we are close to r∞, an arbitrarily slight change in the original func- This means that the function gk de ned by a(k)(f (x) x0)=gk(zk) should k tion f can evidently cause a qualitative change in behavior: a period doubling. be smooth, even though f (2 ) itself is not smooth. Figure 5.1 illustrates the We saw above that each period doubling can be understood by looking at a idea using f (2) and f (4). tiny range of x. We expect the same to be true as we approach r∞. A clever observation shows us what x to focus on. We saw that the derivitive of our (2k) iterate f starts out at the marginally stable value of 1 at r = rk. Then it steadily decreases to -1 as we approach r0. At some r value in between, the derivitive must vanish. We denote this r asr ˜k. When the derivitive vanishes at a xed point, this point is called superstable, as noted above. We may deter- mine a superstable xed point x of any iterate by using the Floquet-multiplier (2k) (2k)0 0 0 0 expression for the derivitive of f : f = f (x1)f (x2)...f (x2k ). At a su- perstable point, this multiplier must vanish. This means that one of the f 0 s in the product must vanish. One point of the 2k cycle must be at a maximum or minimum of f(x). These maxima and minima don’t shift as we change r; r is just a multiplicative factor. For example in the logistic map, there is a single maximum located at x =1/2. From now on, we’ll focus on some chosen maximum or minimum of f located at x0 and consider the region around x0. As k →∞the some xed points at the period-doubling rk must lie close to the superstable xed point at the correspondingr ˜k at x0.
Limiting dependence on x To see how the instabilities pile up as r → r∞ we k clearly need to know the functional form of f (2 )(x) near x = x0. This functional form cannot be very smooth. We know that each k point in the current 2k-cycle must be a xed point of f (2 ). Thus this function must cross the f = x line 2k times. The number of wiggles must double each time k increases by one. We see this happening as we iterate the logistic map: k each time k increases by one f (2 ) becomes a polynomial of twice as high an order. k We cannot hope to have a manageable limit for f (2 ) as k →∞unless we use a scale of x which expands to compensate for the increasing wiggliness Figure 5.1. Dark line: f (2) light line, f (4) near superstable points. A k k of f (2 ). Accordingly we adopt a rescaled co-ordinate for describing f (2 )(x) section of the f (2) is copied below it. This section is reduced by a (4) called zk,and de ned by zk = a(k)(x x0). Evidently a(k) must become huge, factor of 2.74 and inverted and placed near the f curve to show k so that the tiny region of x where f (2 ) looks smooth gets expanded to a nite, that the f (4) curve resembles the f (2) curve except for a change of nonvanishing interval of z. Now when we increase k we can hope to account for horizontal and vertical scale. (2k) the increased wiggliness of f (x) near x0 by the scale factor a(k), so that the (2k) remaining k dependence is smooth. To achieve the hoped-for smooth function, We may use the known behavior of f to infer the behavior of the hy- k k (2 ) we must expand the scale of f (2 ) as well as the scale of x. In words, the pothesized function g. What we know about f is the way it is related to the k+1 behavior we seek is next function in the series,f (2 ):
k k+1 k k expanded f (2 )(x)=[smooth function of](expanded x) f (2 )(x)=f (2 )(f (2 )(x)). (5.1)
Physics 251/CS 279/Math 292 Chapter 5 typeset February 11, 2001 page 3 Physics 251/CS 279/Math 292 Chapter 5 typeset February 11, 2001 page 4 We may express these functions in terms of our reduced functions gk and gk+1: g(0) = 1. With this convention, one may readily relate g to by applying the Feigenbaum equation at z = 0. This gives 1 = g(1). 1 (2k) To get further information about , we take the assumed smoothness of g x0 + gk+1(zk+1)=f (x0 +(1/a(k))gk(zk)) a(k +1) seriously and approximate it as a truncated Taylor expansion: g(z) ' g(0) + (5.2) 1 1 00 2 2 g z . The rst derivitive of g must vanish at x0. We chose x0 because the = x0 + gk(a(k)((1/a(k)gk(zk)). a(k) derivitive of f, and therefore its iterates, vanishes at x0. Our g is merely a rescaled version of the f’s and thus its derivitive vanishes also. Then the We can simplify this by de ning (k)=a(k +1)/a(k) and noting that zk+1 is Feigenbaum equation reads just (k)zk. Then the simpli ed equation reads: 1 1 1 g(0) + g00 2z2 = [g(0) + g00(g(0) + g00z2)2] gk+1( (k)zk)= (k)gk(gk(zk)). 2 2 2
