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: Vn+1 = F(Vn, λ) (1) V being a vector, λ the parameter. The problem consists in the determination of the bifurcation values λ0 of λ, i.e. values such that the qualitative behaviour of a solution of (1) should be different for λ = λ0 ± ε where ε 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. © 1969.