  We study the Cauchy problem for first-order quasi-linear systems of partial
differential equations. When the spectrum of the initial principal symbol is
not included in the real line, i.e., when hyperbolicity is violated at initial
time, then the Cauchy problem is strongly unstable, in the sense of Hadamard.
This phenomenon, which extends the linear Lax-Mizohata theorem, was explained
by G. M\'etivier in [{\it Remarks on the well-posedness of the nonlinear Cauchy
problem}, Contemp.~Math.~2005]. In the present paper, we are interested in the
transition from hyperbolicity to non-hyperbolicity, that is the limiting case
where hyperbolicity holds at initial time, but is violated at positive times:
under such an hypothesis, we generalize a recent work by N. Lerner, Y. Morimoto
and C.-J. Xu [{\it Instability of the Cauchy-Kovalevskaya solution for a class
of non-linear systems}, American J.~Math.~2010] on complex scalar systems, as
we prove that even a weak defect of hyperbolicity implies a strong Hadamard
instability. Our examples include Burgers systems, Van der Waals gas dynamics,
and Klein-Gordon-Zakharov systems. Our analysis relies on an approximation
result for pseudo-differential flows, introduced by B.~Texier in [{\it
Approximations of pseudo-differential flows}, Indiana Univ. Math. J.~2016].
