On the Convergence of Series Solutions of Nonintegrable Systems with Algebraic Singularities

Abstract

It has recently been shown that there exist nonintegrable 2-degree-of-freedom Hamiltonian systems with only algebraic singularities in complex time, which do not cluster on the same Riemann sheet in the t—plane. The general solution x(t), y(t) of these systems around any one of these singularities at t = t _* can be written in the form of series expansions

$$\begin{array}{*{20}{c}} {x(t) = \sum\limits_{{n \geqslant {{n}_{1}}}} {{{a}_{n}}(} t - {{t}_{*}}{{)}^{{np/q}}},} & {y(t) = \sum\limits_{n} {{{a}_{n}}{{{(t - {{t}_{*}})}}^{{nr/q}}}} } \\ \end{array}$$

with n ₁, n ₂ ∈ Z and p, q, r ∈ N. In this paper we prove, for a class of such systems, that these series converge within a finite (non-zero) radius of convergence around t = t _* . We also demonstrate numerically that this radius extends all the way to the singularity nearest to t in the complex t—plane.