Our purpose in this paper is to consider a new class of iterated function systems (IFS) based on the concept of orbital condition introduced by R. Miculescu et al. (Iterated function systems consisting of continuous functions satisfying Banach’s orbital condition, Analele Universităt¸ii de Vest, Timi¸soara, Seria Matematică-Informatică LVI, 2, (2018),71-80). On the given IFS there are imposed some sufficient conditions guaranteeing the existence of an attractor. There are also established some further results which describe the nature of the attractors of the considered type. The introduced theory is supported by some examples of IFSs for which the attractors are also depicted.