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In this paper we study the centers of projective vector fields Q_T of three-dimensional quasi-homogeneous differential system d/dt=Q() with the weight (m,m,n) and degree 2 on the unit sphere S^2. We seek the sufficient and necessary conditions under which Q_T has at least one center on S^2. Moreover, we provide the exact number and the positions of the centers of Q_T. First we give the complete classification of systems d/dt=Q() and then, using the induced systems of Q_T on the local charts of S^2, we determine the conditions for the existence of centers. The results of this paper provide a convenient criterion to find out all the centers of Q_T on S^2 with Q being the quasi-homogeneous polynomial vector field of weight (m,m,n) and degree 2.

electronic journal on the qualitative theory of differential equations

Centers of projective vector fields of spatial quasi-homogeneous systems with weight <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo form="postfix">)</mml:mo> </mml:mrow> </mml:math> and degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> on the sphere

Centers of projective vector fields of spatial quasi-homogeneous systems with weight ( m , m , n ) and degree 2 on the sphere