Structurally unstable regular dynamics in 1D piecewise smooth maps, and circle maps

In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X0 = aX if jXj < z, X0 = bX if jXj > z, where a and b can take any real value, and may have several applications. We show that its dynamic behaviors are those of a linear rotation: either periodic or quasiperiodic, and always structurally unstable. A generalization to piecewise monotone functions X0 = F(X) if jXj < z, X0 = G(X) if jXj > z is also given, proving the conditions leading to a homeomorphism of the circle. 2012 Elsevier Ltd. All rights reserved.