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

Abstract

In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X′=aXif ∣X∣<z, X′=bX if ∣X∣>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 X′=F(X) if ∣X∣<z, X′=G(X) if ∣X∣>z is also given, proving the conditions leading to a homeomorphism of the circle.