= Solution
Put $w=\dot v$. The planar <vector field> and its <equilibrium> <eigenvalue> data are
$$
\dot v=w,\qquad\dot w=-\lambda v+2\varepsilon v^2+v^3+(\kappa-v^2)w,
$$
$$
v_0=0,\quad v_\pm=-\varepsilon\pm\sqrt{\varepsilon^2+\lambda},\qquad
\operatorname{tr}J=\kappa-v_*^2,\quad\det J=\lambda-4\varepsilon v_*-3v_*^2.
$$
For a nonzero <equilibrium>, use $\lambda=v_*^2+2\varepsilon v_*$ to rewrite the <determinant> as $-2v_*(v_*+\varepsilon)$. These formulas determine the local bifurcations of the <imperfect soft Duffing-van der Pol oscillator> directly.
When $\varepsilon=0$, the origin is a <saddle equilibrium> for $\lambda<0$ and has <determinant> $\lambda$ for $\lambda>0$; the two <equilibria> $\pm\sqrt\lambda$ are <saddle equilibria>. They meet the origin in a symmetry-protected <pitchfork bifurcation> at $\lambda=0$ for $\kappa\ne0$. At $\kappa=0,\lambda>0$ the origin undergoes a <supercritical Hopf bifurcation>. To determine criticality, take $v=r\cos(\sqrt\lambda t)+\cdots$: averaging $\dot H=(\kappa-v^2)w^2$ gives $\dot r=\kappa r/2-r^3/8+\cdots$, so the stable small <limit cycle> has $r^2=4\kappa+\cdots$.
The <potential energy> of the conservative part and the exact <energy> balance are
$$
H=\frac12w^2+\frac12\lambda v^2-\frac23\varepsilon v^3-\frac14v^4,
\qquad\dot H=(\kappa-v^2)w^2.
$$
At zero imperfection the stable <limit cycle> grows to a <heteroclinic cycle> connecting the two outer <saddle equilibria>. Near the double-zero point its leading connection curve can be calculated, rather than inferred just from symmetry. In the <Hamiltonian system> limit the <heteroclinic orbit> is $v=\sqrt\lambda\tanh(\sqrt{\lambda/2}\,t)$, and
$$
\int_{-\infty}^{\infty}w^2dt=\frac{4\lambda^{3/2}}{3\sqrt2},\qquad
\frac{\int v^2w^2dt}{\int w^2dt}=\frac\lambda5.
$$
The weak-damping persistence condition is therefore $\kappa=\lambda/5+o(\lambda)$, not an exact formula at order-one parameters.
Now take $\varepsilon>0$; negative imperfection reflects the phase portrait. The <pitchfork bifurcation> unfolds into \b[a <transcritical bifurcation> crossing at $\lambda=0$ and a <saddle-node bifurcation> at $\lambda=-\varepsilon^2$]. The latter occurs at $v=-\varepsilon$. Between those values the origin and $v_-$ are <saddle equilibria>, while $v_+$ lies in $(-\varepsilon,0)$ and is a sink or source according to the <trace>. At $\lambda=0$, $v_+=\lambda/(2\varepsilon)+\cdots$ exchanges its <saddle equilibrium>/non-saddle character with the origin. The double-zero points are now at $(\lambda,\kappa)=(0,0)$ and $(-\varepsilon^2,\varepsilon^2)$.
