Time reversal of an autonomous differential equation ($\dot{z}=F(z)⟼\dot{z}=-F(z)$)

If $z(t)$ solves an autonomous differential equation, $w(t)=z(-t)$ 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 $R$ maps a trajectory $x(t)$ to another trajectory $Rx(-t)$. The chain rule gives the displayed condition. In contrast, an ordinary equivariance symmetry satisfies $F(Sx)=DS(x)F(x)$ and maps trajectories with time unchanged. Oddness $F(-x)=-F(x)$ expresses half-turn equivariance, not reversal by the half-turn. Damping usually breaks the mechanical candidate $R(x,y)=(x,-y)$.