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initial conditions. For example, by writing 0, as a binary number with a finite number of digits as one would in any digital computer-we see that the map amounts to a simple shift operation. When this process is carried out on a real computer, round-off errors replace the right-most bit with garbage after each operation, and each time the map is iterated one bit of information is lost. If the initial condition is known to 48 bits of precision, then after only 48 iterations of the map no information about the initial condition remains. Said another way, despite the completely deterministic nature of the logistic map, the exponential separation of nearby initial conditions means that all long-time information about the motion is encoded in the initial state, whereas none (except for very short times) is encoded in the dynamics.

There is still much more that we can learn from this simple example. One question of obvious interest in nonlinear systems is the mechanism by which such systems move from regular to chaotic motion. In the logistic map, we have seen that this occurs via a period-doubling cascade of bifurcations: that is, by a succession of limit cycles with periods increasing as 2". In a classic contribution to nonlinear science, Mitchell Feigenbaum analyzed the manner in which this cascade occurred. Among his first results was the observation that the values of the parameter r at which the bifurcations occurred converged geometrically: namely, with n defined by

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More important, Feigenbaum was able to show that & did not depend on the details of the logistic map-the function need only have a "generic" maximum, that is, one with a nonvanishing second derivative—and hence & should be universal for all generic maps. Even more, he was able to argue convincingly that whenever a period-doubling cascade of bifurcations is seen in a dissipative dynamical system, the universal number 6, as well as several other universal quantities, should be observed independent of the system's phase-space dimension.

This prediction received dramatic confirmation in an experiment carried out by Albert Libchaber and J. Maurer involving convection in liquid helium at low temperatures. Their observation of the period-doubling cascade and the subsequent extraction of 8 and other universal parameters provided striking proof of universal behavior in nonlinear systems. More recently, similar confirmation has been found in experiments on nonlinear electrical circuits and semiconductor devices and in numerical simulations of the damped, driven pendulum. Further, it is now known rigorously for dissipative systems that the universal behavior of the period-doubling transition to chaos in the logistic map can occur even when the phase-space dimension becomes infinite.

It is important to emphasize that the period-doubling cascade is by no means the only way in which dissipative nonlinear systems move from regular motion to chaos (see, for example, the discussion of the indented trapezoid map on pp. 103– 104). Many other routes—such as quasiperiodic and intermittent-have been identified and universality theories have been developed for some of them. But the conceptual progenitor of all these developments remains the simple logistic map.

Finally, Fig. 5b illustrates one additional obvious feature of the attracting set of Eq. 14: namely, that it contains nontrivial-and, in fact, self-similar-structure under magnification. Indeed, in the mathematical model this self-similar structure occurs on all smaller scales; consequently, Fig. 5b is one example of a class of complex, infinitely ramified geometrical objects called fractals. We shall return to this point later.

The Damped, Driven Pendulum. Armed with the quantitative insight gained from the logistic map, we can confront deterministic chaos in more conventional dynamical

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Fig. 6. A positive value for the Lyapunov expo-
nent (>0) indicates that nearby initial points
separate exponentially, whereas negative val-
ues (< 0) indicate periodic or quasiperiodic
motion. Here the Lyapunov exponent is plot-
ted as a function of the control parameter r
for the logistic map (Fig. 5), and it can be seen
that the periodic windows of Fig. 5 correspond
to regions where <0. (Figure courtesy
of Gottfried Mayer-Kress and Hermann Haken,
Universität Stuttgart, FRG.)

by a thin nonconducting oxide layer. Among the present practical applications of such junctions are high-precision magnetometers and voltage standards. The ability of these Josephson junctions to switch rapidly (tens of picoseconds) and with very low dissipation (less than microwatts) from one current-carrying state to another may provide microcircuit technologies for, say, supercomputers that are more efficient than those based on conventional semiconductors. Hence the nature of the dynamic response of a Josephson junction to the external driving force-the rcos t term in Eq. 4—is a matter of technological, as well as fundamental, interest.

