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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.

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.