A Singularity Analysis of Integrability and Chaos in Dynamical Systems

The analysis of movable singularities in the complex time plane of n first order, ordinary differential equations has led to a better understanding of the real time behavior of the solutions of dynamical systems. The requirement that these singularities be poles, with n-1 free constants, i.e. the so-called Painlevé property, has identified many new completely integrable dynamical systems having, of course, no chaotic behavior whatsoever. On the other hand, the violation of the Painleve property with the introduction of logarithmic terms at higher orders in the series expansions has identified many dynamical systems with only “weakly chaotic” behavior. In this paper, I review these results and discuss the methods of singularity analysis with the aid of illustrative examples.