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In the idealized limit of no dissipative energy loss, these solitons propagate without degradation of shape; they are indeed the natural stable, localized modes for propagation in the fiber. An intrinsically nonlinear characteristic of this soliton, shown explicitly in Eq. 13, is the relation between its amplitude (hence its energy) and its width. In real fibers, where dissipative mechanisms cause solitons to lose energy, the individual soliton pulses therefore broaden (but do not disperse). Thus, to maintain the separation between solitons necessary for the integrity of the signal, one must add optical amplifiers, based on stimulated Raman amplification, to compensate for the loss.

Theoretical numerical studies suggest that the amplification can be done very effectively. An all-optical system with amplifier spacings of 30 to 50 kilometers and

Raman pump power levels less than 100 milliwatts can use solitons of 20 picoseconds duration to send information at a bit rate of over 10 gigahertz. This is more than an order of magnitude greater than the rate anticipated for conventional (linear) systems. Although laboratory experiments have confirmed some of these results, full engineering studies have yet to be carried out. In addition, a critical and still unresolved issue is the relative cost of the repeaters supporting the linear system versus that of the amplifiers in the soliton-based approach. Nonetheless, the prospects for using optical solitons in long-distance communication are exciting and real.

Coherent Structures. Thus far our discussion of the coherent-structure paradigm has focused almost exclusively on solitons. Although this emphasis correctly indicates both the tremendous interest and the substantial progress to which this aspect of the paradigm has led, it obscures the much broader role that nonsoliton coherent structures play in nonlinear phenomena. Vortices in fluids, chemical-reaction waves and nonlinear diffusion fronts, shock waves, dislocations in metals, and bubbles and droplets can all usefully be viewed as instances of coherent structures. As in the case of the solitons, the existence of these structures results from a delicate balance of nonlinear and dispersive forces.

In contrast to solitons, however, these more general coherent structures typically interact strongly and do not necessarily maintain their form or even their separate identities for all times. Fluid vortices may merge to form a single coherent structure equivalent to a single larger vortex. Interactions among shock waves lead to diffraction patterns of incident, reflected, and transmitted shocks. Droplets and bubbles can interact through merging or splitting. Despite these nontrivial interactions, the coherent structures can be the nonlinear modes in which the dynamics is naturally described, and they may dominate the long-time behavior of the system. To exemplify more concretely the essential role of these general coherent structures in nonlinear systems, let me focus on two broad classes of such structures: vortices and fronts.

The importance of vortices in complicated fluid flows and turbulence has been appreciated since ancient times. The giant Red Spot (Fig. 3a) is a well-known example of a fluid vortex, as are tornados in the earth's atmosphere, large ocean circulation patterns called "modons" in the Gulf Stream current, and "rotons" in liquid helium. In terms of practical applications, the vortex pattern formed by a moving airfoil is immensely important. Not only does this pattern of vortices affect the fuel efficiency and performance of the aircraft, but it also governs the allowed spacing between planes at takeoff and landing. More generally, vortices are the coherent structures that make up the turbulent boundary layer on the surfaces of wings or other objects moving through fluids. Further, methods based on idealized point vortices provide an important approach to the numerical simulation of certain fluid flows.

The existence of fronts as coherent structures provides yet another illustration of the essential role of nonlinearity in the physical world. Linear diffusion equations cannot support wave-like solutions. In the presence of nonlinearity, however, diffusion equations can have traveling wave solutions, with the propagating wave front representing a transition from one state of the system to another. Thus, for example, chemical reaction-diffusion systems can have traveling wave fronts separating reacted and unreacted species. Often, as in flame fronts or in internal combustion engines, these traveling chemical waves are coupled with fluid modes as well. Concentration fronts arise in the leaching of minerals from ore beds. Moving fronts between infected and non-infected individuals can be identified in the epidemiology of diseases such as rabies. In advanced oil recovery processes, (unstable) fronts between the injected water and the oil trapped in the reservoir control the effectiveness of the recovery process.

