Organising Centres in a 2D Discontinuous Map

Properties of one-dimensional (1D) piecewise monotone maps with a singleSushko, I.Avrutin, V.Gardini, L. discontinuity point, also known as Lorenz maps, have been the focus of many researchers from various theoretical and applied fields. Several bifurcation structures in the parameter space of these maps have been well described, particularly the period adding bifurcation structure. This structure is formed by periodicity regions related to attracting cycles with the rotation numbers following the well-known Farey sequences. To explore similar structures in two-dimensional (2D) discontinuous maps, we propose a map with an invariant manifold (given by the x-axis) where this map simplifies to a Lorenz map. As a result, the parameter space of the 2D map contains a complete period adding bifurcation structure associated with the cycles on the x-axis, referred to as border cycles. Our 2D map is constructed in such a way that the border cycles lose their transverse stability via a transcritical bifurcation, leading to the appearance of attracting interior (i.e., non-border) cycles whose periodicity regions form a bifurcation structure that is a continuation of the period adding structure. Our goal is to understand how the complete period adding structure is modified when the related attracting cycles no longer belong to the x-axis associated with 1D discontinuous map. Specifically, we describe a novel kind of the organising centre from which this structure issues. As opposed to organising centres leading to appearance of the period adding structures known so far, in the case we report, the bifurcation structure in a neighbourhood of the organising centre is heavily affected by the multistability.