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 dierent 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 magnication factors must be. This result explains the calculate the scaling exponents. observation of universality (dierent 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 dierent 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 rk1 (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) satises (4) (2) 1/2 is just tangent to the f = x line, and f repeats the process that f and f did before it. yk1 yk2 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 magnied 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 innite for some nite r∞. This complexity as r → r∞ dierent for some dierent 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 dened 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 dened 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)).