Jeremy Schiff (Bar-Ilan University, Israel)
I will review the notion of integrability for different kinds of (systems of) ordinary differential equations, and the relationship with the Painlevé property, that the only movable singularities of solutions of a differential equation are poles. I will show that the series solutions used in various tests for the Painlevé property are essentially coordinate transformations linearizing the relevant flow near certain "fixed points at infinity" (another notion to be explained), and show how the criterion of linearizability near these fixed points at infinity can be used to detect integrable cases that cannot be found using existing Painlevé tests. I will illustrate the role fixed points at infinity play in understanding solutions of differential equations, using the examples of 3-dimensional Lotka-Volterra systems and the fourth Painlevé equation.
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