Regilene D. S. Oliveira, Alex C. Rezende, Nicolae Vulpe
Abstract:
In this article we consider the class QS of all
non-degenerate quadratic systems. A quadratic polynomial
differential system can be identified with a single point of
R^{12} through its coefficients. In this paper using
the algebraic invariant theory we provided necessary and
sufficient conditions for a system in QS to have at least
one invariant hyperbola in terms of its coefficients. We also
considered the number and multiplicity of such hyperbolas. We
give here the global bifurcation diagram of the class QS
of systems with invariant hyperbolas. The bifurcation diagram is
done in the 12-dimensional space of parameters and it is
expressed in terms of polynomial invariants. The results can
therefore be applied for any family of quadratic systems in this
class, given in any normal form.
Submitted February 5, 2015. Published June 27, 2016.
Math Subject Classifications: 34C23, 34A34.
Key Words: Quadratic differential systems; invariant hyperbola;
affine invariant polynomials; group action.
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Regilene D. S. Oliveira Instituto de Ciências Matemáticas e de Computação Universidade de São Paulo, Brazil email: regilene@icmc.usp.br |
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Alex C. Rezende Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Brazil email: alexcrezende@gmail.com |
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Nicolae Vulpe Institute of Mathematics and Computer Science Academy of Sciences of Moldova, Moldova email: nvulpe@gmail.com |
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