Calculation of Bifurcation Curves by Map Replacement

Abstract

The complex bifurcation structure in the parameter space of the general piecewise-linear scalar map with a single discontinuity — nowadays known as nested period adding structure — was completely studied analytically by N. N. Leonov already 50 years ago. He used an elegant and very efficient recursive technique, which allows the analytical calculation of the border-collision bifurcation curves, causing the nested period adding structure to occur. In this work, we have demonstrated that the application of Leonov's technique is not resticted to that particular bifurcation structure. On the contrary, the presented map replacement approach, which is an extension of Leonov's technique, allows the analytical calculation of border-collision bifurcation curves for periodic orbits with high periods and complex symbolic sequences using appropriate composite maps and the bifurcation curves for periodic orbits with much lower periods.

Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model

“…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]). 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.…”

“…This method is based on the map replacement technique (see [19,20]). Namely, at first we substitute in Σ 1,1 each symbol by and each symbol by (replacement ), and then we substitute in Σ 1,1 each symbol by and each symbol by (replacement ).…”

“…So, we get 4 families of complexity level two (one more way to construct the families of the complexity level two consists in consecutive application of the concatenation rule to the "neighbour" symbolic sequences of the first complexity level. Symbolic sequences obtained in such a way are shift invariant to those obtained by symbolic replacements (10) (see [19,20])). Further, applying the replacements (10) with = 3 to the families of complexity level two we obtain 2 3 families Σ ,3 , = 1, .…”

“…In particular, the piecewise linear case has been recently reconsidered (after [16]) in [19,20]. In the cited references, it is described as the period adding structure which is characteristic for piecewise increasing maps, when invertible on the absorbing interval.…”

On a bifurcation structure mimicking period adding

“…Similarly, a period-incrementing scenario is formed by one family of periodic orbits and also has one accumulation point. In contrast to this, a periodadding structure contains an infinite number of periodic orbits (for a detailed description of these families and their recursive definition we refer to Avrutin et al (2010)). Consequently, between the existence regions of two consecutive periodic orbits with periods p n and p n+1 , there exists an infinite (uncountable) set of accumulation points of the nested period-adding scenario.…”