An autonomous differential equation has the form , with no explicit dependence of the vector field on time.
If solves an autonomous differential equation, solves the equation with negated vector field. The trajectories and equilibrium points stay in place but their orientations reverse; stable and unstable directions interchange. A saddle equilibrium remains a saddle, and a clockwise centre equilibrium becomes counterclockwise.
A reversing involution maps a trajectory to another trajectory . The chain rule gives the displayed condition. In contrast, an ordinary equivariance symmetry satisfies and maps trajectories with time unchanged. Oddness expresses half-turn equivariance, not reversal by the half-turn. Damping usually breaks the mechanical candidate .
The phase line of a scalar autonomous differential equation marks the zeros of and the direction of motion between them. The sign of gives the arrows. Under local uniqueness, a nonconstant trajectory cannot cross an equilibrium point; inward arrows indicate attraction and outward arrows repulsion.
