Centers and isochronous centers for generalized quintic systems

Agraïments: The first author is partially supported by FEDER-UNAB10-4E-378. The third author has been supported by FCT (grant PTDC/MAT/117106/2010 and through CAMGSDgrant number PEst-OE/EEI/LA0009/2013)In this paper we classify the centers and the isochronous centers of certain polynomial differential systems in R2 of degree d ≥ 5 odd that incomplex notation can be written as z˙ = (λ + i)z + (zz¯)d−52 (Az5 + Bz4z¯ + Cz3z¯2 + Dz2z¯3 + Ezz¯4 + Fz¯5),where λ ∈ R and A, B, C, D, E, F ∈ C. Note that if d = 5 we obtain the full class of polynomial differential systems of the form a linear system with homogeneous polynomial nonlinearities of degree 5. Our study uses algorithms of computational algebra based on the Groebner basis theory and modular arithmetics for simplifying the computations

An approach to solving systems of polynomials via modular arithmetics with applications

The objective of this paper is twofold. First, we describe a method to solve large systems of polynomial equations using modular arithmetics. Then, we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations