The Dynamics of Regularized Discontinuous Maps with Applications to Impacting Systems

One-dimensional piecewise-smooth discontinuous maps (maps with gaps) are known to have surprisingly rich dynamics, including periodic orbits with very high period and bifurcation diagrams showing period-adding or period-incrementing behavour. In this paper we study a new class of maps, which we refer to as regularised one-dimensional discontinuous maps, because they give very similar dynamics to discontinuous maps and closely approximate them, but are yet continuous. We show that regularised discontinuous maps arise naturally as limits of higher dimensional continuous maps when we study the global dynamics of two non-smooth mechanical applications: the camfollower system and the impact oscillator. We review the dynamics of discontinuous maps, study the dynamics of regularised discontinuous maps and compare this to the behaviour of the two mechanical applications which we model. We show that we observe period-adding and period-incrementing behaviour in these two systems respectively but that the effect of the regularisation is to lead to a progressive loss of the higher period orbits.