Sensitivity problems related to certain bifurcations in non-linear recurrence relations

This paper is concerned with certain qualitative aspects of the sensitivity problem in relation to small variations of a parameter of a system, the behaviour of which can be described by an autonomous recurrence relation: V$_{n+1}$ = F(V$_{n}, \lambda$) (1) V being a vector, $\lambda$ the parameter. The problem consists in the determination of the bifurcation values $\lambda_{0}$ of $\lambda$, i.e. values such that the qualitative behaviour of a solution of (1) should be different for $\lambda = \lambda \pm \epsilon$ where $\epsilon$ is a small quantity. Bifurcations that correspond to a critical case in the Liapunov sense, and the crossing through this critical case, are considered. Examples of bifurcations, not connected with the presence of a critical case, and which correspond to a large deformation of the stability domain boundary of an equilibrium point, a fixed point of (1), under the effect of a parameter variation, are given where V is a two dimensional vector.