Study of the Dynamics of a Class of One-dimensional Piecewise Linear Displays with One Gap

Abstract

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.