Persistence and Transition Varieties in Scalar Field Cosmology

We develop a bifurcation-theoretic description of Friedmann–Robertson–Walker cosmologies with a scalar field $\phi$, a barotropic fluid of index $\gamma$, and spatial curvature. For the strict exponential potential $V(\phi )=V_0{e}^{\lambda \phi }$, with $a=\sqrt{3\mathrm{/}2}\lambda$, the local phase portrait is organised by five loci in the $(\gamma ,a)$-plane: $\mathrm{\mid }a\mathrm{\mid }=3$, $a^2=3$, $a^2=9\gamma \mathrm{/}2$, $\gamma =2\mathrm{/}3$, and $\gamma =2$. Near these loci we compute translated jets, centre(-like) reductions, and normal forms governing persistence and transitions. For the quadratic potential $V(\phi )=(1\mathrm{/}2)m^2{\phi }^2$, the effective slope $\lambda$ is dynamical. Using the bounded variable $\zeta =\arctan \lambda$, we obtain a regular autonomous $4$-dimensional system in $(X,Y,{\mathrm{\Omega }}_k,\zeta )$, where ${\mathrm{\Omega }}_k$ is the curvature variable. This reveals invariant gates, robust equilibrium continua, and vertical $\gamma$-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.