1 00 1 00 00 00 for a range of zk around zero. 2 2 Equating the terms in z gives 2 g = 2 g g(0)g ,or = g(0)g . Equating Now we take the limit as k →∞. The g functions go to their assumed 1 00 2 1 the constant terms gives g(0) = [g(0)√ + 2 g g(0) ]or1= [1 + 2 ]. This limiting form g(z). The equation can then only make sense if there is also a quadratic equation implies = 1 3. (Reassuringly, is insensitive to limiting value for (k), which we’ll call . We are left with a simple but strange our convention g(0) = 1.) One of these roots has a magnitude smaller than equation for g called the Feigenbaum equation [1]: 1. This one is incompatible with a smooth g(z), as explained above. Thus our argument leads us to the conclusion that ' 2.73. It is negative! But a little g( z)= g(g(z)). (5.3) k re ection shows that we might have expected this negative value. The f (2 )(x) at x0 alternates between being maximum and a minimum. (The derivitive of Can there be functions g(z) with this property? If so, what are the possible (2k) values for the scale factor ? We can learn something about g by looking at f (x) at the xed point closest to x0 must be negative at the threshold of (2k+1) its xed points. By looking at the de nition of g we see that every xed point a period doubling. But the derivitive of the next iterate f (x) must be k k k+1 (2k) (2 ) (2 ) of f at some x implies a xed point of gk at z = a(k)(x x0). Thus positive there. If f (x) and f (x) are to have the same smooth form, our limiting function g(z) must have at least one xed point. Let’s consider this suggests that the two iterates should switch from a maximum at x0 to a k some xed point z and look at what the Feigenbaum equation (5.3) says: minimum, or vice versa.) Thus the a(k) → (constant) k factor relating f (2 )(x) g( z )= g(g(z )) = g(z )= z . This says that z is also a xed point. to gk must also alternate sign with every k. There must be another xed point a factor away from the original one. By The Feigenbaum equation can be solved to successively better approxima- the same argument there must be a third xed point another factor of away, tions by using re nements of our polynomial procedure. We start by assuming and so forth. This behavior sets constraints on : it cannot have magnitude that g(z) is an even polynomial of order 2n with n-1 unknown coe cients (sup- less than 1. If it did, our sequence of xed points would march into the origin, posing that g(0)=1)†. Then we obtain g(g(z)) as a polynomial of order (2n)2. ending all possibility of the smooth g(z) we have assumed. The Feigenbaum equation doesn’t determine g uniquely. There is an arbi- † It can be shown that there are no odd terms in the Taylor series for g.For 3 3 trariness in g that traces to an arbitrariness in our rescaling procedure. If we example, if one assumes that g(z) has a cubic, a3z , term and examines the z had expanded all the zk scales by the same factor b, it would not have a ected term on each side of Eq. (5.3) to nd the coe cient a3, one readily veri es that our argument for nding g. If we obtained convergence to a g function using the both sides of the equation have factors of a3. Thus one is free to set a3 =0 original zk variables, we would have obtained it in the expanded zk variables as The same thing occurs for each higher odd power. In any case we know that k k well. This shift would have introduced an an extra factor b relating f (2 )(x)to there must be some even solution to Eq. (5.3), since logistic-map iterates f (2 ) gk;thusgk would have been divided by b. Then we would have obtained not are all even and converge to g. Further, by applying the methods of the next g(z) butg ˜(z) g(bz)/b. One can readily show that if g satis es the Feigenbaum section one can show that if a tiny odd part is added to g, it becomes smaller equation, then so doesg ˜. It is conventional to settle this ambiguity by de ning under iteration.
