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Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 2003, Number 2, Pages 13–27
(Mi basm194)
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This article is cited in 11 scientific papers (total in 11 papers)
Research articles
Algebraic equations with invariant coefficients in qualitative study of the polynomial homogeneous differential systems
Valeriu Baltag Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Chisinau, Republic of Moldova
Abstract:
For planar polynomial homogeneous real vector field $X=(P,Q)$ with $\deg(P)=\deg(Q)=n$ some algebraic equations of degree $n+1$ with $GL(2,\mathbb{R})$-invariant coefficients are constructed. A recurrent method for the construction of these coefficients is given. In the generic case each real or imaginary solution $s_i (i=1,2,\ldots,n+1)$ of the main equation is a value of the derivative of the slope function, calculated for the corresponding invariant line. Other constructed equations have, respectively, the solutions $1/s_i$, $1-s_i$, $s_i/(s_i-1)$, $(s_i-1)/s_i$, $1/(1-s_i)$. The equation with the solutions $ (n+1)s_i-1$ is called residual equation. If $X$ has real invariant lines, the values and signs of solutions of constructed equations determine the behavior of the orbits in a neighbourhood at infinity. If $X$ has not real invariant lines, it is shown that the necessary and sufficient conditions for the center existence can be expressed through the coefficients of residual equation.
Keywords and phrases:
algebraic equation, invariant, differential homogeneous system, qualitative study, center problem.
Received: 30.12.2002
Citation:
Valeriu Baltag, “Algebraic equations with invariant coefficients in qualitative study of the polynomial homogeneous differential systems”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2003, no. 2, 13–27
Linking options:
https://www.mathnet.ru/eng/basm194 https://www.mathnet.ru/eng/basm/y2003/i2/p13
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Abstract page: | 470 | Full-text PDF : | 88 | References: | 46 | First page: | 2 |
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