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Number of limit cycles for homogeneous polynomial system

In this paper the bifurcation of limit cycles at infinity for a class of homogeneous polynomial system of degree four is examined. This requires a problem for bifurcation of limit cycles at infinity be converted from the original system to the class of complex autonomous differential system. The evaluation of the conditions from the origin to be a centre and the highest degree fine focus results from the calculation of singular point values. A quartic system is constructed for which it can bifurcate with only one limit cycle at infinity when the normal parameters are constant.