This report presents a detailed bifurcation analysis of a three-dimensional model describing the interaction between tumor cells (\(x_1\)), healthy tissue cells (\(x_2\)), and activated immune cells (\(x_3\)). The model proposed by de Pillis and Radunskaya, and re-written in the dimensionless and the time-scaled form by Itik and Banks, is given by the following system of ODEs:

\[\left\{ \begin{array}{ll}\dot{x}_1 = x_1(1 - x_1) - a_{12}x_1x_2 - a_{13}x_1x_3,\\ \dot{x}_2 = r_2x_2(1 - x_2) - a_{21}x_1x_2,\\ \dot{x}_3 = \frac{r_3x_1x_3}{x_1 + k_3} -a_{31}x_1x_3 - d_3x_3. \end{array} \right.\]

The main results show that chaotic dynamics arise via the Shilnikov scenario, where a homoclinic loop to a saddle-focus equilibrium generates Shilnikov attractor and spiral chaos. Two codimension-two bifurcations act as organizing centers: a degenerate Bogdanov-Takens bifurcation (giving birth to four nontrivial equilibria) and a zero-Hopf bifurcation (leading to an unstable invariant torus). Numerical evidence confirms the existence of homoclinic orbits and Shilnikov attractors. These finding provide a comprehensive bifurcation diagram and explain the emergence of chaotic regimes in the cancer model.

This work was supported by the project Mirror Laboratories at HSE University.