Information Technology Reference
In-Depth Information
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 has been 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 method for fixed points stability
analysis was tested on two different biological models: (1) the FitzHugh-Nagumo
neuron model, (2) a model of circadian cells which produce proteins out of an RNA
transcription and translation procedure. The efficiency of the proposed stability
analysis method has been confirmed through numerical and simulation tests.