Physics 251/CS 279/Math 292 Chapter 5 typeset February 11, 2001 page 5 Physics 251/CS 279/Math 292 Chapter 5 typeset February 11, 2001 page 6 Equating this with the order-2n polynomial on the left side, we obtain n equa- time development can be expressed as iterations of a function, these universal tions for the n 1 unknown coe cients, plus the scale factor . By this means properties must show up in a broad class of physical situations as well as in one can obtain g and to arbitrary accuracy: =2.502907875..... A graph of mathematical examples. g is shown in Figure 5.2 The universal scale factor explains the geometrically spaced xed points observed as one approaches r∞, described at the beginning of the chapter. One may readily verify that the geometrically-spaced xed points z of g imply a spacing of xed points xk satisfying (xk 1 xk 2)/(xk xk 1)= .We have thus explained the convergence behavior of the points yk noted at the beginning of the chapter. The appearing there is simply the negative of the in the Feigenbaum equation. No one has found a way to express in terms of simpler constants. It seems to be a new fundamental constant, like Euler’s constant e. Indeed, there is a certain resemblence [2] between the Feigenbaum equation and the de ning equation for the exponential function exp(x): exp(x + y) = exp(x) exp(y); exp(0) = 1.
Limiting dependence on r Our original motivation for nding the Feigenbaum function g was to understand the sequence of rk that led to the onset of chaos. We have now understood the x dependence near this limit point, but not the r dependence. The treatment above simply ignored any r dependence; thus it is incomplete†. We expect to express the k r dependence of the f (2 ) functions using the same scaling ideas we used to k infer the x dependence. Evidently, the f (2 ) become increasingly sensitive to r just as they do to x. We expect a well-behaved limit when we examine a small enough region around each rk. We must expand the region progressively more as k increases, since we know the behavior becomes arbitrarily sensitive to Figure 5.2. a) Thick line: sketch of solution g(z) of the Feigenbaum changes in r. Thus, to describe the r dependence in terms of smooth functions, equation (5.3) after Ref. 1. Thin line shows g(g(z)). we must rescale. We de ne a scale factor c(k)byrk = r∞ 1/c(k). These c’s must go to in nity with k; they play an analogous role to a(k) used above. As The signi cance of this Feigenbaum equation and its solution is that it (2k) we have argued above, f (r∞,x), when expressed in terms of gk(zk) yields applies to a wide variety of situations. When we explained it, we didn’t use the limiting smooth function g(z). the fact that our iterated function f(x) was the logistic map or any other Now we want to know how the limiting function is altered if we make slight speci c function. Thus we expect similar convergence equally much whenever changes in r. The shift required to obtain corresponding behavior for di erent we iterate a function that behaves like f(x): it has a smooth maximum at some k re ects the diverging sensitivity to r. Thus it must be expanded by the scale x . The resulting function g satis es an equation that makes no reference to 0 factor c(k). Accordingly we de ne a rescaled quantity R analogous to z . the iterated function f that led to it. Thus neither g nor can depend on k k We try the representation: R = c(k)(r r∞). Then our scaling hypothesis this function. A whole class of f(x) functions yield one single behavior at the k onset of chaos. We say that g and are universal. This is the rst example of universality we have encountered. The mere existence of dilation symmetry, † In fact the equation (5.1) that we used to nd the Feigenbaum function g k together with certain qualitative constraints, led to a quantitative behavior that is only true for if r is the same for all the f’s in the equation. Thus the f (2 ) remains unchanged when we change our implementation. Since many forms of only converge to g if r = r∞.
