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 $a=\sqrt{3/2}\,\lambda$, the local phase portrait is organised by five loci in the $(\gamma,a)$-plane: $|a|=3$, $a^{2}=3$, $a^{2}=9\gamma/2$, $\gamma=2/3$, and $\gamma=2$. Near these loci we compute translated jets, centre(-like) reductions, and normal forms governing persistence and transitions. For the quadratic potential $V(\phi)=(1/2)m^{2}\phi^{2}$, the effective slope $\lambda$ is dynamical. Using the bounded variable $\zeta=\arctan\lambda$, we obtain a regular autonomous $4$-dimensional system in $(X,Y,\Omega_{k},\zeta)$, where $\Omega_{k}$ 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.