Since analytic techniques are of limited use in the chaotic regime, we demonstrate the existence of chaos in Eq. 4 by relying on graphical results from numerical simulations. Figure 7 illustrates how the phase plane (Fig. 2) of the pendulum is modified when driving and damping forces are included and, in particular, shows how the simple structure involving fixed points and limit cycles is dramatically altered.





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Fig. 5. (a) The attracting set for the logistic map (Eq. 14 in the main text) generated by plotting 300 values of the iterated function (after the transients have died out) for each of 1150 values of the control parameter r. The map has a cycle of period 2 when the control parameter r is at 3.4 (left edge). This cycle quickly "bifurcates" to cycles of periods 4, 8, 16, and so forth as r increases, generating a perioddoubling cascade. Above re≈ 3.57 the map exhibits deterministic chaos interspersed with gaps where periodic motion has returned. For example, cycles of periods 6, 5, and 3 can be seen in the three larger gaps to the right. (b) A magnified region (shown as a small rectangle in (a)) illustrates the self-similar structure that occurs at smaller scales. (Figure courtesy of Roger Eckhardt, Los Alamos National Laboratory.)



Fig. 7. The motion of a damped, periodically driven pendulum (Eq. 4 in the main text) for certain parameter values is chaotic with the attracting set being a "strange attractor." An impression of such motion can be obtained by plotting the position and velocity of the pendulum once every cycle of the driving force (as shown here for a = 0.3, г = 4.5, and

= 0.6 in units with g// = 4). The fact that the image is repeated at higher and lower values of 0 is a result of the pendulum swinging over the top of its pivot point. (Figure courtesy of James Crutchfield, University of California, Berkeley.)

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which shows how the system depends on the three generalized coordinates: 0, pe, and z. Note further that the presence of damping implies that the system is no longer Hamiltonian but rather is dissipative and hence can have attractors.

Analysis of the damped, driven pendulum neatly illustrates two separate but related aspects of chaos: first, the existence of a strange attractor, and second, the presence of several different attracting sets and the resulting extreme sensitivity of the asymptotic motion to the precise initial conditions.

Figure 7 shows one of the attractors of Eq. 22 for the parameter values a = 0.3, T= 4.5, and N = 0.6 (in units with g/l = 4). As in the case of the logistic map, only the attracting set is displayed; the transients are not indicated. To obtain Fig. 7, which

is a plot showing only the phase-plane variables and , one takes a "stroboscopic snapshot" of the motion once during every cycle of the driving force. The complicated attracting set shown in the figure is in fact a strange attractor and describes a neverrepeating, nonperiodic motion in which the pendulum oscillates and flips over its pivot point (hence the repeated images at 2π-multiples of the angle) in an irregular, chaotic manner. The sensitive dependence on initial conditions implies that nearby points on the attractor will separate exponentially in time, following totally different paths asymptotically. Enlargements of small regions of Fig. 7 show a continuation of the intricate structure on all scales; like the attracting set of the logistic map, this strange attractor is a fractal.

To visualize the motion on this attractor, it may be helpful to recall the behavior of an amusing magnetic parlor toy that has recently been quite popular. This device, for which the mathematical model is closely related to the damped, driven pendulum equation, spins first one way and then the other. At first it may seem that one can guess its behavior. But just when one expects it to spin three times to the right and then go to the left, it instead goes four, five, or perhaps six times to the right. The sequence of right and left rotations is unpredictable because the system is undergoing the aperiodic motion characteristic of a strange attractor.

