BIFURCATIONS OF LIMIT CYCLES IN A CLASS OF QUARTIC PLANAR VECTOR FIELDS

This paper studies the number and the distribution of limit cycles of a class of planar quartic vector fields <p class="disp_formula">$ \dot{x}=-y(ay^2-1)+\varepsilon p(x, y),\; \; \dot{y}=x(ax^2-\sqrt{a}x-1)+\varepsilon q(x, y), $ where $ a&gt;0 $, $ 0&lt;\varepsilon\ll 1 $, $ p(x,y) $ and $ q(x,y) $ are polynomials in $ (x,y) $ of the degree $ 4 $. By the bifurcation theory and qualitative analysis, we obtain four new configurations of limit cycles, two of which can have at least $ 12 $ limit cycles.