Integrating the nonintegrable: Analytic structure of the Lorenz system revisited

A study of the complex time analytic structure of the Lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one (time-dependent) integral of motion exists. Furthermore, the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral. Recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument. The analytic techniques used in these studies, a systematic resummation of a logarithmic psi-series, appears to be quite general and can provide explicit representations of a solution-even in the chaotic regimes-in the neighborhood of a given movable singularity. Furthermore, we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems.