amplifying property of erbium-doped fibers. By splicing short lengths of such fiber at space intervals significantly shorter than the soliton period, it was possible to transmit soliton pulse trains at multigigabit rates over a distance of 12,000 km. Further increases in transmission rates by factors of 100 may be possible by using sliding frequency filters and multiplexing. The last chapter in this collection of technological breakthroughs may occur in the next decade, with the laying of a soliton-based transoceanic cable.
The potential applications of soliton signal processing appear almost unlimited. Following rapidly on the success in fiber optics, the possibility arose of soliton-based nonlinear switches, the key element of an all-optical digital computer. Further, fiber lasers and extended cavities utilizing fibers are being developed to produce mode-locked pulse trains (soliton laser, modulation instability laser, fiber Raman laser, additive pulse mode locking).
The focused goals of the research on soliton telecommunications in the United States, Europe, and Japan have had remarkable feedback in basic nonlinear science as well. The optical fiber became the ultimate analog computer for testing and verifying nonlinear theories (e.g., dark-bright soliton interactions, soliton self-frequency shift, soliton couplers, and polarization switches). The fiber produced the first quantitatively verifiable testbed for nonlinear theories by allowing a clean separation of space and time scales for nonlinear interactions.
The hallmark of deterministic chaos is extreme sensitivity to initial conditions. This characteristic, sometimes called the butterfly effect, makes long-range prediction impossible. In the past few years, a variety of techniques, particularly those of Huebler and of Ott, Grebogi, and Yorke (OGY),4 that build on classical control theory have been developed to exploit this sensitivity in order to control chaos. The OGY technique, for instance, uses small feedback perturbations based on the observed dynamics of a chaotic trajectory and therefore does not require any a priori knowledge of the underlying system structure. Intuitively speaking, a chaotic trajectory can be viewed as comprising an infinite number of (unstable) periodic orbits, each of which is visited repeatedly. Since the butterfly effect causes the chaotic trajectory to depart quickly from any periodic orbit it visits, the time that a trajectory spends near a given orbit tends to be short. However, eventually the system's trajectory returns (for another short visit) to a given orbit along a characteristic path known as the orbit's stable manifold. The OGY technique stabilizes a chaotic system about a chosen periodic orbit by making small, time-dependent perturbations to an accessible system parameter such that the trajectory is directed toward the orbit's stable manifold. The result is that the system is maintained close to the previously unstable periodic orbit—that is, the periodic orbit is effectively stabilized. The OGY technique and its derivatives have demonstrated some success in controlling magnetoelastic ribbons, electronic circuits, thermal convection, lasers, and chemical reactions.
Recently, additional improvements in chaotic control have made further applications possible. Targeting overcomes one important limitation of the OGY technique: namely, because OGY control is not effective until the system's trajectory comes close to the desired unstable periodic orbit, initiation of control sometimes requires a long period. By utilizing the exponential growth of small parameter perturbations to steer the system into the neighborhood of the desired orbit, targeting can reduce this waiting period significantly. It is important to note, however, that targeting can be difficult to apply in certain experimental settings since it requires a global model of the system. Nonetheless, it has proved very effective in a variety of cases. A second modification of the OGY technique, proportional perturbation feedback (PPF), can be applied to systems that lack accessible systemwide parameters. The PPF method perturbs a system variable (not a