| Full text | |
| Author(s): |
de Abreu, Tiago M. P.
;
Martins, Ricardo M.
Total Authors: 2
|
| Document type: | Journal article |
| Source: | Qualitative Theory of Dynamical Systems; v. 23, n. 4, p. 14-pg., 2024-09-01. |
| Abstract | |
In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations x(center dot)=y,y(center dot)=-x-epsilon<middle dot>(f(x)<middle dot>y+sgn(y)<middle dot>g(x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=y, \ \dot{y}=-x-\varepsilon \cdot (f(x)\cdot y +\textrm{sgn}(y)\cdot g(x))$$\end{document}. Using the averaging method, we were able to generalize a previous result for Li & eacute;nard systems. In our generalization, we consider g as a polynomial of degree m. We conclude that for sufficiently small values of |epsilon|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\varepsilon }|$$\end{document}, the number hm,n=n2+m2+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{m,n}=\left[ \frac{n}{2}\right] +\left[ \frac{m}{2}\right] +1$$\end{document} serves as a lower bound for the maximum number of limit cycles in this system, which bifurcates from the periodic orbits of the linear center x(center dot)=y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=y$$\end{document}, y(center dot)=-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{y}=-x$$\end{document}. Furthermore, we demonstrate that it is indeed possible to obtain a system with hm,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{m,n}$$\end{document} limit cycles. (AU) | |
| FAPESP's process: | 18/03338-6 - Global dynamics of piecewise smooth dynamical systems |
| Grantee: | Ricardo Miranda Martins |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 21/08031-9 - Piecewise smooth vector fields on compact manifolds |
| Grantee: | Ricardo Miranda Martins |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 22/07654-5 - Invariant sets and chaotic behavior in piecewise smooth dynamical systems on compact manifolds of low dimension |
| Grantee: | Tiago Miguel Pires de Abreu |
| Support Opportunities: | Scholarships in Brazil - Doctorate |