In this paper by using the Poincaré compactification of R 3 we make a global analysis of the model x = −ax + y + yz, y = x − ay + bxz, z = cz − bxy. In particular we give the complete description of its dynamics on the infinity sphere. For a + c = 0 or b = 1 this system has invariants. For these values of the parameters we provide the global phase portrait of the system in the Poincaré ball. We also describe the α and ω-limit sets of its orbits in the Poincaré ball.
We provide the phase portraits in the Poincaré disk for all the linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis given by the Hamiltonian function H(x, y) = 1 2 (x 2 + y 2 ) + ax 4 y + bx 2 y 3 + cy 5 in function of its parameters.
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