Observational Constraints on Late-Time Cosmological Expansion in Minimal R + Alpha R^2 Gravity

We investigate the numerical dynamics and observational viability of the minimal modified gravity model defined by f(R) = R + alpha R^2 in the late-time Universe. By recasting the higher-derivative gravitational dynamics into an autonomous coupled system of first-order differential equations in terms of the state variables (H, R, U, rho_m), we perform rigorous numerical integrations spanning the redshift range 0 <= z <= 2. The dimensionless coupling parameter beta = alpha * H_0^2 dictates the amplitude and frequency of scalaron oscillations. Using 31 observational Hubble data points derived from cosmic chronometers, we perform a chi-squared likelihood analysis to constrain the model parameters. We establish an upper bound of beta <= 1.4 x 10^-3 at the 95% confidence level, with a best-fit value of H_0 = 68.1 +/- 1.2 km/s/Mpc. In this regime, the background expansion history mimics flat Lambda-CDM to within 1.5%, yielding a minimum chi-squared = 15.38 (reduced chi-squared = 0.530). Larger coupling values (beta >= 10^-2) introduce unphysical rapid oscillations in the deceleration parameter q(z) that are strongly disfavored by late-time expansion observations.