Logarithmic singularities and chaotic behavior in Hamiltonian systems

Abstract

Several connections have been recently discovered between the real time behavior of the solution of dynamical systems and their movable singularities in the complex time plane. One of them has led to a direct method for identifying integrable Hamiltonian and dissipative systems by requiring that they possess the Painleve property, i.e. that their general solutions have no movable singularities other than poles. In this paper we further explore these connections by partially lifting the Painleve property and admitting logarithmic singularities. We find in a number of interesting Hamiltonian examples, that admitting only ln(t−t0) terms still implies globally ’’regular’’ motion and very little chaos, whereas more complicated singularities of the type lnln(t−t0) are associated with the presence of large scale regions of chaotic behavior.