Stability of certain planar unbounded polycycles (Q1604610)

The stability of a certain type of unbounded polycycles which appear in some planar differential equations is investigated. Each of these polycycles has hyperbolic corners, but the product of the hyperbolicity ratios of all its corners does not decide its stability. An explicit convergent integral whose sign gives the stability of the polycycle is obtained. The systems studied include the Kolmogorov equations \[ \dot x= xf(x,y),\quad\dot y= yg(x,y), \] and the system \[ \dot x= yf(x,y)+ g(x,y),\quad\dot y= yq(x,y), \] where \(f\), \(g\) and \(q\) are polynomials.