A Dynamical Systems Approach to the Fourth Painlevé Equation


  Michael Twiton  ,  Jeremy Schiff  
Bar-Ilan University

We use methods from dynamical systems in order to study the fourth Painlevé equation PⅣ. Our starting point is the symmetric form of PⅣ, to which the Poincaré compactification is applied. The motion on the sphere at infinity can be completely characterized. There are 14 equilibrium points, which are classified into three different types. Generic orbits of the full system are curves from 1 of 4 asymptotically unstable points to 1 of 4 asymptotically stable points, with the set of transitions that are allowed depending on the values of the parameters. This allows us to give a qualitative description of a generic real solution of PⅣ.