Physics 251/CS 279/Math 292 Chapter 5 typeset February 11, 2001 page 7 Physics 251/CS 279/Math 292 Chapter 5 typeset February 11, 2001 page 8 becomes h as polynomials, one can obtain successive approximations to . Cvitanovic’s (2k) 1 f (r, x)=x0 + gk(Rk,zk) Introduction gives some explicit prescriptions. a(k) Eigenvalue equations have many solutions. For example if we approximate Translating the equation h by a linear combination of N basis functions such as z,z2, ...zN , there will be N eigenvalues n and eigenvectors hn. We denote the largest of the n as 1. k+1 k k f (2 )(r, x)=f (2 )(r, f (2 )(r, x)) The factor tells us what happens if we iterate a function that is slightly di erent from the original function g(0,z). If our function di ers only slightly into rescaled language as before, we obtain from g, we can express the di erence as a sum of eigenfunctions hn of Eq. (5.6). Typically there will be some contribution of h1 to this sum. Let’s consider this gk+1(Rk+1,zk+1)= gk(Rk,gk(Rk,zk)). contribution by itself. Then the iterated function has the same form as the original one, with R replaced by 1R. It is clearly important whether 1 has As before, we express both sides in terms of common variables Rk,zk by using a magnitude smaller than or greater than one. If it is smaller than one, the → zk+1 =(a(k +1)/a(k))zk zk and Rk+1 = c(k +1)/c(k)Rk. Then departure Rh1 diminishes as one iterates, and the initial perturbation from g(0,z) dies away. If it is greater than one, the slight departure from g(0,z) gets gk+1((c(k +1)/c(k))Rk, zk)= gk(Rk,gk(Rk,zk)) ampli ed as one iterates: our function gets farther and farther from g(0,z). Now, when one determines the ’s numerically, one nds that the largest for any R and z . We’re free to rename these variables R and z. We also k k is greater than 1. In fact is 4.6692016... Thus, one cannot hope to converge observe that for the k →∞limit to make sense in this equation, the ratio 1 to g(0,z) simply by plugging some initially guessed function into the right side c(k +1)/c(k) must go to a nite limit, which we denote as . Then of the Feigenbaum equation. In general, the departure from the the true g g( R, z)= g(R, g(R, z)). (5.4) involves h1 (as well as the other h’s.) This piece involving h1 gets multiplied by 1. It grows without bound. Even if the guess is almost perfect, successive Evidently g(0,z) is the Feigenbaum function g(z). Anticipating that g will have iterations will drive you away from it. Since the 1 contribution grows faster a smooth dependence on R, we write g as a Taylor expansion for small R: than any of the others, the result after a few iterations is a departure from g that is dominated by h1 Thus, the value that matters in the limit k →∞is 2 g(R, z)=g(0,z)+Rh(z)+O(R ). (5.5) 1. The relevant solution to Eq. (5.6) is = 1. Now that we have determined , we know how r must be rescaled to give When we insert this form into the Feigenbaum equation, we obtain an expres- nite limiting behavior as k →∞. From this rescaling we can now see how sion for h: the doubling points rk must approach their limiting value. Since c(k +1)/c(k) k approaches† the constant 1, we must have c(k) → (constant) 1 , so that r∞ { } k g(0, z)+ Rh( z)= g(0, [g(0,z)+Rh(z)]) + Rh([g(0,z)+Rh(z)]) rk = (constant) 1 . This implies a simple scaling for the ratio of successive shifts: (rk 1 rk 2)/(rk rk 1)= 1. This is the behavior noted at the 0 O 2 = g(0,g(0,z)) + g (0,z)Rh(z)+Rh(g(0,z)) + (R ) , beginning of the chapter. We now see that observed ratio is simply the so that constant 1 de ned by our eigenvalue equation (5.6). Since 1 is de ned by g0(g(z))h(z)+h(g(z))=( / ) h( z). (5.6) a prescription that is independent of the chosen iteration function f(x), it is universal, like and g(z). Now that we’ve shown that is our , we’ll use 0 1 Here g(z) means g(0,z) and g means the derivitive with respect to the z ar- the name for both henceforth. It is called the Feigenbaum number. Likewise, gument. Since g(z) and are known, this gives an explicit, linear equation for we’ll use the conventional name in place of our . the function h. (Linearity means that if h1 and h2 were solutions, any linear combination of them would be a solution as well. This is clearly true of Eq. † This amounts to saying that slight shifts in r introduce perturbations in (5.6).) Since the equation is a linear functional of h, it amounts to an eigen- f and g that involve h1. While this is true in general, one could invent cases value equation, whose eigenvalue is / . By approximating g and the unknown where it isn’t true. Then the observed would be di erent and smaller.
