= Solution
The transverse <eigenvalues> $-2$ and $-1$ are stable, so the <center manifold> reduction determines local asymptotic stability. If $a<1$, the leading center term has negative coefficient and $v$ flows toward zero from both sides. If $a>1$, it flows away. At $a=1$ the cubic term vanishes but the quintic coefficient is
$$
\frac{(3-1)(1+3)}8=1>0,
$$
so the origin is again unstable. Therefore
$$
\boxed{\text{the origin is asymptotically stable exactly when }a<1}.
$$
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