Bifurcation in a family of maps

Summary

A bifurcation in a family of maps $f_\mu$ is a parameter value $\mu_0$ at which the qualitative behaviour of the iterates of $f_\mu$ changes. Typical signs are fixed points or periodic orbits that appear, disappear or change their stability. It is the setting of bifurcation theory for discrete-time dynamical systems, that is, for difference equations $x_{n+1} = f_\mu(x_n)$. For one-dimensional maps the basic local types are the saddle-node, the transcritical, the pitchfork and the period-doubling bifurcation.