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