  Renormalization group (RG) evolution of the neutrino mass matrix may take the
value of the mixing angle $\theta_{13}$ very close to zero, or make it vanish.
On the other hand, starting from $\theta_{13}=0$ at the high scale it may be
possible to generate a non-zero $\theta_{13}$ radiatively. In the most general
scenario with non-vanishing CP violating Dirac and Majorana phases, we explore
the evolution in the vicinity of $\theta_{13}=0$, in terms of its structure in
the complex ${\cal U}_{e3}$ plane. This allows us to explain the apparent
singularity in the evolution of the Dirac CP phase $\delta$ at $\theta_{13}=0$.
We also introduce a formalism for calculating the RG evolution of neutrino
parameters that uses the Jarlskog invariant and naturally avoids this singular
behaviour. We find that the parameters need to be extremely fine-tuned in order
to get exactly vanishing $\theta_{13}$ during evolution. For the class of
neutrino mass models with $\theta_{13}=0$ at the high scale, we calculate the
extent to which RG evolution can generate a nonzero $\theta_{13}$, when the low
energy effective theory is the standard model or its minimal supersymmetric
extension. We find correlated constraints on $\theta_{13}$, the lightest
neutrino mass $m_0$, the effective Majorana mass $m_{ee}$ measured in the
neutrinoless double beta decay, and the supersymmetric parameter $\tan\beta$.
