Persistence and Transition Varieties in Scalar Field Cosmology

We develop a unified bifurcation-theoretic description of Friedmann--Robertson--Walker cosmologies with a scalar field, a barotropic fluid of index $\gamma$, and spatial curvature.

For the exponential potential $V(\phi)=V_0e^{\lambda\phi}$, the slope $a=\sqrt{3/2}\,\lambda$ is a distinguished parameter, and the local phase portrait is organised by the loci $|a|=3$, $a^2=3$, $a^2=\tfrac92\gamma$, $\gamma=\tfrac23$, and $\gamma=2$, corresponding to kinetic, curvature, scalar--fluid exchange, and degeneracy thresholds.

For the quadratic potential $V(\phi)=\tfrac12m^2\phi^2$, the effective slope is dynamical.

We introduce the bounded variable $\zeta=\arctan\lambda$, obtaining a closed autonomous four-dimensional system in $(X,Y,\Omega_k,\zeta)$ without time-rescaling.

This exposes invariant gates, robust equilibrium continua, and $\gamma$-thresholds controlling loss and recovery of normal hyperbolicity.

Near the organising loci we compute translated jets, perform centre(-like) reductions, and derive canonical normal forms governing persistence and transitions.

These are assembled into an explicit stratification of the exponential parameter plane and a pull-back stratification for the massive extensions, with physical path maps into the corresponding unfolding charts.

The framework shows how fluid and curvature modes provide deformation directions within the FRW class and yields a regime-level interpretation: slow roll and ultra slow roll arise as persistent attracting balances and nonhyperbolic bottleneck passages, while quadratic invariant slices recover the oscillatory periodic-orbit/invariant-torus sector.

It also organises critical slowing, curvature leakage, tracker exchange, and admissible sequences of such episodes along massive trajectories.