In Section 18.2 we described how, as we approach the period-doubling accumulation para- meter valueà∞=3.5699. . .from below, the periodn+1 of cycles (x0, x1, . . . , xn) with xn+1=x0gets longer. It is also easy to check that the distances
dn=f(n)(x0+ε)−f(n)(x0) (18.6) grow as well for smallε >0. From experience with chaotic behavior we find that this distance increases exponentially withn→ ∞; that is,dn/ε=eλn, or
λ=1 nln
|f(n)(x0+ε)−f(n)(x0)| ε
, (18.7)
whereλis a Lyapunov exponent for the cycle. Forε→0 we may rewrite Eq. (18.7) in terms of derivatives as
λ=1 nln
df(n)(x0) dx
=1
n n i=0
lnf′(xi), (18.8) using the chain rule of differentiation fordf(n)(x)/dx, where
df(2)(x0) dx =dfà
dx
x=fà(x0)
dfà dx
x=x0
=fà′(x1)fà′(x0) (18.9) andfà′ =dfà/dx, etc. Our Lyapunov exponent has been calculated at the point x0, and Eq. (18.8) is exact for one-dimensional maps.
As a measure of the sensitivity of the system to changes in initial conditions, one point is not enough to determineλin higher-dimensional dynamical systems in general, where the motion often is bounded, so thedncannot go to∞. In such cases, we repeat the procedure for several points on the trajectory and average over them. This way, we obtain the average Lyapunov exponent for the sample. This average value is often called and taken as the Lyapunov exponent.
The Lyapunov exponentλis a quantitative measure of chaos: A one-dimensional iterated function similar to the logistic map has chaotic cycles (x0,x1,. . .) for the parameteràif the average Lyapunov exponent is positive for that value ofà. Any such initial point x0 is called a strange or chaotic attractor (the shaded region in Fig. 18.2). For cycles of finite period,λis negative. This is the case forà <3, forà < à∞, and even in the periodic window atà∼3.627 inside the chaotic region of Fig. 18.2. At bifurcation points, λ=0. Forà > à∞the Lyapunov exponent is positive, except in the periodic windows, whereλ <0, andλgrows withà. In other words, the system becomes more chaotic as the control parameteràincreases.
In the chaos region of the logistic map there is a scaling law for the average Lyapunov exponent (we do not derive it),
λ(à)=λ0(à−à∞)ln 2/lnδ, (18.10)
where ln 2/lnδ∼0.445,δ is the universal Feigenbaum number of Section 18.2, andλ0
is a constant. This relation (18.10) is reminiscent of a physical observable at a (second- order) phase transition. The exponent in Eq. (18.10) is a universal number; the Lyapunov exponent plays the role of an order parameter, whileà−à∞is the analog ofT −Tc, whereTcis the critical temperature at which the phase transition occurs.
Fractals
In dissipative chaotic systems (but rarely in conservative Hamiltonian systems) often new geometric objects with intricate shapes appear that are called fractals because of their noninteger dimension. Fractals are irregular geometric objects whose dimension is typi- cally not integral and that exist at many scales, so their smaller parts resemble their larger parts. Intuitively a fractal is a set which is (approximately) self-similar under magnifica- tion. A set of attracting points with noninteger dimension is called a strange attractor.
We need a quantitative measure of dimensionality in order to describe fractals. Unfortu- nately, there are several definitions with usually different numerical values, none of which has yet become a standard. For strictly self-similar, sets, one measure suffices. More com- plicated (for instance, only approximately self-similar) sets require more measures for their complete description. The simplest is the box-counting dimension, due to Kolmogorov and Hausdorff. For a one-dimensional set, we cover the curve by line segments of length R. In two dimensions the boxes are squares of areaR2, in three dimensions cubes of vol- umeR3, etc. Then we count the numberN (R)of boxes needed to cover the set. LettingR go to zero we expectN to scale asN (R)∼R−d. Taking the logarithm the box-counting dimension is defined as
d≡ lim
R→0
lnN (R)
lnR . (18.11)
For example, in a two-dimensional space a single point is covered by one square, so lnN (R)=0 and d =0. A finite set of isolated points also has dimension d=0. For a differentiable curve of lengthL,N (R)∼L/Ras R→0, so d=1 from Eq. (18.11), as expected.
