Slow–fast evolution of scalar fields in higher-order cosmological gravity dynamics inspired by the Pais–Uhlenbeck oscillator

We study scalar field cosmologies in higher-order gravity, inspired by the Pais–Uhlenbeck oscillator, which admits a fourth-order ghost-free sector. By recasting the equations as a slow–fast system, we analyze phase-space evolution under exponential, power-law, and arbitrary potentials using both analytical and geometric methods. The full system exhibits a rational structure with singular surfaces and, under a slow manifold constraint, reduces to a regular four-dimensional form that supports global analysis and perturbative stability. The f-deviser technique reconstructs potential-adapted functions f(λ), enabling attractor classification and center manifold analysis. Two-field extensions yield scaling laws and tracking behavior in the quintom regime. In the quintessence regime, we examine de Sitter stability, incorporating radiation and dust to model realistic transitions, and derive analytic expressions for n_s and r across reconstructed inflationary potentials-obtained from standard, Gaussian, hybrid, extended, and logarithmic expansions of the scale factor-with a scalar field evolving linearly in time. Our results confirm the viability of Pais–Uhlenbeck scalar models for inflation and dark energy, offering tools to study attractors and bifurcations in higher-derivative cosmology.