Nilpotent Singularities and Periodic Perturbation of a GIβ Model: A Pathway to Glucose Disorder

Bifurcations and related dynamical behaviors of a glucose metabolism model are thoroughly studied in this paper. It is shown that the model undergoes transcritical, Hopf, degenerate Hopf, saddle-node, cusp, and zero-Hopf bifurcations, as well as Bogdanov–Takens bifurcations of codimensions 2 and 3. Considering the periodicity of hepatic glucose production and β cells’ glucose tolerance range, four elementary periodic mechanisms are also analyzed. These mechanisms lead to more complex dynamics, including periodic solutions of different periods, quasiperiodic solutions, chaos through torus destruction, or cascade of period doublings. Sensitivity analysis is performed to isolate the high-effect factors and explore a few advanced treatment approaches. The described dynamics explain well several clinical observations, which could provide sound guidance in the therapeutic process.