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In the study of the black holes with Higgs field appears in a natural way the Lotka-Volterra differential systeṁ x = x(y − 1),ẏ = y(1 + y − 2x 2 − z 2),ż = zy, in R 3. Here we provide the qualitative analysis of the flow of this system describing the α-limit set and the ω-limit set of all orbits of this system in the whole Poincaré ball, i.e. we identify R 3 with the interior of the unit ball of R 3 centered at the origin and we extend analytically this flow to its boundary, i.e. to the infinity.

Applying the averaging theory of first, second and third order to one class generalized polynomial Liénard differential equations, we improve the known lower bounds for the maximum number of limit cycles that this class can exhibit.