On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems
We study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However...
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2023-12-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/148/4/mb148_4_13.pdf |
| _version_ | 1797540016696590336 |
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| author | Aziza Berbache |
| author_facet | Aziza Berbache |
| author_sort | Aziza Berbache |
| collection | DOAJ |
| description | We study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However, when the piecewise differential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we have solved the extension of the 16th Hilbert problem to this class of differential systems. |
| first_indexed | 2024-03-10T12:54:59Z |
| format | Article |
| id | doaj.art-439db790a58c4107b909850694ea5a5b |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-03-10T12:54:59Z |
| publishDate | 2023-12-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-439db790a58c4107b909850694ea5a5b2023-11-21T12:00:13ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362023-12-01148461762910.21136/MB.2022.0181-21MB.2022.0181-21On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systemsAziza BerbacheWe study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However, when the piecewise differential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we have solved the extension of the 16th Hilbert problem to this class of differential systems.http://mb.math.cas.cz/full/148/4/mb148_4_13.pdf discontinuous piecewise differential system continuous piecewise differential system first integral non-algebraic limit cycle linear system quadratic center |
| spellingShingle | Aziza Berbache On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems Mathematica Bohemica discontinuous piecewise differential system continuous piecewise differential system first integral non-algebraic limit cycle linear system quadratic center |
| title | On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems |
| title_full | On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems |
| title_fullStr | On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems |
| title_full_unstemmed | On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems |
| title_short | On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems |
| title_sort | on limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems |
| topic | discontinuous piecewise differential system continuous piecewise differential system first integral non-algebraic limit cycle linear system quadratic center |
| url | http://mb.math.cas.cz/full/148/4/mb148_4_13.pdf |
| work_keys_str_mv | AT azizaberbache onlimitcyclesofpiecewisedifferentialsystemsformedbyarbitrarylinearsystemsandaclassofquadraticsystems |