Phase space and Data analyses of a non-minimally coupled scalar field system with decaying dark energy model

We investigate a class of scalar field dark energy models non-minimally coupled to gravity, characterized by a double exponential potential and parameterized coupling {\xi}. First, we study the cosmological dynamics for a recently proposed Q-SC-CDM model. Initially, we choose two distinct values of {\xi}: 0.1 and 0.5. In case of {\xi}=0.1, the evolution of the universe is split up into three different phases: deceleration, acceleration and slow-contraction, and provide Big Crunch Singularity at distant future. However, in case of {\xi}=0.5, the phase of slow-contraction vanishes, correspondingly Big Crunch Singularity is redundant. Second, we perform the phase space analysis for Q-SC-CDM, bring new asymptotic regimes and find stable de-Sitter solution. Finally, fixing {\xi} to two representative values: {\xi} = 0.1 and 0.05, we perform a comprehensive Bayesian analysis using recent late-time cosmological observations, including Cosmic Chronometers (CC), Type Ia Supernovae (Pantheon+ and DES-SN5YR), and Baryon Acoustic Oscillation (DESI DR2) data. Our results demonstrate that both models yield constraints on key cosmological parameters \Omega_{0m}, H_0 and the sound horizon r_d that are consistent with {\Lambda}CDM within 68\% confidence level, yet exhibit mild tension with Pantheon+ measurements. We analyze the evolution of the effective equation of state, showing that the model transitions from a stiff matter phase at high redshift to a dark energy dominated phase with effective equation of state less than -0.5 at late times. Additionally, we employ the Om(z) diagnostic to distinguish our model from {\Lambda}CDM, finding minimal deviation up to redshift z \sim 2. Statistical model comparison using Akaike (AIC) and Bayesian (BIC) Information Criteria reveals moderate support for the model with {\xi} = 0.1, though {\Lambda}CDM remains statistically preferred.