  Finite-size scaling is a key tool in statistical physics, used to infer
critical behavior in finite systems. Here we use the analogous concept of
finite-time scaling to describe the bifurcation diagram at finite times in
discrete dynamical systems. We analytically derive finite-time scaling laws for
two ubiquitous transitions given by the transcritical and the saddle-node
bifurcation, obtaining exact expressions for the critical exponents and scaling
functions. One of the scaling laws, corresponding to the distance of the
dynamical variable to the attractor, turns out to be universal. Our work
establishes a new connection between thermodynamic phase transitions and
bifurcations in low-dimensional dynamical systems, and opens new avenues to
identify the nature of dynamical shifts in systems for which only short time
series are available.
