Transition from mode-locked periodic orbit to chaos in a 2D piecewise   smooth non-invertible map
  In this work we report a new route to chaos from a resonance torus in a piecewise smooth non-invertible map of the plane into itself. The closed invariant curve defining the resonance torus is formed by the union of unstable manifolds of saddle cycle and the points of stable cycle and saddle cycle. We have found that a cusp torus cannot develop before the onset of chaos, though the loop torus appears. The destruction of the two-dimensional torus occurs through homoclinic bifurcation in the presence of an infinite number of loops on the invariant curve. We show that owing to the non-invertible nature of the map, the structure of the basin of attraction changes from simply connected to a nonconnected one. We also describe how the mechanism of transition to chaos differs from the scenario of appearance of chaos in invertible maps as well as in smooth non-invertible maps. 
Soma De, Soumitro Banerjee, Akhil Ranjan Roy