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Deterministic chaos sounds like an oxymoron but it is a well defined mathematical concept. Deterministic means that there are exact rules that govern the evolution and chaos here means only that it is impossible to predict the outcome in the long run. The key to the resolution of this paradox is the practical limitation that is always present in our ability to process information. For chaotic systems the resources needed to make a prediction grow extremely fast with the extent of time for which the prediction is to be valid.

This chapter continues the discussion of nonlinear dynamics in plasmas by exploring nonlinear particle interactions with coherent waves, including soliton wave-wave interactions (WWI) and chaotic dynamics. It turns to results from computational physics which draws from examples developed for, and carried over from, solved problems in nonlinear physics and engineering of the past and present. These results are helpful in understanding solitons and deterministic chaos. The study here can be separated in two major parts, with the first part giving the treatment of a variety of simple models, each accounting for specific nonlinear plasma manifestations in both wave-particle interactions (WPI) and WWI. Also included is an analysis of nonlaminarity in boundary dynamics and a general and detailed derivation of the Manley–Rowe relations. The second part addresses two specific nonlinear evolutions in plasmas—solitons in nonlinear waves and WWI, and deterministic chaos in WPI.

The majority of experiments in neurophysiology are based upon spike trains recorded from individual or multiple nerve cells. If all the action potentials are taken to be identical and only the times at which they are generated are considered, the experimentalist obtains a discrete series of time events {t1,···, tn}, where t¡ corresponds to the occurrence of the i th spike, characterizing the spike train. This spike train is transmitted down the axon to all of the target cells of the neuron, and it is this spike train that contains all of the relevant information that the cell is representing (assuming no dendro-dendritic connections). As alluded to in the preceding chapter, there are two opposing views of neuronal coding, with many interim shades. One view holds that it is the firing rate, averaged over a suitable temporal window (Eqs. 14.1 or 14.2), that is relevant for information processing. The dissenting view, correlation coding, argues that the interactions among spikes, at the single cell as well as between multiple cells, encodes information. A key property of spike trains is their seemingly stochastic or random nature, quite in contrast to switching in digital computers. This randomness is apparent in the highly irregular discharge pattern of a central neuron to a sensory stimulus whose details are rarely reproducible from one trial to the next. The apparent lack of reproducible spike patterns has been one of the principal arguments in favor of the hypothesis that neurons only care about the firing frequency averaged over very long time windows. Such a mean rate code is very robust to “sloppy” hardware but is also relatively inefficient in terms of transmitting the maximal amount of information per spike. Encoding information in the intervals between spikes is obviously much more efficient, in particular if correlated across multiple neurons. Such a scheme does place a premium on postsynaptic neurons that can somehow decode this information. Because little or no information can be encoded into a stream of regularly spaced action potentials, this raises the question of how variable neuronal firing really is.

This chapter introduces the reader to canonical perturbation theory as a tool for studying near-integrable systems. Many problems in physics and chemistry do not have exact analytical solutions; these systems are in direct opposition to integrable systems and action-angle variables. The chapter starts by considering tiny perturbations to integrable Hamiltonians. Poincaré in 1893 claimed this was the fundamental question of classical mechanics and, fittingly, Hamilton–Jacobi theory is the starting point. The chapter develops Poincaré’s fundamental equation as well as Delaunay’s small divisor problem. Resonant, near–resonant and non-resonant tori are investigated in the context of Poincaré’s theorem and KAM theory is described in detail. Chaos and Poincaré maps are presented before discussing determinism, deterministic chaos and Laplace’s demon.