Dynamics of a piecewise linear map with a gap

In this paper, we consider periodic solutions of discontinuous non-smooth maps. We show how the fixed points of a general piecewise linear map with a discontinuity (‘a map with a gap’) behave under parameter variation. We show in detail all the possible behaviours of period 1 and period 2 solutions. For positive gaps, we find that period 2 solutions can exist independently of period 1 solutions. Conversely, for negative gaps, period 1 and period 2 solutions can coexist. Higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation. Finally, we compare our results with those of Jain & Banerjee (Jain & Banerjee 2003 Int. J. Bifurcat. Chaos 13 , 3341–3351) and Banerjee et al . (Banerjee et al . 2004 IEEE Trans. Circ. Syst. II 51 , 649–654) and explain their numerical simulations.