= Solution
Let $F(x,y)=(y,x-\Omega y-x^3-x^2y)$. 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>.
\b[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.
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