Linear and structural stability of a cell divisionprocess model

Abstract

The paper investigates the linear stability of mammalian physiology time-delayed flow for three distinct cases (normal cell cycle, a neoplasmic cell cycle, and multiple cell arrest states), for the Dirac, uniform, and exponential distributions. For the Dirac distribution case, it is shown that the model exhibits a Hopf bifurcation for certain values of the parameters involved in the system. As well, for these values, the structural stability of the SODE is studied, using the five KCC-invariants of the second-order canonical extension of the SODE, and all the cases prove to be Jacobi unstable.

Jet Geometrical Extension of the KCC-Invariants

“…In this Section we generalize on the 1-jet space J 1 (R, M ) the basics of the KCC-theory ( [1], [4], [6], [13]). In this respect, let us consider on J 1 (R, M ) a second-order system of differential equations of local form…”

“…is called the first h−KCC-invariant on the 1-jet space J 1 (R, M ) of the SODEs (2.1), which is interpreted as an external force [1], [6].…”

Multi-Time KCC-Invariants

“…In this Section we construct some multi-time generalizations on the 1-jet space J 1 (T, M ) for the basic objects of the KCC-theory ([1], [2], [3], [10]). In this respect, let us consider on J 1 (T, M ) a second-order system of partial differential equations of local form…”

“…is called the first multi-time h−KCC-invariant on the 1-jet space J 1 (T, M ) of the PDEs (2.1), which can be interpreted as an external force [1], [3].…”