Codimension-2 Border Collision, Bifurcations in One-Dimensional, Discontinuous Piecewise Smooth Maps

Abstract

We consider a two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point. The bifurcation structures in a parameter plane of the map are investigated, related to codimension-2 bifurcation points defined by the intersections of two border collision bifurcation curves. We describe the case of the collision of two stable cycles of any period and any symbolic sequences. For this case, we prove that the local monotonicity of the functions constituting the first return map defined in a neighborhood of the border point at the parameter values related to the codimension-2 bifurcation point determines, under suitable conditions, the kind of bifurcation structure originating from this point; this can be either a period adding structure, or a period incrementing structure, or simply associated with the coupling of colliding cycles.

Symmetry breaking in a bull and bear financial market model

“…It is known (see e.g., [9]) that for this class of maps (also called gap maps) the periodicity regions are organized in the period adding structure. Fig.…”

“…In order to obtain 2 K families of complexity level K ≥ 3, similar concatenation procedures can be applied to sequences of families of complexity level K − 1 (see [18,19,9]). Another way to determine all families of the period adding structure is to apply the map replacement technique, as described in Avrutin et al [1,10].…”

“…Hence, the point (m,s 2 ) = ( − 11/6, − 5/6) can be seen as an intersection point of two BCB boundaries: one is related to cycle LR and the other to cycle LM + RM − . It is known that such a point, called the big bang bifurcation point (or organizing center),is an issue point of a period adding structure under certain additional conditions which are satisfied for the considered case (for details, see [9]). Similar structures are also described in Tramontana et al [25].…”

“…Obviously, in the presence of two discontinuity points, the bifurcation structure may be much more complicated than those observed in the parameter space of a map with one discontinuity point. In fact, the dynamics of a generic 1D PWL map with one discontinuity point, say, map g, are relatively well understood (see, e.g., [16,10,9], and references therein). For instance, there are two basic bifurcation structures for the periodicity regions of map g, namely the period adding structure and the period incrementing structure.…”

Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model

“…The dynamics of 1D discontinuous piecewise monotone maps have been studied by many researchers (see, e.g., [17,18,28,29]). In particular, the piecewise linear case has been recently reconsidered (after [16]) in [19,20].…”