Study of the dynamics of a class of one-dimensional piecewise linear displays with one gap

In the paper, the dynamics of a class of one-dimensional piecewise linear displays with one gap is studied. Stable conditions of equilibrium as well as other attractors are found by numerical methods. During the investigation two basic cases to which all remaining ones come down are considered. In the space of parameters, the areas responding to these or those phase reorganizations are selected. In particular, it was ascertained that for this class of functions, under condition of a continuity on the considered display, there is no set of parameters of it that in case of the given restrictions on the function there were at least two attractors. In case of the existence of a gap is there are infinitely many areas in which two attracting cycles coexist, and if in the area there are two attracting cycles, their periods differ exactly by a unit, and there are no areas where there would be three or more attractors. Besides, it was revealed that in case of three-dimensional motion of parameters along a straight line steady cycles of the various periods with the following important feature are watched: each area supports exactly one or exactly two attracting cycles, and the area containing $k$ attracting cycles adjoin to the areas containing $3-k$ attracting cycles, and sets of values of the periods of any two adjoining areas have a nonzero intersection.