Classifications and bifurcations of tangent points and their loops of planar piecewise-smooth systems

Tangent points, especial dynamics existing only in piecewise-smooth systems, usually have dynamical properties like equilibria of smooth systems. Loops connecting tangent points own partly properties of limit cycles and homoclinic loops of smooth systems. In this paper we give classifications for tangent points by tangency degree and for loops connecting them by configuration, and investigate their bifurcations. The classic method is to construct functional parameters for the case of low tangency degree but, is no longer valid for the case of general tangency degree, which leads to complicated interlacement of sliding and crossing motions on the switching manifold. We provide an explicit unfolding for tangent points of general tangency degree and their loops, in which explicit functional functions are constructed to replace functional parameters. We mainly obtain relations between original tangency degree and numbers of bifurcating tangent points, bifurcating tangent orbits and bifurcating loops for this unfolding. Some of these relations are generalizations to general tangency degree and others are new for previous publications.