Metamagical Themas, page 49
In short, as λ approaches λc, at special A-values predicted by Feigenbaum's constant 8, f's attractor doubles in population, and its increasingly many elements are geometrically arranged on the x-axis according to a simple recursive plan, the main determining parameter of which is Feigenbaum's other constant, a.
Then for λ beyond λc -called the chaotic regime-the results of iterating f can, for some seed values, yield orbits that converge to no finite attractor. These are aperiodic orbits. For most seed values, the orbit will remain periodic, but the periodicity will be very hard to detect. First of all, the period will be extremely high. Secondly, the orbit will be much more chaotic than before. A typical periodic orbit, instead of quickly converging to a geometrically simple attractor, will meander all over the interval [0,1 ], and its behavior will appear indistinguishable from total chaos. Such behavior is termed ergodic. Furthermore, neighboring seeds may, within a very small number of iterations, give rise to utterly different orbits. In short, a statistical view of the phenomena becomes considerably more reasonable beyond λc.
Figure 16-6 beautifully portrays the period-doubling route to chaos, as well as what happens after you've gotten there. The bifurcations are clear to the eye, and since the horizontal distance from each set of them to the next shrinks geometrically, the onset of chaos at λ is plainly visible. But the regularity of the structure to the right of λc,-that is, in the chaotic regime -is quite unexpected. It is certain that there are many deep mathematical secrets locked up in this elegant graph.
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Now, what do such novel concepts as the iteration of folded functions, period doubling, chaotic regime, and so on have to do with the study of turbulence in hydrodynamic flow, the erratic population fluctuations in predator-prey relations, and the instability of laser modes? The basic idea is embedded in the contrast between laminar flow and turbulent flow. In a peacefully flowing fluid, the flow is laminar-a soft and gentle word that means that all the molecules in the fluid are moving like cars on a multilane freeway. The key features are: (1) that each car follows the same path as its predecessor, and (2) that two nearby cars, whether they are in the same lane or in different ones, will, as time passes, slowly separate from each other essentially in proportion to the difference in their velocities-which is tb say, linearly. These features also apply to molecules of fluid in laminar flow; there, the lanes are called streamlines or laminas.
By contrast, when a fluid is churned up by some external force, this smooth behavior turns into turbulent behavior, as is seen in breakers at the beach and cream being stirred into coffee. Even the word "turbulent" .sounds much harsher and more angular than the soft word "laminar". Here, the image of a multilane freeway no longer holds; the streamlines separate from each other and tangle in the most convoluted of ways, as shown in Figure 16-7. In such systems there are eddies and vortices and all sorts of unnamable whorls on many size-scales at once, and consequently, two points that were initially very close may soon wind up in totally different regions of the fluid. Such quickly diverging paths are the hallmark of turbulence. The distance between points can increase exponentially with time, instead of just linearly, and the coefficient of time in the exponent is called the Lyapunov number. When one speaks of chaos in turbulent flow, it is this rapid, nearly unpredictable separation of neighbors that is meant. Such behavior is strikingly reminiscent of the rapid separation, in the chaotic regime of X, of two orbits whose seeds might originally have been very close together.
FIGURE 16-7. Showing the approach to turbulence. In the upper two pictures, a rod was drawn through a viscous liquid once, setting up trains of vortices behind it. In the lower two, the rod was drawn more than once, and the forms are therefore more complicated and recursive seeming. It is provocative to compare this figure with Figure 13-4. [From Sensitive Chaos, by Theodor Schwenk.)
This suggests that the "scenario" (as it is called) by which pretty, periodic orbits gradually give way to the messy, chaotic orbits of our parabolic function might conceivably be mathematically identical to the scenario underlying the transition to turbulence in a fluid or other system. Exactly how this connection is established, though, requires some more detailed setting of context. In particular, we must briefly consider how the spatio-temporal flow of a fluid or some other entity, such as population density or money, is mathematically modeled.
