| Full text | |
| Author(s): |
Carmona, Victoriano
;
Fernandez-Sanchez, Fernando
;
Novaes, Douglas D.
Total Authors: 3
|
| Document type: | Journal article |
| Source: | Applied Mathematics Letters; v. 137, p. 8-pg., 2023-03-01. |
| Abstract | |
The existence of a uniform upper bound for the maximum number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line has been subject of interest of hundreds of papers. After more than 30 years of investigation since Lum-Chua's work, it has remained an open question whether this uniform upper bound exists or not. Here, we give a positive answer for this question by establishing the existence of a natural number L* <= 8 for which any planar piecewise linear differential system with two zones separated by a straight line has no more than L* limit cycles. The proof is obtained by combining a newly developed integral characterization of Poincare half-maps for linear differential systems with an extension of Khovanskii's theory for investigating the number of intersection points between smooth curves and a particular kind of orbits of vector fields. (c) 2022 Elsevier Ltd. All rights reserved. (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: | 21/10606-0 - On limit cycles in piecewise linear vector fields with algebraic discontinuity variety |
| Grantee: | Douglas Duarte Novaes |
| Support Opportunities: | Scholarships abroad - Research |
| FAPESP's process: | 18/13481-0 - Geometry of control, dynamical and stochastic systems |
| Grantee: | Marco Antônio Teixeira |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 22/09633-5 - Averaging theory for studying invariant tori and periodic behavior in differential equations and inclusions |
| Grantee: | Douglas Duarte Novaes |
| Support Opportunities: | Regular Research Grants |