Revealing bifurcation mechanisms in a 2D nonsmooth map by means of the first return map

In this work, our goal is to show how a properly constructed first return map can help to describe the dynamics of a two-dimensional piecewise smooth map. In particular, to understand some formation principles of the bifurcation structure of its parameter space. The considered map is important from the applied point of view: it models the effects of fraud in a public procurement procedure and allows one to investigate whether honest or dishonest behavior prevails in society. We present the bifurcation structure of the map formed by the periodicity regions associated with attracting cycles of different periods. By means of the first return map, we show that the boundaries of these regions are related to border collision bifurcations typical for nonsmooth maps, as well as standard fold and flip bifurcations. The main economic result is related to the emergence of unexpected changes in qualitative non-compliant behavior over time.