= Phase portrait of x dot equals two x times y minus a
{title2=$\dot x=2x(y-a),\quad\dot y=1-x^2-y^2$}
This reflection-symmetric planar system has equilibria $(0,\pm1)$ and, for $|a|\leq1$, $(\pm\sqrt{1-a^2},a)$. Its divergence is the constant $-2a$, excluding periodic orbits when $a\ne0$ by the <Bendixson-Dulac criterion>. At $a=0$ it is Hamiltonian with
$$
H(x,y)=xy^2+\frac{x^3}{3}-x,
$$
and the nonzero equilibria are nonlinear centers. At $a=\pm1$ the symmetric equilibria undergo pitchfork bifurcations.
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