In this work we classify the first integrals of all planar polynomial differential systems of deg... more In this work we classify the first integrals of all planar polynomial differential systems of degree one with real constant coefficients. Additionally we characterize when these first integrals are either polynomial, or rational , or non-algebraic. 2010 Mathematics Subject Classification. Primary 34C05, 34A34.
We study the crossing periodic orbits and limit cycles of the planar piecewise-continuous differe... more We study the crossing periodic orbits and limit cycles of the planar piecewise-continuous differential systems separated by the straight-line [Formula: see text] having in [Formula: see text] the general quadratic isochronous center [Formula: see text], [Formula: see text] after an affine transformation, and in [Formula: see text] an arbitrary quadratic isochronous center except for the quadratic isochronous center [Formula: see text], [Formula: see text] which has been studied in [Ghermoul et al., 2021]. For these piecewise-continuous differential systems the upper bound of crossing limit cycles is 2, and there are specific examples having one crossing limit cycle.
La mécanique des ‡uides est un domaine très vaste qui nécessite un large spectre des connaissance... more La mécanique des ‡uides est un domaine très vaste qui nécessite un large spectre des connaissances physiques et mathématiques pour modéliser et puis résoudre un problème d'écoulement posé. Dans notre mémoire, on va faire une synthèse fondamentale nécessaire à l'étude des écoulements des ‡uides. On donne une notion brève sur le modèle de la mécanique des milieux continus en générale, puis on se limite aux ‡uides dits newtoniens pour l'établissement des équations mathématiques.
In the name of Allah the Merciful, Praise to Allah, Lord of the Worlds, Praise be to the Lord of ... more In the name of Allah the Merciful, Praise to Allah, Lord of the Worlds, Praise be to the Lord of all worlds. Prayers and peace be upon our Prophet, Muhammad, his family and all of his companions. Firstly, I would like to express my sincere gratitude to my supervisor Prof. Ahmed Bendjeddou for the continuous support of my PhD thesis and related research, for his patience and motivation. Secondly, I would like to thank Prof. Jaume Llibre from Universitat Autònoma de Barcelona for his guidance in some problems during preparation of my PhD thesis, without forgetting my colleague Dr. Tayeb Salhi from Mohamed El-Bachir El-Ibrahimi university. Besides all of them, I would like to thank members of my thesis committee: Prof.
In this paper, we consider a class of planar quintic differential systems, for which a non-algebr... more In this paper, we consider a class of planar quintic differential systems, for which a non-algebraic limit cycle around a non-elementary critical point is given and it is the unique limit cycle. The non-algebraic limit cycle is constructed explicitly by using polar coordinates.
International Journal of Bifurcation and Chaos, 2022
First, we study the planar continuous piecewise differential systems separated by the straight li... more First, we study the planar continuous piecewise differential systems separated by the straight line [Formula: see text] formed by a linear isochronous center in [Formula: see text] and an isochronous quadratic center in [Formula: see text]. We prove that these piecewise differential systems cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles. Second, we study the crossing periodic orbits and limit cycles of the planar continuous piecewise differential systems separated by the straight line [Formula: see text] having in [Formula: see text] the general quadratic isochronous center [Formula: see text], [Formula: see text] after an affine transformation, and in [Formula: see text] an arbitrary quadratic isochronous center. For these kind of continuous piecewise differential systems the maximum number of crossing limit cycles is one, and there are examples having one crossing limit cycles. In short for these families of continuous piecewise diffe...
In this work we classify the first integrals of all planar polynomial differential systems of deg... more In this work we classify the first integrals of all planar polynomial differential systems of degree one with real constant coefficients. Additionally we characterize when these first integrals are either polynomial, or rational , or non-algebraic. 2010 Mathematics Subject Classification. Primary 34C05, 34A34.
We study the crossing periodic orbits and limit cycles of the planar piecewise-continuous differe... more We study the crossing periodic orbits and limit cycles of the planar piecewise-continuous differential systems separated by the straight-line [Formula: see text] having in [Formula: see text] the general quadratic isochronous center [Formula: see text], [Formula: see text] after an affine transformation, and in [Formula: see text] an arbitrary quadratic isochronous center except for the quadratic isochronous center [Formula: see text], [Formula: see text] which has been studied in [Ghermoul et al., 2021]. For these piecewise-continuous differential systems the upper bound of crossing limit cycles is 2, and there are specific examples having one crossing limit cycle.
La mécanique des ‡uides est un domaine très vaste qui nécessite un large spectre des connaissance... more La mécanique des ‡uides est un domaine très vaste qui nécessite un large spectre des connaissances physiques et mathématiques pour modéliser et puis résoudre un problème d'écoulement posé. Dans notre mémoire, on va faire une synthèse fondamentale nécessaire à l'étude des écoulements des ‡uides. On donne une notion brève sur le modèle de la mécanique des milieux continus en générale, puis on se limite aux ‡uides dits newtoniens pour l'établissement des équations mathématiques.
In the name of Allah the Merciful, Praise to Allah, Lord of the Worlds, Praise be to the Lord of ... more In the name of Allah the Merciful, Praise to Allah, Lord of the Worlds, Praise be to the Lord of all worlds. Prayers and peace be upon our Prophet, Muhammad, his family and all of his companions. Firstly, I would like to express my sincere gratitude to my supervisor Prof. Ahmed Bendjeddou for the continuous support of my PhD thesis and related research, for his patience and motivation. Secondly, I would like to thank Prof. Jaume Llibre from Universitat Autònoma de Barcelona for his guidance in some problems during preparation of my PhD thesis, without forgetting my colleague Dr. Tayeb Salhi from Mohamed El-Bachir El-Ibrahimi university. Besides all of them, I would like to thank members of my thesis committee: Prof.
In this paper, we consider a class of planar quintic differential systems, for which a non-algebr... more In this paper, we consider a class of planar quintic differential systems, for which a non-algebraic limit cycle around a non-elementary critical point is given and it is the unique limit cycle. The non-algebraic limit cycle is constructed explicitly by using polar coordinates.
International Journal of Bifurcation and Chaos, 2022
First, we study the planar continuous piecewise differential systems separated by the straight li... more First, we study the planar continuous piecewise differential systems separated by the straight line [Formula: see text] formed by a linear isochronous center in [Formula: see text] and an isochronous quadratic center in [Formula: see text]. We prove that these piecewise differential systems cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles. Second, we study the crossing periodic orbits and limit cycles of the planar continuous piecewise differential systems separated by the straight line [Formula: see text] having in [Formula: see text] the general quadratic isochronous center [Formula: see text], [Formula: see text] after an affine transformation, and in [Formula: see text] an arbitrary quadratic isochronous center. For these kind of continuous piecewise differential systems the maximum number of crossing limit cycles is one, and there are examples having one crossing limit cycles. In short for these families of continuous piecewise diffe...
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Papers by Bilal Ghermoul