Past exam of the topics in applied mathematics course of the University of Oxford / 2010 / B / 4 / i / Solution

Let $F(x,y)=(y,x-\Omega y-x^3-x^{2}y)$. Every term changes sign when both variables change sign, so $F(-x,-y)=-F(x,y)$. Consequently the derivative of $(-x(t),-y(t))$ equals $F(-x(t),-y(t))$, proving the stated rotational symmetry.

This is equivariance under a half-turn, not time reversibility. Reversibility of an autonomous differential equation requires a reverser $R$ satisfying $F(Rz)=-DR(z)F(z)$, so that $Rz(-t)$ is a solution. The half-turn $R=-I$ would require $F(-z)=F(z)$, the opposite of what holds. The usual mechanical candidate $R(x,y)=(x,-y)$ also fails when damping is present: its second equation would require $(\Omega +x^2)y=-(\Omega +x^2)y$ identically. The symmetry relates two orbits with the same direction of time, and does not reverse their arrows.