Joan C. Artes, Jaume Llibre, Nicolae Vulpe
Abstract:
When one considers a quadratic differential system, one realizes
that it depends on 12 parameters of which one can be fixed by
means of a time change. One also can notice that fixing 4 finite
real singular points plus 3 infinite real ones (all its possible
singular points) implies to fix 11 conditions, that is, 11
equations that the parameters must satisfy. Since these conditions
are linear with respect to the parameters, it is obvious to think
that the system will be determined, except that the fixed conditions
are incompatible with a quadratic differential system having
finitely many singular points.
In this paper we prove exactly this. That is, if we fix the
position of the 7 singular points of a quadratic differential system
in a distribution that does not force an infinite number of finite
singular points, then the system is completely determined, and
consequently its phase portrait is also determined. This determination
includes the local behavior of all singular points, even if they are weak
focus or centers, the global behavior of separatrices, and even
the existence or not of limit cycles. This also implies that limit
cycles are sensitive to small perturbations of the coordinates of
singular points, even if they are far from the singular points.
The result of the paper goes far beyond this, since we state that
this result is independent of the fact that the fixed singular points are
real or complex, and it does not mind if the infinite singular points
are simple or multiple due to the collision of several infinite singular
points. Only when some data is lost due to the collision of finite
singular points or to the collision of some finite singular points with
infinite ones, this adds free parameters to the set of parameters at
the same rate than the number of finite singular points are
lost.
Submitted September 10, 2006. Published May 30, 2008.
Math Subject Classifications: 34C05, 34C08.
Key Words: Quadratic systems; singular points.
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Joan C. Artés Departament de Matemátiques, Universitat Autónoma de Barcelona 08193 Bellaterra, Barcelona, Catalonia, Spain email: artes@mat.uab.cat |
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Jaume Llibre Departament de Matemátiques, Universitat Autónoma de Barcelona 08193 Bellaterra, Barcelona, Catalonia, Spain email: jllibre@mat.uab.cat |
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Nicolae Vulpe Institute of Mathematics and Computer Science, Academy of Science of Moldova 5 Academiei str, Chisinau, MD-2028, Moldova email: nvulpe@mail.md |
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