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 $tanbeta$.
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