| Full text | |
| Author(s): |
Braun, Francisco
;
da Cruz, Leonardo Pereira Costa
;
Torregrosa, Joan
Total Authors: 3
|
| Document type: | Journal article |
| Source: | NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS; v. 79, p. 15-pg., 2024-05-04. |
| Abstract | |
We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to nonsmooth perturbations of the four families of isochronous centers of the Loud family, S1, S2, S3, and S4, as well as to non-smooth perturbations of non-smooth centers given by putting different Si's in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper. (AU) | |
| FAPESP's process: | 22/14484-9 - Applications in biology of piecewise differential equations |
| Grantee: | Leonardo Pereira Costa da Cruz |
| Support Opportunities: | Scholarships abroad - Research Internship - Post-doctor |
| FAPESP's process: | 19/07316-0 - Singularity theory and its applications to differential geometry, differential equations and computer vision |
| Grantee: | Farid Tari |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 21/14987-8 - Bifurcation of limit cycles in smooth piecewise systems and an application in Medicine |
| Grantee: | Leonardo Pereira Costa da Cruz |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| FAPESP's process: | 20/14498-4 - Global injectivity of maps and related topics |
| Grantee: | Francisco Braun |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 23/00376-2 - Global injectivity of maps and related topics |
| Grantee: | Francisco Braun |
| Support Opportunities: | Regular Research Grants |