A complex dynamical behaviour called ‘chaos” is observed in mathematical models expressed in the form of recurrences with a non-unique inverse. The chaotic solutions of such second-order point-mappings are located in bounded domains of the phase plane, called chaotic areas. All the attractive limit sets of an endomorph-ism, whatever the nature may be, are located in phase plane domains designated by absorptive areas. The variation of a parameter can modify the nature of these areas. Different bifurcations occurring in absorptive and chaotic areas are described in this paper. These bifurcations are illustrated by the study of an example related to a linear piecewise recurrence.