RT Journal Article
T1 On normal forms and return maps for pseudo-focus points
A1 Esteban, Marina
A1 Freire, Emilio
A1 Ponce, Enrique
A1 Torres, Francisco
K1 Piecewise smooth systems
K1 Bifurcations
K1 Normal forms
K1 Pseudo-focus
AB 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.
PB Elsevier
YR 2021
FD 2021-10-27
LK https://hdl.handle.net/10433/22244
UL https://hdl.handle.net/10433/22244
LA en
NO Marina 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.
NO Departamento de Economía, Métodos Cuantitativos e Historia Económica. Universidad Pablo de Olavide.
DS RIO
RD May 9, 2026