On the upper bound of the number of functionally independent focal quantities of the Lyapunov differential system

Mihail Popa ; Victor Pricop

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2019), pp. 99-112.

Résumé

Denote by $N_1=2\sum _{i=1}^{\ell }(m_i+1)+2$ the maximal possible number of non-zero coefficients of the Lyapunov differential system $\dot{x}=y+\sum _{i=1}^{\ell }{P}_{m_i}(x,y)$, $\dot{y}=-x+\sum _{i=1}^{\ell }{Q}_{m_i}(x,y)$, where $P_{m_i}$ and $Q_{m_i}$ are homogeneous polynomials of degree $m_i$ with respect to $x$ and $y$, and $1$ $(\ell \mathrm{\infty })$. Then the upper bound of functionally independent focal quantities in the center and focus problem of considered system does not exceed $N_1-1$.

Bibliographie

[1] Popa M. N., Pricop V. V., “Applications of algebraic methods in solving the center-focus problem”, Bul. Acad. Ştiinţe Repub. Moldova, Matematica, 2013, no. 1(71), 45–71, arXiv: 1310.4343 [math.DS] | MR | Zbl

[2] Popa M. N., Pricop V. V., The center and focus problem: algebraic solutions and hypotheses, Academy of Sciences of Moldova, Chişinău, 2018 (in Russian) | MR

[3] Popa M. N., Algebraic methods for differential systems, The Flower Power Edit, 2004 (in Romanian) | Zbl

[4] Sibirsky K. S., Introduction to the algebraic theory of invariants of differential equations, Translated from Russian, Nonlinear Science: Theory and Applications, Manchester University Press, Manchester, 1988 | MR | Zbl

[5] Pontryagin L. S., Ordinary differential equations, Addison Wesley Publishing Company, London, 1962, vi+298 pp. | MR | Zbl

[6] Sibirsky K. S., Algebraic Invariants of Differential Equations and Matrices, Shtiintsa, Kishinev, 1976 (in Russian) | MR

[7] Sadovsky A. P., Polynomial ideals and varieties, BSU, Minsk, 2008 (in Russian)