On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems

Aziza Berbache

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 4, page 617-629
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Berbache, Aziza. "On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems." Mathematica Bohemica 148.4 (2023): 617-629. <http://eudml.org/doc/299487>.

@article{Berbache2023,
abstract = {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.},
author = {Berbache, Aziza},
journal = {Mathematica Bohemica},
keywords = {discontinuous piecewise differential system; continuous piecewise differential system; first integral; non-algebraic limit cycle; linear system; quadratic center},
language = {eng},
number = {4},
pages = {617-629},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems},
url = {http://eudml.org/doc/299487},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Berbache, Aziza
TI - On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 4
SP - 617
EP - 629
AB - 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.
LA - eng
KW - discontinuous piecewise differential system; continuous piecewise differential system; first integral; non-algebraic limit cycle; linear system; quadratic center
UR - http://eudml.org/doc/299487
ER -

References

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