The main goal is to study the dynamics of certain parametrized families \((f_\mu)_\mu\) of one-dimensional maps with infinite critical set. The authors show that they exhibit global chaotic behaviour in a persistent way: For a positive Lebesgue measure set of parameter values the map is transitive and almost every orbit has positive Lyapunov exponent. This class of systems is motivated by a problem in the dynamics of flows in three dimensions: the unfolding of saddle-focus homoclinic connections.