Routes to Chaos in a Three-Dimensional Cancer Model

We provide a detailed bifurcation analysis in a three-dimensional system describing interaction between tumor cells, healthy tissue cells, and cells of the immune system. As is well known from previous studies, the most interesting dynamical regimes in this model are associated with the spiral chaos arising due to the Shilnikov homoclinic loop to a saddle-focus equilibrium [1, 2, 3]. We explain how this equilibrium appears and how it gives rise to Shilnikov attractors. The main part of this work is devoted to the study of codimension-two bifurcations which, as we show, are the organizing centers in the system. In particular, we describe bifurcation unfoldings for an equilibrium state when: (1) it has a pair of zero eigenvalues (Bogdanov – Takens bifurcation) and (2) zero and a pair of purely imaginary eigenvalues (zero-Hopf bifurcation). It is shown how these bifurcations are related to the emergence of the observed chaotic attractors.