magnetisation (P) ${\bf m}_0={\bf p}$ we have $v_{\parallel}=vg_{\parallel}(0)$, $h=vg_{\perp}(0)+h_{\rm
  ext}$, $\alpha_0=0$ and $Q=1$. The stability conditions (\[stab\]) and (\[F0\]) become $$\label{stab2}
\left[g_{\parallel}(0)\right]^2v^2+(vg_{\perp}(0)+h_{\rm
  ext}+1)^2+h_{\rm p}\left[vg_{\perp}(0)+h_{\rm
    ext}+1\right]\geqslant 0$$

$$\label{F0-2}
g_{\parallel}(0)v+\gamma\left[vg_{\perp}(0)+h_{\rm
  ext}+1+\frac12 h_{\rm p}\right]\geqslant 0.$$

In the state of anti-parallel magnetisation (AP) ${\bf m}_0=-{\bf p}$ we have $v_{\parallel}=vg_{\parallel}(\pi)$, $h=vg_{\perp}(\pi)+h_{\rm
  ext}$, $\alpha_0=\pi$ and $Q=-1$. The stability conditions for the AP state are thus $$\label{stab2AP}
\left[g_{\parallel}(\pi)\right]^2v^2+(-vg_{\perp}(\pi)-h_{\rm
  ext}+1)^2+h_{\rm p}\left[-vg_{\perp}(\pi)-h_{\rm
    ext}+1\right]\geqslant 0$$

$$\label{F0-2AP}
g_{\parallel}(\pi)v+\gamma\left[vg_{\perp}(\pi)+h_{\rm
  ext}-1-\frac12 h_{\rm p}\right]\leqslant 0.$$

In the regime of low external field ($h_{\rm ext}\approx 1$, [ *i.e.*]{} $H_{\rm ext}\approx H_{u0}$) we have $H_{\rm p}>>H_{\rm ext}$ ($h_{\rm p}\approx 100$). Eqs. (\[stab2\]) and (\[stab2AP\]) may be then approximated by $$\label{stab2AP-approx}
vg_{\perp}(0)+h_{\rm ext}+1>0$$

$$\label{F0-2AP-approx}
vg_{\perp}(\pi)+h_{\rm ext}-1<0.$$

Equation (\[stab2AP-approx\]) corresponds to P stability and (\[F0-2AP-approx\]) to AP stability. It is convenient to define scalar quantities $T_{\perp}$, $T_{\parallel}$ by $T_{\perp}=g_{\perp}(\psi)\sin\psi$, $T_{\parallel}=g_{\parallel}(\psi)\sin\psi$, these being scalar components of spin-transfer torque in units of $eV_{\rm b}$ (cf. Eq. (\[torque2\])). Then $g_i(0)=[{\rm d}T_i/{\rm
    d}\psi]_{\psi=0}$ and $g_i(\pi)=-[{\rm d}T_i/{\rm
    d}\psi]_{\psi=\pi}$ with $i=\perp,\parallel$. Model calculations [@13] show that both $g_{\perp}$ and $g_{\parallel}$ can be of either sign, although positive values are more common. Also there is no general rule about the relative magnitude of $g_i(0)$ and $g_i(\pi)$.

We now illustrate the consequences of the above stability 