Bifurcation Structures in a Bimodal Piecewise Linear Map: Regular Dynamics

Abstract

We consider a family of piecewise linear bimodal maps having the outermost slopes positive and less than one. Three types of bifurcation structures incorporated into the parameter space of the map are described. These structures are formed by periodicity regions related to attracting cycles, namely, the skew tent map structure is associated with periodic points on two adjacent branches of the map, the period adding structure is related to periodic points on the outermost branches, and the fin structure is contiguous with the period adding structure and is associated with attracting cycles having at most one point on the middle branch. Analytical expressions for the periodicity region boundaries are obtained using the map replacement technique.

Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model

“…The period adding structure is formed by periodicity regions related to cycles organized according to the Farey summation rule applied to the rotation numbers of the related cycles. Besides this, in bimodal piecewise linear maps the so-called fin structure is also observed (see [15]). We will describe these regions in the parameter space of the considered map, recalling how these two structures are organized and giving formulas of the boundaries of related regions.…”

“…As noticed in [15], the overall bifurcation structure of the parameter space for a generic 1D bimodal piecewise linear map is characterized by several substructures among which we recall the skew tent map structure, the period adding structure, and a particular one called fin structure (due to the shape of the periodicity regions, as it will be clear below). The simplest one is the skew tent map structure associated with absorbing intervals involving only two adjacent partitions, so that on these absorbing intervals the map is reduced to a skew tent map.…”

“…Clearly, a map with two border points possesses more complicated dynamics, and all the possible outcomes are not yet fully investigated; some results can be found in [14,15]. In particular, the bifurcation structure of its parameter space includes both regions which belong to the known period adding structure (called also Arnold tongues or mode-locking tongues), which is characteristic for piecewise increasing discontinuous maps and also for certain circle maps (see, e.g., [16][17][18][19][20]).…”

“…As explained in [15], for an ⋅ -cycle whose periodicity region has the common boundary with the region −1 , the symbolic sequences of the cycles in the fins are ( on the other side. Interchanging and in these sequences we get the symbolic sequences of the cycles related to ⋅ -fins whose trunks are −1 regions.…”

“…To describe the period adding structure observed in the parameter space of map consider first the basic cycles 1 and 1 belonging to the families Σ 1,1 and Σ 2,1 of complexity level one defined in (9). As stated in [15], the periodicity regions 1 and 1 of map are defined as These formulas are valid for the map given in (2) substituting…”