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Aspects of the Nos\'e and Nos\'e-Hoover dynamics developed in 1983-1984 alongwith Dettmann's closely related dynamics of 1996, are considered. We emphasizeparadoxes associated with Liouville's Theorem. Our account is pedagogical,focused on the harmonic oscillator for simplicity, though exactly the sameideas can be, and have been, applied to manybody systems. Nos\'e,Nos\'e-Hoover, and Dettmann flows were all developed in order to access Gibbs'canonical ensemble directly from molecular dynamics. Unlike Monte Carloalgorithms dynamical flow models are often not ergodic and so can fail toreproduce Gibbs' ensembles. Accordingly we include a discussion of ergodicity,the visiting of all relevant microstates corresponding to the desired ensemble.We consider Lyapunov instability too, the usual mechanism for phase-spacemixing. We show that thermostated harmonic oscillator dynamics can besimultaneously expanding, incompressible, or contracting, depending upon thechosen "phase space". The fractal nature of nonequilibrium flows is alsoillustrated for two simple two-dimensional models, the hard-disk-based GaltonBoard and the time-reversible Baker Map. The simultaneous treatment of flows asone-dimensional and many-dimensional suggests some interesting topologicalproblems for future investigations.

Aspects of Dynamical Simulations, Emphasizing Nos\'e and Nos\'e-Hoover Dynamics and the Compressible Baker Map