[en] Nosé's pioneering 1984 work inspired a variety of time-reversible deterministic thermostats. Though several groups have developed successful doubly-thermostated models, single-thermostat models have failed to generate Gibbs' canonical distribution for the one-dimensional harmonic oscillator. A 2001 doubly-thermostated model, claimed to be ergodic, has a singly-thermostated version. Though neither of these models is ergodic this work has suggested a successful route toward singly-thermostated ergodicity. We illustrate both ergodicity and its lack for these models using phase-space cross sections and Lyapunov instability as diagnostic tools. - Highlights: • We develop cross-section and Lyapunov methods for diagnosing ergodicity. • We apply these methods to several thermostatted-oscillator problems. • We demonstrate the nonergodicity of previous work. • We find a novel family of ergodic thermostatted-oscillator problems.