Cosmological dynamics and bifurcation analysis of the general non-minimal coupled scalar field models

Non-minimally coupled scalar field models are well-known for providing interesting cosmological features. These include a late time dark energy behavior, phantom evolution without singularity, an early inflationary universe, scaling solutions, convergence to the standard $\Lambda$CDM, etc. While usual stability analysis helps us determine of model geometrically, bifurcation theory allows precisely locate parameters' values describing global dynamics fine-tuning initial conditions. Using center manifold and analysis, we show that general undergoes transcritical bifurcation, which predicts tune our have certain desired dynamics. We obtained class range parameters capable cosmic from radiation era towards over wide There is also possible scenario crossing divide line. find where attractor mechanism indistinguishable structurally stable relativity based model; thus, can elude big rip singularity generically. Therefore, select viable with observations.