Alessandro Sergi ∗ School of Physics, University of KwaZulu-Natal, Pietermaritzburg, Private Bag X01 Scottsville, 3209 Pietermaritzburg, South Africa Gregory S. Ezra † Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA 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 Nos´ e variable, and another where each demon is coupled to an independent Nos´ e-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 Nos´ e-Hoover controlled dynamics sample the canonical distribution correctly. Abstract arXiv:1002.0657v2 [cond-mat.stat-mech] 22 Feb 2010 $^{∗}$Electronic address: sergi@ukzn.ac.za $^{†}$Electronic address: gse1@cornell.edu 1 Bulgac-Kusnezov-Nos´e-Hoover thermostats October 10, 2018