LAUSR

We study the maximum number of limit cycles of the polynomial differential systems of the form $$\begin{aligned} \dot{x}=-y+l(x), \,\dot{y}=x-f(x)-g(x)y-h(x)y^{2}-d_{0}y^{3}, \end{aligned}$$x˙=-y+l(x),y˙=x-f(x)-g(x)y-h(x)y2-d0y3,where $$l(x)=\varepsilon l^{1}(x)+\varepsilon ^{2}l^{2}(x),$$l(x)=εl1(x)+ε2l2(x),$$f(x)=\varepsilon f^{1}(x)+\varepsilon ^{2}f^{2}(x),$$f(x)=εf1(x)+ε2f2(x),$$g(x)=\varepsilon g^{1}(x)+\varepsilon ^{2}g^{2}(x),$$g(x)=εg1(x)+ε2g2(x),$$h(x)=\varepsilon h^{1}(x)+\varepsilon ^{2}h^{2}(x)$$h(x)=εh1(x)+ε2h2(x) and $$d_{0}=\varepsilon d_{0}^{1}+\varepsilon ^{2}d_{0}^{2}$$d0=εd01+ε2d02 where $$l^{k}(x),$$lk(x),$$f^{k}(x),$$fk(x),$$g^{k}(x)$$gk(x) and $$h^{k}(x)$$hk(x) have degree m,  $$n_{1},$$n1,$$n_{2}$$n2 and $$n_{3}$$n3 respectively, $$d_{0}^{k}\ne 0$$d0k≠0 is a real number for each $$k=1,2,$$k=1,2, and $$\varepsilon $$ε is a small parameter. We provide an upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centre $$\dot{x}=-y,\, \dot{y}=x$$x˙=-y,y˙=x using the averaging theory of first and second order.