Logarithmic structure of the generalized bifurcation set

Let ${\displaystyle G:{ℂ}^n\times {ℂ}^r\to ℂ}$ be a holomorphic family of functions. If ${\displaystyle \Lambda \subset {ℂ}^n\times {ℂ}^r}$, ${\displaystyle {\pi }_r:{ℂ}^n\times {ℂ}^r\to {ℂ}^r}$ is an analytic variety then   ${\displaystyle Q_{\Lambda }\left(G\right)=\{(x,u)\in {ℂ}^n\times {ℂ}^r:G(·,u)}$ has a critical point in ${\displaystyle \Lambda \cap {\pi }_r^{-1}\left(u\right)\}isanatural\ge \ne ralizationofthebifurcationvarietyofG.We\in vestigatethelocalstructureof}$Q_{Λ}(G)${\displaystyle f\text{or}locallytrivialdef\text{or}mationsof}$Λ₀ = π_{r}^{-1}(0)!$!. In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.