| Full text | |
| Author(s): |
Buzzi, Claudio
;
Gasull, Armengol
;
Santana, Paulo
Total Authors: 3
|
| Document type: | Journal article |
| Source: | Journal of Differential Equations; v. 429, p. 32-pg., 2025-02-25. |
| Abstract | |
Let X be a planar smooth vector field with a polycycle Gamma n with n sides and all its corners, that are at most n singularities, being hyperbolic saddles. In this paper we study the cyclicity of Gamma n in terms of the hyperbolicity ratios of these saddles, giving explicit conditions that ensure that it is at least k, for any k <= n. Our result extends old results and also provides a more accurate proof of the known ones because we rely on some recent powerful works that study in more detail the regularity with respect to initial conditions and parameters of the Dulac map of hyperbolic saddles for families of vector fields. We also prove that when Xis polynomial there is a polynomial perturbation (in general with degree much higher that the one of X) that attains each of the obtained lower bounds for the cyclicities. Finally, we also study some related inverse problems and provide concrete examples of applications in the polynomial world. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies. (AU) | |
| FAPESP's process: | 19/10269-3 - Ergodic and qualitative theories of dynamical systems II |
| Grantee: | Claudio Aguinaldo Buzzi |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 23/02959-5 - Non-Smooth Systems and Singular Perturbations |
| Grantee: | Paulo Ricardo da Silva |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 22/14353-1 - Study of polycycles with applications in game theory |
| Grantee: | Paulo Henrique Reis Santana |
| Support Opportunities: | Scholarships abroad - Research Internship - Doctorate |
| FAPESP's process: | 21/01799-9 - The study of vector fields with applications at game theory |
| Grantee: | Paulo Henrique Reis Santana |
| Support Opportunities: | Scholarships in Brazil - Doctorate |