Bifurcations in piecewise smooth dynamical systems and applications

Abstract

This work investigates the maximum number of limit cycles in hybrid and non-hybrid piecewise smooth dynamical systems (PSDS) in ℝ2, formed by rigid subsystems and separated by a straight line, as well as the effect of a perturbation on a piecewise linear smooth system in ℝ3. It begins with the analysis of a system formed by a linear rigid center and a rigid homogeneous polynomial center of even degree 𝑛, separated by the line 𝑥=0, after an affine change of variables that preserves the rigid centers. This result complements the work of Carvalho (2022), in which the case for 𝑛 odd is studied. The present study advances the analysis of this family by addressing the even-degree case under certain conditions on the affine transformation. The research extends to PSDS formed by two rigid systems of degrees 𝑚 and 𝑛, with 2≤𝑚≤𝑛≤6, establishing that, while systems of equal degrees have at most one limit cycle, combinations of distinct degrees may exhibit specific upper bounds, reaching up to 8 limit cycles in the case (𝑚,𝑛)=(5,6). By introducing hybrid dynamics through a reset map, it is observed that the inclusion of discrete dynamics preserves the upper bounds of the continuous system, with the exception of the cases (𝑚,𝑛)=(3,6) and (𝑚,𝑛)=(4,6), in which the bound increases from 5 and 6 to 6 and 10 cycles, respectively. Finally, the study extends to three-dimensional space by investigating the effect of a perturbation on systems with parallel tangency lines, revealing that such a perturbation causes the lines to intersect at a point characterized as a parabolic two-fold singularity, which results in the emergence of new pseudo-equilibrium points without altering the qualitative structural stability of the original system.