We investigate the dynamics of a 2-dimensional piecewise-smooth map containing -type singularity with physical system derived from tapping-mode atomic-force microscopy model. With considering the soft impact between cantilever tip and specimen surface, we obtain -type singularity in the maps. Traditionally, if a system’s vector field at least has discontinuity in its Jacobian, people call this system non-smooth system. The map we obtained from tapping-mode atomic-force microscopy has discontinuity in derivative of Jacobian, here we call it a system with lower non-smooth degree. Though it is a lower non-smooth degree map, the bifurcation structures often reveal characteristic features of non-smooth maps. We explore the reasons for such behavior in  maps, and show that this character is caused by the discontinuous rate of the change of the eigenvalues following border collision. The piecewise-smooth system with lower non-smooth degree reveals the transitional phenomena between smooth system and non-smooth system.

non-smooth system, bifurcations, dynamic analysis.