» amen/ Quadratic Systems with Homoclinic Cycles Described by Quintic Curves ¤

We ¯nd quadratic systems with homoclinic cycles described by quintic curves. The determination of homoclinic bifurcations of quadratic systems is not known in general [1,5,6,7]. In [2-4], cubic or quartic homoclinic cycles are found. In this paper, we present quadratic systems with homoclinic cycles which are described by quintic curves. Consider the quadratic system dx dt = P (x; y); dy dt = Q(x; y); (1) where P and Q are second order bivariate polynomials with real coe ± cients. If the system (1) has a homoclinic cycle, then without loss of any generality, we may suppose that (1) the homoclinic cycle passes through the origin, which is a hyperbolic saddle; (2) the stable and unstable manifold of the origin are tangent to the lines x2 ¡ y2 = 0 and the homoclinic orbit is located in the region D = f(x; y)jjyj < jxj; x> 0g; and (3) one of the in¯nite singular points is ìn the y-axis direction'. Under these assumptions, the corresponding normal form is dx dt = P (x; y); dy dt = Q(x; y); (2) where P (x; y) = cx + y + a1x 2 + a2xy; Q(x; y) = x + cy ¡ x2 ¡ b2xy ¡ b1y2; jcj < 1 and a1; a2; b1; b2 are real. It is easy to see that the quintic algebraic homoclinic cycle through the origin of the system must take the following form: F (x;y) = x2 ¡ y2 + F3 + F4 + F5 = 0; (3) where F3; F4 and F5 are homogeneous polynomials of degrees 3; 4 and 5 respectively. By Batins ' formula, we are able to ¯nd, by means of the software Mathematica, an invariant quintic curve of the above system