Global structure for a class of dynamical systems (Q1573914)

The authors consider the ordinary differential equations in the plane \[ {dx\over dt}= X(x,y),\quad {dy\over dt}= Y(x,y),\tag{1} \] where \(X(x,y)\), \(Y(x,y)\) are continuous, and (1) satisfies the existence and uniqueness of solutions for the initial value problem. They are mainly interested in the global structure of positive bounded systems on the plane which have \(m\) singular points, but no closed orbits and singular closed orbits. The authors prove that these systems have at least \(m-1\) connecting orbits and all the connecting orbits, homoclinic orbits and singular points form a compact simply connected set. They study the asymptotic behaviour of these orbits as \(t\to\pm\infty\).