In such real-world problems, the most successful equations yet found to model the phenomena are differential equations. A differential equation connects the continuous rate of variation of some quantity to that quantity's current size and the current sizes of other quantities. Moreover, the time variable is itself continuous, not jerking from one discrete instant to the next as some strange clocks and watches occasionally do, but indivisibly flowing, like a liquid. One way to visualize the patterns defined by differential equations is to imagine a multidimensional space-it could have thousands of dimensions, or merely a few-in which a point is continuously tracing out a curve. At any one moment, the single point contains all the information about the state of the physical system. Its projections along the various axes give the values of all the relevant quantities that pin down a unique state. Clearly the space-called phase space-would need to have an enormous number of dimensions for a mere point to store the entire shape of a wave breaking on a beach. On the other hand, in a simple predator-prey relation, only two dimensions suffice: one coordinate, say x, giving the predator population and the other, say y, giving the prey population. Two dimensions are more easily visualized, and so we will stick with that case for the time being. The ideas generalize, however, to higher-dimensional cases.
As time progresses, x and y determine each other in an intertwined manner. For example, a large population of predators will tend to reduce the population of prey, whereas a small population of prey will tend to reduce the population of predators. In such a system, x and y constitute a single point (x,y) that swirls around smoothly in a continuous orbit on the plane. (Here the sense of "orbit" is different from the preceding one-that of the discrete, or jumping, orbits we saw when our parabolic function was iterated.) One such possible orbit appears in Figure 16-8; it is generated by a differential equation called "Duffing's equation". It looks like the path of a buzzing fly in your bedroom-or rather, it looks like the shadow of the fly's path on a wall. As a matter of fact, this self-intersecting two-dimensional curve is the shadow of a non-self-intersecting three-dimensional curve. The motion of a point in phase space must always be non-self-intersecting. This arises from the fact that a point in phase space representing the state of a system encodes all the information about the system, including its future history, so that there cannot be two different pathways leading out of one and the same point.
In particular, in Duffing's equation there is a third variable, z, that I have not mentioned so far. If you think of x and y as representing predator and prey populations, then you can think of z. as representing a periodically varying external influence, such as the sun's azimuth or the amount of snow on the ground. Now, if you will allow me to mix my buzzing-fly image with the predator-prey example, imagine a bedroom with a fly buzzing periodically back and forth between two walls. Let us say it takes the fly a year to cross the room and come back. (Perhaps it is a rather large bedroom, or maybe just a slow fly.) In any case, as the fly flies, its shadow on one of the two walls traces out the curve shown in Figure 16-8a. If the fly ever chances to come back to a point in the room which it has passed through before, it is doomed to loop forever, following the path it took the preceding time over and over again. This gives you a picture of the continuous orbit of a point in phase space representing the state of dynamic system controlled by differential equations.
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FIGURE 16-8. If values of x and y mutually determine each other according to Duffing's equation as time passes, then the point (x,y) will trace out a curve (a). If a strobe light blinks periodically and shows (x,y) at selected instants, then a group of isolated points will start appearing as in (b), and gradually filling out a region of their own-a Poincare map.
Now suppose we wanted to establish some connection of these systems to discrete orbits. How might we do so? Well, the values of x, y, and z need not be watched at all moments; they can be sampled periodically, at some natural frequency. In the case of animal populations, a year is the obvious natural period. The sun's azimuth is exactly periodic, and the weather at least tries to repeat itself a year later. Thus a natural sequence of discrete points (x,, y,, z, ), (x2, y2, z2), . . . can be singled out-one per year. It is as if a strobe light blinked regularly and froze the fly on special annual occasions-perhaps at midnight every Halloween. Or you can think of a firefly that flashes on for just a split second once every year. At all other times its peregrinations around the room are unseen. Figure 16-8b shows a sequence of discrete points along the fly-path's shadow, marked by numbers telling when they occurred. Gradually, as many "years" elapse, enough of these discrete points will accumulate that they will start to form a, recognizable shape of their own. This pattern of points is a discrete "orbit", and so it is closely related to the discrete orbits defined by the iteration of our parabola f(x). In that parabolic case, we had a simple one-dimensional recurrence relation (or an iteration):
xn+l=f(xn)
Here we have a two-dimensional recurrence:
xn+i =fl(xn,yn)
Yn+I =f2(xn,yn)
This is a system of coupled recurrence relations, in which output values of the nth generation (xn,yn)are fed right back into f, and f2 as new inputs, to produce the n + 1st generation. On and on it goes, generation after generation. In higher-dimensional cases, of course, there are more such '~ equations. Nevertheless, the skeleton of all these systems remains the same: a multidimensional point (xn,yn, z) jumps from one discrete location in phase space to another, as a discrete variable, n, representing time jumping ahead in discrete units, is incremented.
