Bifurcations in four-dimensional switched systems

In this paper, the focus is on a bifurcation of period-K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{K}$\end{document} orbit that can occur in a class of Filippov-type four-dimensional homogenous linear switched systems. We introduce a theoretical framework for analyzing the generalized Poincaré map corresponding to switching manifold. This provides an approach to capturing the possible results concerning the existence of a period-K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{K}$\end{document} orbit, stability, a number of invariant cones, and related bifurcation phenomena. Moreover, the analysis identifies criteria for the existence of multi-sliding bifurcation depending on the sensitivity of the system behavior with respect to changes in parameters. Our results show that a period-two orbit involves multi-sliding bifurcation from a period-one orbit. Further, the existence of invariant torus, crossing-sliding, and grazing-sliding bifurcation is investigated. Numerical simulations are carried out to illustrate the results.