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Degenerated Bogdanov-Takens bifurcations in an immuno-tumor model

Author(s): Mariana P. Trifan | Adelina Georgescu

Journal: Atti dell'Accademia Peloritana dei Pericolanti - Classe di Scienze Fisiche, Matematiche e Naturali
ISSN 0365-0359

Volume: LXXXVII;
Issue: 1;
Start page: C1A0901005;
Date: 2009;
Original page

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.
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