Notice that we have finessed our way around the continuous time variable that is involved in differential equations. We have done it by focusing on the' way the point is connected to its predecessor one "year" earlier (or whatever natural period is involved). But is there always a "natural period" at which to look at a system of mutually intertwined differential equations? Not always. In some situations, however, there is, and this happens to be the case in all situations where turbulent behavior occurs.
Why is this so? All systems that exhibit turbulent behavior are dissipative, which means that they dissipate, or degrade, energy from more usable forms such as electricity into the less usable form of heat. In the case of hydrodynamic flow, this dissipation is caused by friction, and in the other systems we have been considering, by abstract analogues of friction. A familiar consequence of friction is that objects in motion will grind to a halt unless energy is pumped in. Now if we "drive" a dissipative system with a periodic driving force (you can imagine, for example, stirring a cup of coffee with a spoon in a periodic, circular way), then, of course, the system will not grind to a halt; it will head for some kind of steady state. Such a steady state is a stable orbit-or in our terms, an attractor in phase space. And since we have driven the system with a periodic spoon, we have defined a natural frequency at which to flash our strobe light and freeze the system's statenamely, each time the spoon comes swinging around and passes some fixed mark on the cup, such as its handle. This will constitute our "year". In this way, continuous time can be replaced by a series of discrete instants, as long as we are dealing with a dissipative system driven by a periodic force. And so continuous orbits can be replaced by discrete orbits, which brings iteration back into the picture.
If the driving force itself has no natural period (it may be simply a constant force), there is still a way to define a natural period, as long as some variable in the system swings back and forth between extremes. Just flash your strobe whenever that variable hits its extreme value, and the fly will still be caught at discrete instants. This type of discrete representation of the fly's motion in a multidimensional space is called a Poincare map.
This stirring argument is only hand-waving, of course, and needs much more rigor to be convincing to a mathematician. It nonetheless gives the flavor of how the study of a set of coupled differential equations can be replaced by the study of a set of coupled discrete recurrence relations. This is the vital step that brings us back to the recent discoveries about the parabola.
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In 1975, Feigenbaum discovered that his numbers a and S do not depend on the details of the shape of the curve defined by f(x). Almost any smooth convex shape that peaks in the same spot will do as well. Inspired by the structural universality discovered by Metropolis, Stein, and Stein, Feigenbaum tried working with a sine curve instead of a parabola. He was flabbergasted by the reappearance of the same numerical values, to many decimal places, of the numbers a and S, which had characterized the period-doubling and the onset of chaos for the parabola. For the sine curve just as for the parabola, there is a height-parameter A and a set of special A-values that converge to a critical point Ac. Moreover, the onset of chaos at Ac is governed by the same numbers a and S. Feigenbaum began to suspect that there was something universal going on here. In other words, he suspected that what is more important than f itself is the mere fact that f is being iterated over and over. In fact, he suspected that f itself might play no role in the onset of chaos.
