Categories of Conservative Flows

In this note, we define four main categories of conservative flows: (a) those in which the dissipation is identically zero, (b) those in which the dissipation depends on the state of the system and is zero on average as a consequence of the orbits being bounded, (c) those in which the dissipation depends on the state of the system and is zero on average, but for which the orbit need not be bounded and a different proof is required, and (d) those in which the dissipation depends on the initial conditions and cannot be determined from the equations alone. We introduce a new 3D conservative jerk flow to serve as an example of the first two categories and show what might be the simplest examples for each category. Also, we categorize some of the existing known systems according to these definitions.