Persistence and Transition Varieties in Scalar Field Cosmology

Abstract: We develop a bifurcation-theoretic description of Friedmann--Robertson--Walker cosmologies with a scalar field $φ$, a barotropic fluid of index $γ$, and spatial curvature. For the strict exponential potential $V(φ)=V_{0}e^{λφ}$, with $a=\sqrt{3/2}\,λ$, the local phase portrait is organised by five loci in the $(γ,a)$-plane: $|a|=3$, $a^{2}=3$, $a^{2}=9γ/2$, $γ=2/3$, and $γ=2$. Near these loci we compute translated jets, centre(-like) reductions, and normal forms governing persistence and transitions. For the quadratic potential $V(φ)=(1/2)m^{2}φ^{2}$, the effective slope $λ$ is dynamical. Using the bounded variable $ζ=\arctanλ$, we obtain a regular autonomous $4$-dimensional system in $(X,Y,Ω{k},ζ)$, where $Ω{k}$ is the curvature variable. This reveals invariant gates, robust equilibrium continua, and vertical $γ$-thresholds for loss and recovery of normal hyperbolicity. We then construct an explicit stratification for the exponential class and a pull-back stratification for the massive case, together with the corresponding physical path maps into unfolding space. The resulting framework also organises slow-roll, ultra slow-roll, and oscillatory regimes.