Upper bound of the number of limit cycles for symmetric scalar piecewise linear differential equations with three zones
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916834404466688 |
|---|---|
| author | Bravo, J. L. Carmona, V. Fernández, M. Ojeda, I. |
| author_facet | Bravo, J. L. Carmona, V. Fernández, M. Ojeda, I. |
| contents | The study of the dynamics of a continuous observable and non-controllable three-dimensional symmetric piecewise linear system with three zones can be reduced to the study of the existence of limit cycles for the piecewise differential equation $x'=ax+(b-a)\mathop{\rm sat}(x)+μ\sin t$, where $\operatorname{sat}$ stands for the normalized saturation function. This paper proves that the number of limit cycles of the equation is finite independently of $a,b,μ$. Moreover, it is proven that the maximum number of limit cycles is exactly one or three for certain values of the parameters. Using Melnikov theory for a certain deformation of the equation, it is proven that for small values of the perturbation parameter, there are exactly three, five or one limit cycles, depending on the values of $μ$. This strengthens the conjecture that the equation has at most five limit cycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06705 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper bound of the number of limit cycles for symmetric scalar piecewise linear differential equations with three zones Bravo, J. L. Carmona, V. Fernández, M. Ojeda, I. Dynamical Systems Classical Analysis and ODEs Primary 34C25. Secondary: 34A34, 37C27, 37G15 The study of the dynamics of a continuous observable and non-controllable three-dimensional symmetric piecewise linear system with three zones can be reduced to the study of the existence of limit cycles for the piecewise differential equation $x'=ax+(b-a)\mathop{\rm sat}(x)+μ\sin t$, where $\operatorname{sat}$ stands for the normalized saturation function. This paper proves that the number of limit cycles of the equation is finite independently of $a,b,μ$. Moreover, it is proven that the maximum number of limit cycles is exactly one or three for certain values of the parameters. Using Melnikov theory for a certain deformation of the equation, it is proven that for small values of the perturbation parameter, there are exactly three, five or one limit cycles, depending on the values of $μ$. This strengthens the conjecture that the equation has at most five limit cycles. |
| title | Upper bound of the number of limit cycles for symmetric scalar piecewise linear differential equations with three zones |
| topic | Dynamical Systems Classical Analysis and ODEs Primary 34C25. Secondary: 34A34, 37C27, 37G15 |
| url | https://arxiv.org/abs/2507.06705 |