| Full text | |
| Author(s): |
Gouveia, Marcio R. A.
;
Llibre, Jaume
;
Roberto, Luci Any
Total Authors: 3
|
| Document type: | Journal article |
| Source: | DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS; v. N/A, p. 9-pg., 2024-04-05. |
| Abstract | |
While all the phase portraits of the quadratic polynomial Hamiltonian systems in the Poincare disc were classified in 1994 (see Artes and Llibre (J Differ Equ 107: 80-95, 1994)), we are far from the classification of the phase portraits of the cubic polynomial Hamiltonian systems in the Poincare disc. In this paper, we deal with the one-parameter family of cubic polynomial Hamiltonian systems x=y-y(y2+3x2 mu),y=x+x(x2+3y2 mu),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\dot{x}}=y-y(y<^>2+3x<^>2\mu ), \;\;\quad {\dot{y}}=x+x(x<^>2+3y<^>2\mu ), \end{aligned}$$\end{document}where (x,y)is an element of R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)\in \mathbb {R}<^>2$$\end{document} are the variables and mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} is a real parameter. We classify in the Poincare disc the topological phase portraits of this family of systems when the parameter mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} varies, describing the bifurcations which take place. (AU) | |
| FAPESP's process: | 22/04040-6 - Dynamical properties of some classes of interval maps |
| Grantee: | Márcio Ricardo Alves Gouveia |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 19/10269-3 - Ergodic and qualitative theories of dynamical systems II |
| Grantee: | Claudio Aguinaldo Buzzi |
| Support Opportunities: | Research Projects - Thematic Grants |