Late-time cosmic dynamics in $f(R,L_{m})$ gravity with recent observations

In this work, we investigate the late-time cosmic dynamics in the framework of non-linear $f(R, L_m)$ gravity, adopting the functional form $f(R,L_m)=\frac{R}{2}+L_m^2$. To explore the dark energy behavior, we assume an oscillatory parametric equation of state, $\omega(z) = \omega_0 + b \sin[\log(1+z)]$, which allows smooth deviations from the cosmological constant. Using a joint MCMC analysis with the latest Hubble 31 chronometer data, DESI DR2 BAO measurements, and Type Ia supernova samples (Pantheon+, DES-SN5Y and Union 3), we obtain well-constrained parameters around $H_0 \simeq 67.2~\text{km s}^{-1}\text{Mpc}^{-1}$ and $\omega_0\approx-0.5$, consistent with Planck 2018 and other current observations. The model exhibits a clear transition from deceleration to acceleration with $z_{\rm tr} \sim 0.7$--$0.8$, satisfies the NEC and DEC while violating the SEC and yields present EoS values close to $-1$, reproducing $\Lambda$CDM behavior at late times. The derived Universe ages ($t_0 \approx 13.3~\text{Gyr}$) agree well with CMB and stellar constraints, confirming that the proposed oscillatory $f(R, L_m)$ model provides an observationally consistent and dynamically viable alternative to $\Lambda$CDM cosmology.