= Imperfect soft Duffing-van der Pol oscillator
{title2=$\ddot v=-\lambda v+2\varepsilon v^2+v^3+(\kappa-v^2)\dot v$}
The <imperfect soft Duffing-van der Pol oscillator>
$$
\ddot v=-\lambda v+2\varepsilon v^2+v^3+(\kappa-v^2)\dot v
$$
combines a softening restoring force, velocity-dependent <damping> and the symmetry-breaking term $2\varepsilon v^2$. The <equilibrium points> are $v=0$ and $v=-\varepsilon\pm\sqrt{\varepsilon^2+\lambda}$, whenever the latter are real. The associated <Jacobian matrix> has <trace> $\kappa-v_*^2$ and <determinant> $\lambda-4\varepsilon v_*-3v_*^2$.
At $\varepsilon=0$, the equilibrium equation has the reflection symmetry $v\mapsto-v$. For $\varepsilon\ne0$, the branch near zero crosses the zero branch at $\lambda=0$, giving the generic <transcritical bifurcation> when $\kappa\ne0$. The two nonzero roots meet at $\lambda=-\varepsilon^2$, $v=-\varepsilon$, giving a generic <saddle-node bifurcation> when $\kappa\ne\varepsilon^2$. The excluded trace-zero cases have an additional linear degeneracy and require further analysis.
The mechanical <energy>
$$
H=\frac12\dot v^2+\frac\lambda2v^2-\frac{2\varepsilon}{3}v^3-\frac14v^4
$$
has the exact derivative $\dot H=(\kappa-v^2)\dot v^2$. Thus the sign of energy input can change with displacement even at a fixed value of $\kappa$.
For $\lambda>0$, the two outer equilibria are <saddle equilibria>. Their potential barriers are generally unequal when $\varepsilon\ne0$, so a symmetric saddle connection can split into distinct directed connections under an imperfection. The <weak-damping heteroclinic splitting of a tilted quartic oscillator> calculates their leading parameter curves. Local equilibrium formulas alone do not guarantee a <limit cycle> or determine its global termination.
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