Nose-Hoover attractor
The Nosé-Hoover attractor comes from a thermostat used in molecular dynamics: a spring-like oscillator coupled to a variable that nudges its energy toward a set temperature.
Equations
Read x as a position and y as a momentum: without the yz term this is a plain harmonic oscillator. The third variable is the thermostat. It grows when y2 is below 1 and shrinks when it is above, and through yz it speeds the oscillator up or slows it down to hold the average of y2 at 1.
The divergence of the flow is z. Along a bounded orbit z averages to zero, because d/dt of (x2 + y2 + z2)⁄2 is exactly z. So unlike the other eight, it does not shrink volume on average, and strictly speaking it has no attractor. The name is used loosely.
History
Shuichi Nosé introduced the thermostat in 1984 and William Hoover reformulated it in 1985, as a way to hold a molecular-dynamics simulation at a fixed temperature. J. C. Sprott listed this oscillator as case A in his 1994 survey of simple chaotic flows.
What to look for
Smooth loops that look orderly for a long time. Some starting points wander through a chaotic region; others circle a smooth tube indefinitely. Chaos Studies opens on this attractor, because its motion is easy to follow.