International Journal of Bifurcation and Chaos, Vol. 22, No. 3 (2012) 1250068 (30 pages) c  World Scientific Publishing Company DOI: 10.1142/S021812741250068X PERIOD ADDING IN PIECEWISE LINEAR MAPS WITH TWO DISCONTINUITIES FABIO TRAMONTANA Department of Economics and Quantitative Methods, University of Pavia, Italy fabio.tramontana@unipv.it LAURA GARDINI Department of Economics, Society and Politics, University of Urbino, Italy laura.gardini@uniurb.it VIKTOR AVRUTIN ∗ and MICHAEL SCHANZ † Institute of Parallel and Distributed Systems, University of Stuttgart, Germany ∗ Viktor.Avrutin@ipvs.uni-stuttgart.de † Michael.Schanz@informatik.uni-stuttgart.de Received January 31, 2011; Revised June 19, 2011 In this work we consider the border collision bifurcations occurring in a one-dimensional piece- wise linear map with two discontinuity points. The map, motivated by an economic application, is written in a generic form and considered in the stable regime, with all slopes between zero and one. We prove that the period adding structures occur in maps with more than one discontinuity points and that the Leonov’s method to calculate the bifurcation curves forming these struc- tures is applicable also in this case. We demonstrate the existence of particular codimension-2 bifurcation (big-bang bifurcation) points in the parameter space, from which infinitely many bifurcation curves are issuing associated with cycles involving several partitions. We describe how the bifurcation structure of a map with one discontinuity is modified by the introduction of a second discontinuity point, which causes orbits to appear located on three partitions and organized again in a period-adding structure. We also describe particular codimension-2 bifur- cation points which represent limit sets of doubly infinite sequences of bifurcation curves and appear due to the existence of two discontinuities. Keywords : Piecewise linear discontinuous maps; period adding bifurcation structure; Leonov’s approach; two discontinuity points. 1. Introduction Since the pioneering works by Richard Day focused on piecewise linear maps (see e.g. [Day, 1982, 1994]), several applications to economics ultimately lead to models which are described by piecewise linear or piecewise smooth maps [Hommes, 1991, 1995; Hommes & Nusse, 1991; Hommes et al., 1995; Galle- gati et al., 2003; Puu & Sushko, 2002, 2006; Sushko et al., 2003, 2005, 2006; Gardini et al., 2006a, 2006b, 2008]. In particular, several systems are modeled via discontinuous maps, often with several discon- tinuity points [Puu et al., 2002, 2005; Puu, 2007; Sushko et al., 2004; Tramontana et al., 2009; Gar- dini et al., 2011]. However, the bifurcations occur- ring in discontinuous models with more than two partitions have not yet been investigated. In this 1250068-1