It is not quite that simple, in reality. Feigenbaum soon discovered that what does matter about f is just the nature of the peak at its very center. The long-term behavior of orbits depends only on an infinitesimal segment at the crest of the graph, and ultimately, it depends only on the behavior at the very point where the maximum occurs! The rest of the shape, even the region close to the peak, is irrelevant. A parabola has what is called a quadratic maximum, as do a sine wave, a circle, and an ellipse. In fact, the behavior of a randomly-produced smooth function at a typical maximum would be expected to be of the quadratic type, in the absence of any special coincidences. So the parabolic case, rather than being a quirky exception, begins to seem like the rule. This empirical discovery by Feigenbaum, involving two fundamental scaling factors a and S that characterize the onset of chaos through period-doubling attractors, represents a new kind of universality, known as metrical universality, to distinguish it from the earlier-known structural universality. This empirically demonstrated metrical universality was later proved to be correct (in the more orthodox sense of proof) in the one-dimensional case by Oscar Lanford.
A truly exciting development occurred when Feigenbaum's constants unexpectedly turned up in some messy models of actual physical systems that exhibit turbulence, not just in pretty and idealized mathematical systems. Valter Franceschini of the University of Modena in Italy adapted the Navier-Stokes equation, which governs all hydrodynamic flow, for computer simulation. To do so, he turned it into a set of five coupled differential equations whose Poincare maps he could then study numerically on his computer. He first found that the system exhibited attractors with repeated period-doubling as its governing parameters approached the values where turbulence was expected to set in. Unaware of Feigenbaum's work, he showed his results to Jean-Pierre Eckmann of the University of Geneva, who immediately urged him to go back and determine the rate of convergence of the A-values at which period-doubling occurred. To their amazement, Feigenbaum's a- and 8-values-accurate to about four decimal places-appeared seemingly out of nowhere! For the first time, an accurate mathematical model of true physical turbulence revealed that its structure was intimately related to the humble chaos lurking in the humble parabola y=4Ax(1-x). Subsequently, Eckmann, Pierre Collet, and H. Koch showed that in the behavior of a multidimensional driven dissipative system, all 'dimensions but one tend to drop out after a sufficiently long period of time, and so one should expect the characteristic of one-dimensional behaviour namely Feigenbaum's metrical universality-to reappear.
Since then, experimentalists have been keeping their eyes peeled for period-doubling behavior in actual physical systems (not just in computer models). Such behavior has been observed in certain types of convective flow, but so far the measurements are too imprecise to lend very strong support to the idea that the parabola contains the clues revealing the nature of genuine physical turbulence. Still, it is tantalizing to think that somehow, all that really matters is that a dissipative set of coupled recurrence relations is being iterated-but that the detailed properties of those recurrences can be entirely ignored if one is concentrating on understanding the route to turbulence.
Feigenbaum puts it this way. One often sees a pattern of clouds in the sky -a celestial trellis composed of a myriad of small white puffs stretching from horizon to horizon-that clearly did not happen "by accident". Some systematic hydrodynamic law has got to be operating. Yet, says Feigenbaum, it must be a law operating at a higher level, or on a larger scale, than the Navier-Stokes equation, which is based on infinitesimal volumes of fluid and not on large "chunks". It seems that in order to understand such beautiful sky patterns, one must somehow bypass the details of the Navier-Stokes equation, and come up with some coarser-grained but more relevant way of analyzing hydrodynamic flow. The discovery that iteration gives rise to universality-that is, independence of the details of the function (or functions) being iterated-offers hope that such a view of hydrodynamics may be well on its way to emerging.
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Well, we have covered attractors and turbulence; what about strange attractors? We have now built up the necessary concepts to understand this idea. When a periodically driven two-dimensional (or higher-dimensional) dissipative system is modeled by a set of coupled iterations, the successive points lit up by the flashes of the periodic strobe light trace out a shape that plays the role, for this system, that a simple orbit did for our parabola. But when one is operating in a space of more than one dimension, the possibilities are richer. Certainly it is possible to have a stable fixed point (an attractor of period one). This would just mean that at every flash of the strobe, the point representing the system's state is exactly where it was last time. It is also possible to have a periodic attractor: one where after some finite number of flashes, the point has returned to a preceding position. This would be analogous to the 2-cycles, 4-cycles, and so on that we saw occurring for the parabola.
