= Imperfect soft Duffing-van der Pol oscillator
{title2=$\ddot v=-\lambda v+2\varepsilon v^2+v^3+(\kappa-v^2)\dot v$}
The <equilibria> are $0$ and $-\varepsilon\pm\sqrt{\varepsilon^2+\lambda}$. Their linear <trace> is $\kappa-v_*^2$ and <determinant> $\lambda-4\varepsilon v_*-3v_*^2$. For nonzero $\varepsilon$ the symmetry-protected <pitchfork bifurcation> splits into a <transcritical bifurcation> at $\lambda=0$ and a <saddle-node bifurcation> at $\lambda=-\varepsilon^2$. The <energy> $H=\dot v^2/2+\lambda v^2/2-2\varepsilon v^3/3-v^4/4$ 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>.
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