Level sets of polynomials in several real variables.
Heiberg, C.H. (1983)
Publications de l'Institut Mathématique. Nouvelle Série
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Heiberg, C.H. (1983)
Publications de l'Institut Mathématique. Nouvelle Série
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Ivanov, Vladimir V., Volokitin, Evgenii P. (2005)
Electronic Journal of Differential Equations (EJDE) [electronic only]
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Hamadi Jerbi (2003)
ESAIM: Control, Optimisation and Calculus of Variations
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We construct explicitly an homogeneous feedback for a class of single input, two dimensional and homogeneous systems.
Jean Moulin Ollagnier (1996)
Colloquium Mathematicae
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Given a 3-dimensional vector field V with coordinates , and that are homogeneous polynomials in the ring k[x,y,z], we give a necessary and sufficient condition for the existence of a Liouvillian first integral of V which is homogeneous of degree 0. This condition is the existence of some 1-forms with coordinates in the ring k[x,y,z] enjoying precise properties; in particular, they have to be integrable in the sense of Pfaff and orthogonal to the vector field V. Thus, our theorem...
Volokitin, Evgenii P. (2003)
Electronic Journal of Differential Equations (EJDE) [electronic only]
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Javier Chavarriga, Jaume Giné (1997)
Publicacions Matemàtiques
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In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.
A. Cima, A. Gasukk, F. Mañosas (1997)
Applicationes Mathematicae
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We study planar polynomial differential equations with homogeneous components. This kind of equations present a simple and well known dynamics when the degrees (n and m) of both components coincide. Here we consider the case and we show that the dynamics is more complicated. In fact, we prove that such systems can exhibit periodic orbits only when nm is odd. Furthermore, for nm odd we give examples of such differential equations with at least (n+m)/2 limit cycles.
Javier Chavarriga, Jaume Giné (1996)
Publicacions Matemàtiques
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In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones.
Javier Chavarriga (1994)
Applicationes Mathematicae
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We study integrability of two-dimensional autonomous systems in the plane with center type linear part. For quadratic and homogeneous cubic systems we give a simple characterization for integrable cases, and we find explicitly all first integrals for these cases. Finally, two large integrable system classes are determined in the most general nonhomogeneous cases.
Peretz, R. (1993)
International Journal of Mathematics and Mathematical Sciences
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