In this paper, we formulate Bulgac-Kusnezov constant temperature dynamics in phase space by means of non-Hamiltonian brackets. Two generalized versions of the dynamics are similarly defined, one where the Bulgac-Kusnezov demons are globally controlled by means of a single additional Nosé variable, and another where each demon is coupled to an independent Nosé-Hoover thermostat. Numerically stable and efficient measure-preserving time-reversible algorithms are derived in a systematic way for each case. The chaotic properties of the different phase space flows are numerically illustrated through the paradigmatic example of the one-dimensional harmonic oscillator. It is found that, while the simple Bulgac-Kusnezov thermostat is apparently not ergodic, both of the Nosé-Hoover controlled dynamics sample the canonical distribution correctly.
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