Esteban, MarinaFreire, EmilioPonce, EnriqueTorres, Francisco2025-01-082025-01-082021-10-27Marina Esteban, Emilio Freire, Enrique Ponce, Francisco Torres, On normal forms and return maps for pseudo-focus points, Journal of Mathematical Analysis and Applications, Volume 507, Issue 1, 2022, 125774, ISSN 0022-247X, https://doi.org/10.1016/j.jmaa.2021.125774.10.1016/j.jmaa.2021.125774https://hdl.handle.net/10433/22244For 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.application/pdfenAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Piecewise smooth systemsBifurcationsNormal formsPseudo-focusjournal articleopen access