= Solution
For $\mu>0$, the cubic equation has the same sign as $a$ everywhere away from zero: small positive or negative disturbances grow without saturation, and the cubic truncation even predicts finite-time blow-up. An <amplitude> of convection cannot grow indefinitely under that local approximation. The conclusion is that additional nonlinear physics and higher-order terms are needed, not that the small-amplitude reduction is valid at arbitrarily large <amplitude>.
For the modified equation put $\eta=C_1\mu$ and $C=C_2>0$. It can be written
$$
a'=a[\eta-a+Ca^2-a^4].
$$
The quintic term points inward for sufficiently large $|a|$, so all forward solutions are bounded and approach <equilibria> in this scalar problem. The quadratic term breaks $a\mapsto-a$ <symmetry>. Besides $a=0$, the <equilibrium> diagram is
$$
\boxed{\eta=g(a)=a-Ca^2+a^4,\qquad
\text{a nonzero equilibrium is stable iff }ag'(a)>0.}
$$
This stability criterion follows from $f'(a)=-ag'(a)$ on the <equilibrium> curve, where $g'=1-2Ca+4a^3$. Near zero the new branch is $a=\eta+O(\eta^2)$, crossing the trivial branch transcritically. It is unstable for small negative $\eta$ and stable for small positive $\eta$; the trivial branch has the opposite stability.
The negative-amplitude part of $g$ has exactly one minimum. Above its negative fold value but below zero, a stable finite-amplitude negative state coexists with the stable origin, separated by an unstable negative <equilibrium>. This gives <hysteresis> and finite-amplitude triggering of convection. For positive $\eta$ the origin is unstable and stable nonzero states provide saturation. Additional positive-amplitude folds depend on $C$: $g'$ has two positive roots for $C>3/2$, a double root at $C=3/2$ and none for $C<3/2$, so further multistability can occur. The right-hand figure illustrates $C=1$, without those extra positive folds; it does not assume that every possible $C_2$ gives that same detailed diagram. These are the qualitative changes in <quintic saturation of a symmetry-broken pitchfork>.
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