The original <Hopf bifurcation> line $\kappa=0,\lambda>0$ remains, still supercritical. A second <Hopf bifurcation> curve occurs on the intermediate nonzero <equilibrium>:
$$
\boxed{\lambda=q^2+2\varepsilon q,\quad\kappa=q^2,\quad-\varepsilon<q<0.}
$$
It is important not to assign the same criticality to its entire length. Write $s=\sqrt{\varepsilon^2+\lambda}$, so $q=-\varepsilon+s$. Around that center the restoring quadratic coefficient is $g_2=-\varepsilon+3s$, the damping linear coefficient is $f_1=2(\varepsilon-s)$, and $\omega^2=2s(\varepsilon-s)$. The <Hopf criticality for an asymmetric Lienard center> calculation gives cubic amplitude coefficient
$$
\frac18\left(-1+\frac{f_1g_2}{\omega^2}\right)=\frac18(2-\varepsilon/s).
$$
Hence it is supercritical for $s<\varepsilon/2$ and subcritical for $s>\varepsilon/2$, with a <generalized Hopf bifurcation> point at $\lambda=-3\varepsilon^2/4,\kappa=\varepsilon^2/4$. Higher-order terms are needed exactly there. For small imperfection the leading quintic coefficient is positive: writing $q_1=v-q$ and using the coordinate that normalizes the <potential energy>, $y=q_1\sqrt{1-aq_1-bq_1^2}$, with $a=2g_2/(3\omega^2)$ and $b=1/(2\omega^2)$, the coefficient of $y^4$ in $(f_1q_1-q_1^2)dq_1/dy$ at vanishing cubic coefficient is $10b/3>0$. Weak-damping averaging therefore supplies the nearby <saddle-node bifurcation> of stable and unstable <limit cycles> in the <generalized Hopf bifurcation> unfolding. None of these small $O(\varepsilon^2)$ features intersects either requested parameter line.
Globally, the two <saddle equilibrium> barriers are no longer equal. Their leading <energy> difference is $H(v_+,0)-H(v_-,0)=-4\varepsilon\lambda^{3/2}/3+\cdots$. The two one-way connections consequently separate; matching their <energy> differences with the connection integral gives
$$
\kappa=\lambda/5-\sqrt2\,\varepsilon+\cdots\quad(v_-\to v_+),\qquad
\kappa=\lambda/5+\sqrt2\,\varepsilon+\cdots\quad(v_+\to v_-).
$$
These <weak-damping heteroclinic splitting of a tilted quartic oscillator> formulas apply for small $\lambda$ with $|\varepsilon|\ll\sqrt\lambda$. Between the separated connection curves, the central stable <limit cycle> terminates in a <homoclinic orbit> to the lower-barrier <saddle equilibrium>, $v_+$ for positive imperfection. The simultaneous symmetric heteroclinic-cycle destruction is therefore replaced by two basin-changing one-way connections and a distinct <homoclinic orbit> loss of the <limit cycle>. Its period diverges at that loss.
Along $\kappa+\lambda=1$, decrease $\lambda$ from above one. The sequence is \b[<supercritical Hopf bifurcation> at $\lambda=1$; first <heteroclinic orbit>; <homoclinic orbit> loss of the stable <limit cycle>; second <heteroclinic orbit>; <transcritical bifurcation> crossing at zero; <saddle-node bifurcation> at $-\varepsilon^2$]. The connection locations are global, not obtained by treating the Melnikov approximation as exact. For illustration, direct stable/unstable manifold matching at $\varepsilon=0.04$ places them at approximately $\lambda=0.882,0.858,0.783$, respectively. The diagram below gives the qualitative ordering without prescribing those example values for arbitrary imperfection.
Along $\kappa+\lambda=-1$, the sequence is simply \b[<transcritical bifurcation> crossing at zero, followed by <saddle-node bifurcation> at $-\varepsilon^2$]. Before the latter, $\kappa<0$ in the relevant region, so $H$ decreases strictly on nonstationary trajectories and there is no <periodic orbit> or <saddle equilibrium> loop. After it only a <saddle equilibrium> remains, which cannot be enclosed by a <periodic orbit> by the <Poincare-index obstruction to a periodic orbit>. The origin does not undergo a <Hopf bifurcation> at $\lambda=-1$, since its <determinant> there is negative.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-60-oscillator-sequences.png]
{title=Bifurcation sequences along the two diagonal parameter lines after a small positive reflection-symmetry imperfection; arrows indicate decreasing lambda}
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