Information Technology Reference
In-Depth Information
Chapter 3
Bifurcations and Limit Cycles in Models
of Biological Systems
Abstract The chapter proposes a systematic method for fixed point bifurcation
analysis in circadian cells and similar biological models using interval polynomials
theory. The stages for performing fixed point bifurcation analysis in such biological
systems comprise (i) the computation of fixed points as functions of the bifurcation
parameter and (ii) the evaluation of the type of stability for each fixed point through
the computation of the eigenvalues of the Jacobian matrix that is associated with the
system's nonlinear dynamics model. Stage (ii) requires the computation of the roots
of the characteristic polynomial of the Jacobian matrix. This problem is nontrivial
since the coefficients of the characteristic polynomial are functions of the bifurcation
parameter and the latter varies within intervals. To obtain a clear view about the
values of the roots of the characteristic polynomial and about the stability features
they provide to the system, the use of interval polynomials theory and particularly
of Kharitonov's stability theorem is proposed. In this approach the study of the
stability of a characteristic polynomial with coefficients that vary in intervals is
equivalent to the study of the stability of four polynomials with crisp coefficients
computed from the boundaries of the aforementioned intervals. The efficiency of the
proposed approach for the analysis of fixed points bifurcations in nonlinear models
of biological neurons is tested through numerical and simulation experiments.
3.1
Outline
The chapter analyzes the use of interval polynomial theory for the study of fixed
point bifurcations and the associated stability characteristics in circadian cells
and similar biological models. The bifurcation parameter is usually one of the
coefficients of the biological system or a feedback control gain that is used to modify
the system's dynamics. The bifurcation parameter varies within intervals; therefore,
assessment of the stability features for all fixed points of the system (each fixed
point depends on a different value of the bifurcation parameter) requires extended
computations. Therefore, it is significant to develop methods that enable to draw
Search WWH ::




Custom Search