On the nonexistence, existence and uniqueness of limit cycles

We present two new criteria for studying the nonexistence, existence and uniqueness of limit cycles of planar vector fields. We apply these criteria to some families of quadratic and cubic polynomial vector fields, and to compute an explicit formula for the number of limit cycles which bifurcate out of the linear centre x = -y, y, = x, when we deal with the system x = -y + epsilon Sigma(i+j=1)(n) a(ij)x(i)y(j), y = x + epsilon Sigma(i+j=1)(n) b(ij)x(i)y(j). Moreover, by using the second criterion we present a method to derive the shape of the bifurcated limit cycles from a centre.