Given their ubiquity and obvious importance in nonlinear phenomena, it is gratifying that recent years have witnessed remarkable progress in understanding and modeling these general coherent structures. Significantly, this progress has been achieved by pre

cisely the synergy among computation, theory, and experiment that we have argued characterizes nonlinear science. Further, as a consequence of this progress, coherent structures and solitons have emerged as an essential paradigm of nonlinear science, providing a unifying concept and an associated methodology at the theoretical, computational, and experimental levels. The importance of this paradigm for technological applications, as well as its inherent interest for fundamental science, will guarantee its central role in all future research in this subject.

Deterministic Chaos and Fractals

Deterministic chaos is the term applied to the aperiodic, irregular, unpredictable, random behavior that in the past two decades has been observed in an incredible variety of nonlinear systems, both mathematical and natural. Although the processes are strictly deterministic and all forces are known, the long-time behavior defies prediction and is as random as a coin toss.

That a system governed by deterministic laws can exhibit effectively random behavior runs directly counter to our normal intuition. Perhaps it is because intuition is inherently “linear;” indeed, deterministic chaos cannot occur in linear systems. More likely, it is because of our deeply ingrained view of a clockwork universe, a view that in the West was forcefully stated by the great French mathematician and natural philosopher Laplace. If one could know the positions and velocities of all the particles in the universe and the nature of all the forces among them, then one could chart the course of the universe for all time. In short, from exact knowledge of the initial state (and the forces) comes an exact knowledge of the final state. In Newtonian mechanics this belief is true, and to avoid any possible confusion, I stress that we are considering only dynamical systems obeying classical, Newtonian mechanics. Subsequent remarks have nothing to do with "uncertainties" caused by quantum mechanics.

However, in the real world exact knowledge of the initial state is not achievable. No matter how accurately the velocity of a particular particle is measured, one can demand that it be measured more accurately. Although we may, in general, recognize our inability to have such exact knowledge, we typically assume that if the initial conditions of two separate experiments are almost the same, the final conditions will be almost the same. For most smoothly behaved, "normal" systems this assumption is correct. But for certain nonlinear systems it is false, and deterministic chaos is the result.

At the turn of this century, Henri Poincaré, another great French mathematician and natural philosopher, understood this possibility very precisely and wrote (as translated in Science and Method):

"A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that that effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the later. Prediction becomes impossible, and we have the fortuitous phenomenon."

Despite Poincaré's remarkable insight, deterministic chaos remained virtually unexplored and unknown until the early 1960s. As the ensuing discussion will reveal, the reason for this long hiatus is that chaos defies direct analytic treatment. The seeds

planted by Poincaré could only germinate when the advances in interactive computation made experimental mathematics a reality.

The Logistic Map. One remarkable instance of a successful experimental mathematical study occurred in a nonlinear equation simple enough to explain to an elementary school child or to analyze on a pocket calculator yet subtle enough to capture the essence of a whole class of real world phenomena. It is arguably the simplest model of a system displaying deterministic chaos, and as such has been studied by a host of distinguished researchers, including Ulam, von Neumann, Kac, Metropolis, Stein, May, and Feigenbaum (see “Iteration of Maps, Strange Attractors, and Number theory-An Ulamian Potpourri"). As we shall see, this focus of talent has been fully justified, for the simple model provides remarkable insight into a wealth of nonlinear phenomena. Thus it is a natural place to begin our quantitative study of deterministic chaos.

The model, known as the logistic map, is a discrete-time, dissipative, nonlinear dynamical system. The value of a variable x at time n is mapped to a new value Xn+1 at time n + 1 according to the nonlinear function

Xn+1 = rxn(1 − Xn),

where the control parameter r satisfies 0 < r < 4 and the allowed values-loosely speaking, the phase space of the x are 0 ≤ n ≤ 1. The map is iterated as many times as desired, and one is particularly interested in the behavior as time—that is, n, the number of iterations-approaches infinity. Specifically, if an initial condition is picked at random in the interval (0, 1) and iterated many times, what is its motion after all transients have died out?

The behavior of this nonlinear map depends critically on the control parameter and exhibits in certain regions sudden and dramatic changes in response to small variations in r. These changes, technically called bifurcations, provide a concrete example of our earlier observation that small changes in the parameters of a nonlinear system can produce enormous qualitative differences in the motion.

