  Systems where time evolution follows a multiplicative process are ubiquitous
in physics. We study a toy model for such systems where each time step is given
by multiplication with an independent random $N\times N$ matrix with complex
Gaussian elements, the complex Ginibre ensemble. This model allows to
explicitly compute the Lyapunov exponents and local correlations amongst them,
when the number of factors $M$ becomes large. While the smallest eigenvalues
always remain deterministic, which is also the case for many chaotic quantum
systems, we identify a critical double scaling limit $N\sim M$ for the rest of
the spectrum. It interpolates between the known deterministic behaviour of the
Lyapunov exponents for $M\gg N$ (or $N$ fixed) and universal random matrix
statistics for $M\ll N$ (or $M$ fixed), characterising chaotic behaviour. After
unfolding this agrees with Dyson's Brownian Motion starting from equidistant
positions in the bulk and at the soft edge of the spectrum. This universality
statement is further corroborated by numerical experiments, multiplying
different kinds of random matrices. It leads us to conjecture a much wider
applicability in complex systems, that display a transition from deterministic
to chaotic behaviour.
