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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 writtenThe 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 isThis stability criterion follows from $f^{\prime }(a)=-a{g}^{\prime }(a)$ on the equilibrium curve, where $g^{\prime }=1-2Ca+4a^3$. 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 $C$: $g^{\prime }$ 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.