  For an algebraic family $(f_t)$ of regular quadratic polynomial endomorphisms
of $\mathbb{C}^2$ parametrized by $\mathbb{D}^*$ and degenerating to a H\'enon
map at $t=0$, we study the continuous (and indeed harmonic) extendibility
across $t=0$ of a potential of the bifurcation current on $\mathbb{D}^*$ with
the explicit computation of the non-archimedean Lyapunov exponent associated to
$(f_t)$. The individual Lyapunov exponents of $f_t$ are also investigated near
$t=0$. Using $(f_t)$, we also see that any H\'enon map is accumulated by the
bifurcation locus in the space of quadratic holomorphic endomorphisms of
$\mathbb {P}^2$.