Figure 8 illustrates the important point that the strange attractor of Fig. 7 is not the only attractor that exists for Eq. 22. Specifically, for a = 0.1, r = 7/4, and 2 = 1 (now in units of g/l = 1), the system is attracted to periodic limit cycles of clockwise or counterclockwise motion. Figure 8 demonstates this with another variant of our familiar phase-plane plot in which a color code is used to indicate the long-time behavior of all points in the plane. More precisely, this plot is a map of every initial state (0, 0) onto a "final state" corresponding to one of the attractors. A blue dot is plotted at a point in the plane if the solution that starts from that point at t = 0 is attracted asymptotically to the limit cycle corresponding to clockwise rotation of the pendulum. Similarly, a red dot is plotted for initial points for which the solution asymptotically approaches counterclockwise rotation.

In Fig. 8 we observe large regions in which all the points are colored red and, hence, whose initial conditions lead to counterclockwise rotations. Similarly, there are large blue regions leading to clockwise rotations. In between, however, are regions in which the tiniest change in initial conditions leads to alternations in the limit cycle eventually reached. In fact, if you were to magnify these regions even more, you would see further alternations of blue and red-even at the finest possible level. In other words, in these regions the final state of the pendulum-clockwise or counterclockwise motion-is an incredibly sensitive function of the exact initial point.

There is an important subtlety here that requires comment. For the red and blue regions the asymptotic state of the pendulum does not correspond to chaotic motion, and the two attracting sets are not strange attractors but are rather just the clockwise and counterclockwise rotations that exist as allowed motions even for the free pendulum (Fig. 2). The aspect of chaos that is reflected by the interwoven red and blue regions is the exquisite sensitivity of the final state to minute changes in the initial state. Thus, in regions speckled with intermingled red and blue dots, it is simply impossible to predict the final state because of an incomplete knowledge of initial conditions.

In addition to the dominant red and blue points and regions, Fig. 8 shows much smaller regions colored greenish white and black. These regions correspond to still other attracting limit sets, the greenish-white regions indicating oscillatory limit cycles (no rotation) and the black regions indicating points that eventually go to a strange


From the example of Fig. 8 we learn the important lesson that a nonlinear dissipative system may contain many different attractors, each with its own basin of attraction, or range of initial conditions asymptotically attracted to it. A subtle further consequence of deterministic chaos is that the boundaries between these basins can




Fig. 8. In this variation of the phase plot for the damped, driven pendulum, a blue dot is plotted at a point in the plane if the solution that starts from that point at t = 0 is attracted to clockwise rotation, whereas a red dot represents an attraction to counterclockwise rotation, and a greenish-white dot represents an attraction to any oscillatory limit cycle without rotation. Only a portion of the phase plane is shown. The conditions used to show these limit-cycle attractors are a = 0.1, r = 7/4, and = 1 (in units of g/l = 1). Despite the nonchaotic motion of the limit cycles, sensitive dependence on initial conditions is still quite evident from the presence of extensive regions of intermingled red and blue. Further, the black regions indicate initial conditions for which the limiting orbit is a strange attractor. (Figure courtesy of Celso Grebogi, Edward Ott, James Yorke, and Frank Varosi, University of Maryland.)

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To develop a clearer understanding of these admittedly bizarre objects and the dynamical motions they depict, we turn to another simple nonlinear dynamical model. Known as the Lorenz equations, this model was developed in the early 1960s by Edward Lorenz, a meteorologist who was convinced that the unpredictability of weather forecasting was not due to any external noise or randomness but was in fact compatible with a completely deterministic description. In this sense, Lorenz was attempting to make precise the qualitative insight of Poincaré, who, in another prescient commentall the more remarkable for its occurring in the paragraph immediately following our earlier quotation from Science and Method-observed:

"Why have meteorologists such difficulty in predicting the weather with any certainty?... We see that great disturbances are generally produced in regions where the atmosphere is in unstable equilibrium. The meteorologists see very well that the equilibrium is unstable, that a cyclone will be formed somewhere, but exactly where they are not in a position to say; a tenth of a degree more or less at any given point, and the cyclone will burst here and not there, and extend its ravages over districts it would otherwise have spared. ... Here, again, we find the same contrast between a very trifling cause that is inappreciable to the observer, and considerable effects, that are sometimes terrible disasters."

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