Published August 4, 2023
| Version v1
Publication
Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region
Contributors
Others:
- Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI)
- Universidad de Sevilla. TIC130: Investigación en Sistemas Dinámicos en Ingeniería
- Ministerio de Ciencia, Innovación y Universidades (MICINN). España
- Fondo Europeo de Desarrollo Regional (FEDER)
- Ministerio de Economia, Industria y Competitividad (MINECO). España
- Consejería de Educación y Ciencia. Junta de Andalucía
- Consejería de Economía, Conocimiento, Empresas y Universidad. Junta de Andalucía
Description
In this paper, we consider the family of planar piecewise linear differential systems
with two zones separated by a straight line without sliding regions, that is, differential
systems whose flow transversally crosses the switching line except for at most one
point. In the research literature, many papers deal with the problem of determining the
maximum number of limit cycles that these differential systems can have. This problem
has been usually approached via large case-by-case analyses which distinguish the many
different possibilities for the spectra of the matrices of the differential systems. Here,
by using a novel integral characterization of Poincaré half-maps, we prove, without
unnecessary distinctions of matrix spectra, that the optimal uniform upper bound for
the number of limit cycles of these differential systems is one. In addition, it is proven
that this limit cycle, if it exists, is hyperbolic and its stability is determined by a simple
condition in terms of the parameters of the system. As a byproduct of our analysis, a
condition for the existence of the limit cycle is also derived.
Abstract
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Additional details
Identifiers
- URL
- https://idus.us.es/handle//11441/148406
- URN
- urn:oai:idus.us.es:11441/148406