Abstract:
This talk concerns the problem of local integrability at the origin of some families in three dimensions. First I present some results about the local integrability at the origin of a nine parametric family of a three dimensional Lotka--Volterra differential systems with 3:-1:2-resonance. More concrete I present the necessary and sufficient conditions on the parameters of the family that guarantee the existence of two independent local first integrals at the origin of coordinates. Additionally, I present the cases where the origin is linearizable. Then, I present a study of another nine parametric family of quadratic systems with 1:-2:1 resonance at the origin and the axes planes do not need to be invariant. For some subfamily I present the conditions that guarantee the non-existence of a polynomial first integral.
The first part of the talk is a joint work with W. Aziz(Salahaddin University-Erbil, Iraq), C. Christopher(Plymouth University, UK) and J. Llibre(UAB, Catalonia,Spain). The second part of the talk is a joint work with W. Aziz(Salahaddin University-Erbil, Iraq) and A. Amen(Salahaddin University-Erbil, Iraq).