Physics 251/CS 279/Math 292 Chapter 5 typeset February 11, 2001 page 9 Physics 251/CS 279/Math 292 Chapter 5 typeset February 11, 2001 page 10 Menu Project. Calculating and . This chapter sketched meth- 1) The map is sensitive to initial conditions. Neighboring points spread apart ods of obtaining and by approximating g and h as polynomials. with successive iterations. (Since i+1 =2 i, the distance between neigh- The aim of this project is to obtain successive approximations to boring points doubles at each iteration.) and using this analytical scheme or another one. Obtain at least 2) The map mixes up the entire interval. Within any small interval of x values two successive approximations for both and , to assess how fast some x’s will get mapped into any other interval. Devaney calls maps with the approximations are convering. Explain your procedure. Feel free this property topologically transitive. to use calculational aids such as Mathematica or Matlab. The Intro- 3) Despite this complexity, there are periodic orbits arbitrarily close to any duction in Cvitanovic is a useful starting point. It points to detailed x. That is, the periodic orbits are dense in the interval. information in the rest of the book, which includes a reprint of Ref. These three properties embody the paradoxical coexistence of simplicity and 1. Hilborn, section 5.7 has a heuristic discussion. complexity. A dynamical system with these three properties is called chaotic in Devaney’s de nition. B. Description of chaos in iterative maps When r is between r∞ and 4, there are also regions where chaos occurs. In We have seen above that the approach to the point of in nite periodicity is this region of r, as elsewhere, there is only one attractor for the interval between regulated by a symmetry under dilation. Since the symmetry involves changes 0 and 1. Any starting point that is not part of an n-cycle comes arbitrarily close k of scale to put the functions f (2 ) into a standard or normal form, this is to every point in the attractor. often called renormalization symmetry, and the set of transformations that We can gain insight into the chaotic regions by following the images of the take gk into gk+1 (or gk+2 etc.) is called the renormalization group. If the extremal point of f(x), namely x =1/2. If f is chaotic, f(1/2) must lie in the transformation leaves the function invariant, we say that the function is at a chaotic band. There must be a small interval interval around 1/2 that also lies xed point of the renormalization group. The limiting function g(z)issucha in it. A small-enough interval must transform smoothly under a given number 0 xed point. of iterations n. But since f (1/2) = 0, this interval gets strongly contracted While this symmetry is a powerful and elegant feature of iterative maps, it with each iteration. Thus points of the attractor get concentrated near f(1/2), doesn’t begin to describe the richness we have seen in our numerical calculations f (2)(1/2), etc. These concentrated regions are readily visible as “scars” on a the when our amplitude parameter r exceeds r∞. In this section we’ll note a few of bifurcation diagram of iterates of f vs. r. The rst image of 1/2, i.e., r/4isa these features and suggest properties that can be proven. Two good sources of straight line on the diagram. It is an upper bound of the attractor for all r. The further information are the descriptive book by Hilborn and the readable math second image, f (2)(1/2), is a quadratic function of r, namely r(r/4)(1 r/4). text by Devaney, on the course reference list. It is a lower bound of the attractor. For r