Degenerated Bogdanov-Takens bifurcations in an immuno-tumor model
A mathematical immuno-tumor model proposed by A. Kavaliauskas [Nonlinear Anal. Model. Control 8, 55 (2003)] and consisting of a Cauchy problem for a system of two first-order ordinary differential equations is studied. For some particular parameters values, this model has saddle-node, Hopf and Bogdanov-Takens (BT) singularities. In the case of the BT singularities, we herein derive the normal forms of the governing equations by using ideas and a method from S.-N. Chow, C. Li, and D. Wang [Normal forms and bifurcation of planar vector fields (1994)] and Yu. A. Kuznetsov [Elements of applied bifurcation theory (1994)], based on an appropriate splitting of associated Hilbert spaces. It is found that a limit case of parameters associated with medicine administration corresponds to degenerate BT bifurcations and, so, to a large variety of responses to the medical treatments for admissible parameters near the limit ones.