Imperfect soft Duffing-van der Pol oscillator ($\ddot{v}=-\lambda v+2\epsilon v^2+v^3+(\kappa -v^2)\dot{v}$)

The equilibria are $0$ and $-\epsilon \pm \sqrt{{\epsilon }^2+\lambda }$. Their linear trace is $\kappa -v_{\ast }^2$ and determinant $\lambda -4\epsilon v_{\ast }-3v_{\ast }^2$. For nonzero $\epsilon$ the symmetry-protected pitchfork bifurcation splits into a transcritical bifurcation at $\lambda =0$ and a saddle-node bifurcation at $\lambda =-{\epsilon }^2$. The energy $H={\dot{v}}^2/2+\lambda v^2/2-2\epsilon v^3/3-v^4/4$ obeys $\dot{H}=(\kappa -v^2){\dot{v}}^2$. For positive $\lambda$, the two outer saddle equilibria have unequal barrier heights, so the symmetric heteroclinic cycle unfolds into separated one-way connections and a homoclinic orbit termination of the central stable limit cycle.