On normal forms and return maps for pseudo-focus points

For planar systems with a pseudo-equilibrium point of focus type within its discontinuity line, computable normal forms are introduced. Thus, the classical theory of normal forms is adapted for dealing with piecewise smooth systems having a common invisible tangency from each side. The methodology looks for removing unessential terms in the expression of the vector field, and is based upon expanding the vector field as a sum of quasi-homogeneous terms, next applying adequate changes of variables that preserve every point of the discontinuity line. From these normal forms, it is easier to compute the associated half-return maps and to determine the maximal number of periodic orbits than can bifurcate from a pseudo-focus. Furthermore, a recent conjecture in this journal on the behaviour of the displacement function around an invisible fold-fold singularity is shown to be true. We illustrate the obtained results by considering some relevant examples concerning piecewise linear and linear-quadratic systems, also revisiting a piecewise smooth vector field of quasi-homogeneity degree four studied by A.F. Filippov.