Abstract. We characterize the centers of the quasi-homogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order.
We investigate the local integrability and linearizability of three dimensional Lotka-Volterra equations at the origin. Necessary and sufficient conditions for both integrability and linearizability are obtained for (1, −1, 1), (2, −1, 1) and (1, −2, 1)-resonance. To prove sufficiency, we mainly use the method of Darboux with extensions for inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable.
We provide necessary and sufficient conditions for both integrability and linearizability of a three dimensional vector field with quadratic nonlinearities. For our investigation we consider the case of (1 : −2 : 1)-resonance at the origin and in general non of the axes planes is invariant.Hence, we deal with a nine parametric family of quadratic systems. Some techniques like Darboux method are used to prove the sufficiency of the obtained conditions. For a particular three parametric subfamily we provide conditions to guarantee the non existence of a polynomial first integral.
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