  To follow up recent work of Xiao-Song Yang on the Nos\'e-Hoover oscillator we
consider Dettmann's harmonic oscillator, which relates Yang's ideas directly to
Hamiltonian mechanics. We also use the Hoover-Holian oscillator to relate our
mechanical studies to Gibbs' statistical mechanics. All three oscillators are
described by a coordinate $q$ and a momentum $p$. Additional control variables
$(\zeta, \xi)$ vary the energy. Dettmann's description includes a time-scaling
variable $s$, as does Nos\'e's original work. Time scaling controls the rates
at which the $(q,p,\zeta)$ variables change. The ergodic Hoover-Holian
oscillator provides the stationary Gibbsian probability density for the
time-scaling variable $s$. Yang considered {\it qualitative} features of
Nos\'e-Hoover dynamics. He showed that longtime Nos\'e-Hoover trajectories
change energy, repeatedly crossing the $\zeta = 0$ plane. We use moments of the
motion equations to give two new, different, and brief proofs of Yang's
long-time limiting result.
