Nonlinear Science (1997)

Nonlinear dynamics has had proven impact on emerging and commercialized technologies, from which useful lessons can be extracted on how best to facilitate the transfer from basic research, to advanced development, to novel applied technologies. Two specific examples are presented.

A contending technology for transoceanic communication for both AT&T and NTT is based on soliton transport, a nonlinear phenomena first suggested in 1973. In that year, Hasegawa and Tappert^1,2 predicted that an optical fiber could support both bright and dark temporal soliton pulses, an intrinsically nonlinear phenomenon. At the time, their result was, for the most part, regarded as the satisfaction of an idle mathematical curiosity with few applications possibilities.

In the 1970s, the entire linear fiber optics research community was facing the same challenge of reducing the 20 dB/km losses of the then-best available fibers. The great potential seen in linear fiber optics technology was the driving force behind the huge investment in developing low-loss single-mode fibers.

Ironically, it was the availability of the low-loss fibers developed for linear optics that in 1980 enabled Mollenauer, Stolen, and Gordon³ to test experimentally the earlier prediction of Hasegawa and Tappert for bright solitons. This experiment catalyzed the explosive growth of the field of nonlinear fiber optics.

The key to stable soliton pulses is the balance between weak material or waveguide dispersion and the intrinsic (very fast but weak) optical nonlinearity of the glass core. By operating at wavelengths in the anomalous region, close to the zero dispersion of glass, low-power soliton pulses could be generated over distances of the order of a kilometer. Dispersion-shifted fibers with lower losses were developed rapidly, allowing experimentalists to tune the carrier frequency of the optical soliton pulse and utilize tiny semiconductor laser diodes as light sources. The finite loss in the fiber, although very small, still posed a major impediment to long-haul telecommunications.

Remarkably, another nonlinear phenomenon, called stimulated Raman scattering, provided the critical breakthrough. It was known from bulk nonlinear optics that stimulated scattering involving nonlinear (three-wave) interactions was a common occurrence above some threshold. It was found that the inverse Raman effect could be exploited to provide a regular boost in energy to compensate loss. The idea was to inject a weak continuous wave signal at a frequency corresponding to the Stokes-downshifted Raman frequency to restore the depleted soliton pulse energy. Although the soliton interaction properties could be inferred from inverse scattering theory (IST) or analytical soliton perturbation methods, the actual demonstration relied heavily on computer simulations.

An equally remarkable breakthrough followed from the discovery of the remarkable

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

parameter) so as to force the system trajectory directly onto the stable manifold of a desired unstable periodic orbit. Consequently, the system walks toward the unstable periodic orbit along the stable manifold. Eventually, it is repelled from that orbit, at which point PPF control again intervenes to constrain the system. The PPF technique has been used, for example, to control the interspike intervals of arrhythmic cardiac tissue and spontaneously bursting neuronal networks.