Integrability conditions of a resonant saddle perturbed with homogeneous quintic nonlinearities

Abstract

In this work we complete the integrability conditions (i.e. conditions for the existence of a local analytic first integral) for a family of a resonant saddle perturbed with homogeneous quintic nonlinearities studied in a previous work. In order to obtain the necessary conditions we use modular arithmetic computations.

On the maximal saddle order of <inline-formula><tex-math id="M1">\begin{document}$ p:-q $\end{document}</tex-math></inline-formula> resonant saddle

“…Since it seems too difficult to decide the exact M (p, q, n) or M h (p, q, n) for general p, q and n, it is quite natural to examine the complexity of this problem by looking for their lower bounds. By computing the saddle values with the help of computer algebra systems, many authors have presented a lot of incomplete integrable conditions for some small p, q and n. Obviously, these conditions imply the corresponding lower bounds of M (p, q, n), e.g., for p = q = 1, from the references [10,13,14,15,16,18,21,22,25,27,29], one can conclude…”

Integrability conditions of a resonant saddle in generalized Liénard-like complex polynomial differential systems

“…, where the V 2k are in fact the saddle quantities that depend on the parameters of system (3). From the recursive equations that generate V 2k we can see that these V 2k are polynomials in the parameters of system (3), see [1,3,6]. Due to the Hilbert Basis theorem, the ideal J =< V 4 , V 6 , ... > generated by the saddle quantities is finitely generated, i.e.…”

“…We compute the next saddle quantity and we obtain V 8 = a2 1 Hence we take, as before, a 4 = k 4 where k 4 is an arbitrary constant. The next saddle quantity is V 10 = a3 1 Hence we take, as before, a where k 5 is an arbitrary constant. Under these conditions V 12 = a 4 1…”

Centers and isochronicity of some polynomial differential systems

“…There are also some works on the center problem for families of polynomial systems of higher degrees, see e.g. [17,18,45,46,54,56] and references given there.…”