Let us now construct a more irregular set, the Koch curve. We start with a line segment of unit length in Fig. 18.3 and remove the middle third. Then we replace it with two seg- ments of length 1/3, which form a triangle in Fig. 18.3. We iterate this procedure with each segment ad infinitum. The resulting Koch curve is infinitely long and is nowhere dif- ferentiable because of the infinitely many discontinuous changes of slope. At thenth step each line segment has lengthRn=3−n and there are N (Rn)=4n segments. Hence its dimension isd=ln 4/ln 3=1.26. . . ,which is more than a curve but less than a surface.
Because the Koch curve results from iteration of the first step, it is strictly self-similar.
For the logistic map the box-counting dimension at a period–doubling accumulation point à∞ is 0.5388. . . , which is a universal number for iterations of functions in one variable with a quadratic maximum. To see roughly how this comes about, consider the pairs of line segments originating from successive bifurcation points for a given parameter àin the chaos regime (see Fig. 18.2). Imagine removing the interior space from the chaotic bands. When we go to the next bifurcation, the relevant scale parameter isα=2.5029. . . from Eq. (18.5). Suppose we need 2nline segments of lengthRto cover 2nbands. In the
FIGURE18.3 Construction of the Koch curve by iterations.
next stage then we need 2n+1 segments of lengthR/αto cover the bands. This yields a dimensiond= −ln(2n/2n+1)/lnα=0.4498. . .. This crude estimate can be improved by taking into account that the width between neighboring pairs of line segments differs by 1/α(see Fig. 18.2). The improved estimate, 0.543, is closer to 0.5388. . .. This example suggests that when the fractal set does not have a simple self-similar structure, then the box-counting dimension depends on the box-construction method.
Finally, we turn to the beautiful fractals that are surprisingly easy to generate and whose color pictures had considerable impact. For complexc=a+ib, the correspond- ing quadratic complex map involving the complex variablez=x+iy,
zn+1=z2n+c, (18.12)
looks deceptively simple, but the equivalent two-dimensional map in terms of the real variables
xn+1=xn2−yn2+a, yn+1=2xnyn+b (18.13) reveals already more of its complexity. This map forms the basis for some of Mandelbrot’s beautiful multicolor fractal pictures (we refer the reader to Mandelbrot (1988) and Peitgen and Richter (1986) in the Additional Readings), and it has been found to generate rather intricate shapes for variousc=0. For example, the Julia set of a mapzn+1=F (zn)is defined as the set of all its repelling fixed or periodic points. Thus it forms the boundary between initial conditions of a two-dimensional iterated map leading to iterates that diverge and those that stay within some finite region of the complex plane. For the casec=0 and F (z)=z2, the Julia set can be shown to be just a circle about the origin of the complex plane. Yet, just by adding a constantc=0, the Julia set becomes fractal. For instance, for c= −1 one finds a fractal necklace with infinitely many loops (see Devaney (1989) in the Additional Readings).
While the Julia set is drawn in the complex plane, the Mandelbrot set is constructed in the two-dimensional parameter spacec=(a, b)=a+bi. It is constructed as follows.
Starting from the initial valuez0=0=(0,0)one searches Eq. (18.12) for parameter val- uescso that the iterated{zn}do not diverge to∞. Each color outside the fractal boundary of the Mandelbrot set represents a given number of iterationsm, say, needed for thezn
to go beyond a specified absolute (real) value R,|zm|> R >|zm−1|. For real parame- ter valuec=a, the resulting map,xn+1=x2n+a, is equivalent to the logistic map with period-doubling bifurcations (see Section 18.2) asa increases on the real axis inside the Mandelbrot set.
Exercises
18.3.1 Use a programmable pocket calculator (or a personal computer with BASIC or FOR- TRAN or symbolic software such as Mathematica or Maple) to obtain the iteratesxi of an initial 0< x0<1 andfà′(xi)for the logistic map. Then calculate the Lyapunov exponent for cycles of period 2,3, . . .of the logistic map for 2< à <3.7. Show that forà < à∞the Lyapunov exponentλis 0 at bifurcation points and negative elsewhere, while forà > à∞it is positive except in periodic windows.
Hint. See Fig. 9.3 of Hilborn (1994) in the Additional Readings.
18.3.2 Consider the mapxn+1=F (xn)with F (x)= +
a+bx, x <1, c+dx, x >1,
forb >0 andd <0. Show that its Lyapunov exponent is positive whenb >1, d <−1.
Plot a few iterations in the(xn+1, xn)plane.