Integrability versus chaos in differential systems

We analyze ordinary differential equations from the point of view of integrability, in the broad sense of existence of well behaved global conserved quantities (constants of motion). A century old and very powerful practical criterion of integrability is the Painleve property, the absence of branched movable (initial condition-dependent) singularities. We discuss why global integrals exist in this case, and their implications on global control of solutions.

Simple examples such as Abel's equation, y'=y3+t, and in fact generic equations do not satisfy Painleve property. Using Borel summability methods we show that this implies a form of ergodicity: for an open set of complex initial conditions, every trajectory is dense in an open set of solutions. When this is the case, solvability in any explicit sense, and even precise control of the solution in large complex regions of the phase space are virtually precluded.

Work in collaboration with R.D. Costin, L. Zhang, and F. Fauvet.