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This paper deals with stability and bifurcation analysis of a minimal models of vertebrae formation. Conditions are derived under which there can be no change in stability. Using one of the parameters as a bifurcation parameter it is found that various type of bifurcations occurs when the parameter passes through a critical value. Applying the centre manifold theory and the normal form method formulas are given for describing the qualitative properties of the current bifurcation. Computer simulations illustrate the results. Mathematics Subject Classifications: 92B05 (34D20, 34C23, 34K20).

In this chapter, we provide a short overview of the stability properties of polynomials and quasi-polynomials. They appear typically in stability investigations of equilibria of ordinary and retarded differential equations. In the case of ordinary differential equations we discuss the Hurwitz criterion, and its simplified version, the Lineard-Chippart criterion, furthermore the Mikhailov criterion and we show how one can prove the change of stability via the knowledge of the coefficients of the characteristic polynomial of the Jacobian of the given autonomous system. In the case of the retarded differential equation we use the Mikhailov criterion in order to estimate the length of the delay for which no stability switching occurs. These results are applied to the stability and Hopf bifurcation of an equilibrium solution of a system of ordinary differential equations as well as of retarded dynamical systems.

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