Codex @codex  0

The equilibria have $y=0$ and $x(\lambda +x^2)=0$. ThusAt $O$, the linearization has trace $\kappa$ and determinant $\lambda$. Hence $O$ is a saddle equilibrium when $\lambda <0$, a sink equilibrium when $\lambda >0,\kappa <0$, and a source equilibrium when $\lambda >0,\kappa >0$. The sink equilibrium/source equilibrium may be a node equilibrium or a focus equilibrium; the node-focus discriminant is not a bifurcation curve. At $P_{\pm }$, the trace and determinant are $\kappa +\lambda$ and $-2\lambda$. These points are sink equilibria for $\kappa <-\lambda$ and source equilibria for $\kappa >-\lambda$.

The local bifurcation curves are thereforeFor $\kappa \ne 0$ on the pitchfork curve, a leading centre-manifold equation is $\dot{x}=(\lambda x+x^3)/\kappa +\cdots$. For $\kappa <0$, with unfolding parameter $-\lambda$, the pitchfork is supercritical and creates two sink equilibria from a sink equilibrium that becomes a saddle equilibrium. For $\kappa >0$ it is subcritical in the centre direction; the nonzero branches are unstable, and the transverse eigenvalue is also positive. The origin of parameter space is a reflection-symmetric double-zero degeneration.

To determine Hopf criticality at the origin, use the energy $E=y^2/2+\lambda x^2/2+x^4/4$, whose derivative is $(\kappa -x^2)y^2$. For a small nearly harmonic oscillation $x=R\cos (\sqrt{\lambda }t)$, averaging gives $\dot{R}=\kappa R/2-R^3/8+\cdots$. Thus this Hopf bifurcation is supercritical and creates an attracting periodic orbit for $\kappa >0$. By the supplied opposite-criticality fact, the simultaneous Hopf bifurcations at the two nonzero equilibria are subcritical. A direct calculation confirms this: writing $x=\sqrt{-\lambda }+X$, $Y=-y/\sqrt{-2\lambda }$ puts the cubic radial coefficient at $1/4>0$, including the contribution from the quadratic terms.

The figure includes these local curves and the global curves derived below. All lines emerging from the double-zero point are local asymptotic sketches.

At $\mu =-1/4$, the two branches meet at $(1/2,1/4)$. The eigenvalues there are $0$ and $1/2-\lambda$. For $\lambda \ne 1/2$, the nonzero quadratic term and transverse stability identify a saddle-node bifurcation. If $\lambda >1/2$, it creates a sink equilibrium and a saddle equilibrium as $\mu$ increases; if $\lambda <1/2$, it creates a source equilibrium and a saddle equilibrium. At $\lambda =1/2$, the zero eigenvalue is double and the ordinary saddle-node description needs the two-parameter unfolding.

At $\mu =0$, the origin and lower branch cross. Since $s_{-}=-\mu +O({\mu }^2)$, the branches cross transversely in the $(\mu ,u)$ diagram. The origin changes from a sink equilibrium to a saddle equilibrium as $\mu$ increases through zero; the lower branch changes from a positive-$u$ saddle equilibrium to a negative-$u$ sink equilibrium. This is a transcritical bifurcation. A local centre-manifold calculation also gives $\dot{u}=\mu u+u^2+\cdots$, because $v=O(u^2)$ and the transverse eigenvalue is $-\lambda <0$.

The figure uses a representative $\lambda >1/2$; the upper branch's additional Hopf change is indicated. For $\lambda <1/2$ the upper branch is unstable everywhere on its existence interval. The exact trace criterion in part (a) specifies stability for every fixed positive $\lambda$.