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In the last twenty years, the work was done on the different problems related to the Qualitative Theory of differential equations. But during the last few years, the interest surrounded around the wellknown Hilbert's Sixteenth Problem which he posed at Paris Conference of International Congress of Mathematicians in 1900, together with other twentytwo problems [17]. In this book we are mainly concerned in the second part of Hilbert's sixteenth Problem, which poses the question of maximal number and relative position of limit cycles of the polynomial system of the form: (A) in which P and Q are polynomials in x and y. We write the system A in the form of (B) Where , and , are homogeneous quadratic and cubic polynomials in x and y. Chapter No. 1 comprises the basic concepts for general theory of limit cycles and Hilbert's Sixteenth Problem. Chapter No. 2 contains an Algorithm for determining so called focal basis. This can be implemented on the computer to get the estimate for the number of smallamplitude limit cycles. Chapter No. 3 deals with some classes of system (B) with several smallamplitude limit cycles.
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