For 0 < r < 1, the value of x, drops to 0 as n approaches infinity no matter what its inital value. In other words, after the transients disappear, all points in the interval (0, 1) are attracted to the fixed point x* at x = 0. This fixed point is analogous to the fixed point in Fig. 2 at (0 = 0, de/dt = 0) with the very important distinction that the fixed point in the logistic map is an attractor: the dissipative nature of the map causes the "volume" in phase space to collapse to a single point. Such attractors are impossible in Hamiltonian systems, since their motion preserves phase-space volumes (see "Hamiltonian Chaos and Statistical Mechanics"). The mathematical statement of this behavior then is

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We can easily calculate the (linearized) stability of this fixed point by considering how small deviations from it behave under the map. In Eq. 14 we set x = x* + €ŋ and Xn+1 = x* +En+1 and consider only terms linear in ε, and En+1. The resulting equation is

€n+1 = r(1 − 2x*)€n + O(e}),

so that for x = 0, the n's will remain small for all iterations-provided r < 1.

This last comment suggests that something interesting happens as r passes 1, and indeed for 1 < r < 3 we find an attracting fixed point with a value that depends on r. This value is readily calculated, since at a fixed point x = Xn+1 = x*. Substituting this relation into Eq. 14, we find

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Hence as the value of r moves from 1 toward 3, the value of the fixed point x moves from 0 toward 2/3. Notice that the linear stability analysis given above shows that this r-dependent fixed point is stable for 1 < r < 3. Notice also that while x* = 0 is still a fixed point in this region, the linear stability analysis shows that it is unstable. Hence the point x = 0 is now analogous to the unstable fixed point in Fig. 2 at 0 = π, de/dt = 0; the slightest perturbation will drive the solution away from x = 0 to the stable fixed point at x*(r).

A more interesting bifurcation occurs at r = 3. Suddenly, instead of approaching a single fixed point, the long-time solution oscillates between two values: thus the model has an attracting limit cycle of period 2! This limit cycle is the discrete analogue of the closed periodic oscillations shown in the phase plane of the pendulum (Fig. 2), again, of course, subject to the distinction that the logistic-map limit cycle is an attractor.

Although one can still continue analytically at this stage, it is easier to refer to the results of an experimental mathematical simulation (Fig. 5) that depicts the attracting set in the logistic map as a function of r. Here we see clearly the bifurcation to the period-2 limit cycle at r = 3. But more striking, as r moves toward 3.5 and beyond, period-4 and then period-8 limit cycles occur, followed by a region in which the attracting set becomes incredibly complicated. A careful anlysis of the logistic map shows that the period-8 cycle is followed by cycles with periods 16, 32, 64, and so forth. The process continues through all values 2" so that the period approaches infinity. Remarkably, all this activity occurs in the finite region of r below the value re~ 3.57.

Abover the attracting set for many (but not all) values of r shows no periodicity whatsoever. In fact, the set consists of a sequence of points x, that never repeats itself. For these values of r, the simple logistic map exhibits deterministic chaos, and the attracting set-far more complex than the fixed points and limit cycles seen below re—is called a strange attractor. Beyond the critical value r, the logistic map exhibits a transition to chaos.

Although this complicated, aperiodic behavior clearly motivates the name "chaos," does it also have the crucial feature of sensitive dependence on initial conditions that we argued was necessary for the long-time behavior to be as random as a coin toss? To study this question, one must observe how two initially nearby points separate as they are iterated by the map. Technically, this can be done by computing the Lyapunov exponent X. A value of › greater than 0 indicates that the nearby initial points separate exponentially (at a rate determined by X). If we plot the Lyapunov exponent as a function of the control parameter (Fig. 6), we see that the chaotic regions do have > > 0 and, moreover, the periodic windows in Fig. 5 that exist above r, correspond to regions where <0. That such a filigree of interwoven regions of periodic and chaotic motion can be produced by a simple quadratically nonlinear map is indeed remarkable.

In view of the complexity of the attracting sets above rẹ, it is not at all surprising that this model, like the typical problem in chaotic dynamics, defied direct analytic approaches. There is, however, one elegant analytic result-made all the more relevant here by its having been discovered by Ulam and von Neumann-that further exemplifies the sensitive dependence that characterizes deterministic chaos.

For the particular value r = 4, if we let xn = sin2 0, the logistic map can be


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