The use of complex time singularity analysis in dynamical systems

Abstract

Two new general results about dynamical systems are obtained using the characteristics of their complex time series solutions. These series are obtained locally around movable singularities in the complex time domain via methods which are an extension of the Painleve-Kovalevskaya test for integrability and which therefore have the advantage of being algorithmic in nature. The first of these results applies to autonomous polynomial vector fields and provides necessary and sufficient conditions for the existence of an open set of initial conditions for which the solutions will diverge to infinity as time (i.e. the independent variable) approaches some finite real value. The conditions for blow-up involve only the asymptotic leading order coefficient of the local series representation for the general solution around the complex time singularities. Additional analyses lead to the second result, which involves exponentially small separatrix splitting. When an autonomous system of ODE's possessing a homoclinic or heteroclinic orbit is perturbed by a rapidly oscillating non-autonomous term, the resulting splitting distance of the separatrix becomes exponentially small. Therefore, any first order approximation technique for measuring this splitting, e.g. the Melnikov vector, apparently loses its validity. An accurate expression for the splitting distance is valuable because it can be used to detect the presence of chaos in the system. Using only the local asymptotic forms of the solutions to the linearized variational equations and of the perturbation term, sufficient conditions on the perturbation amplitude such that the Melnikov vector gives the proper leading order splitting distance are found. This result applies to autonomous polynomial vector fields with periodic perturbations for which the amplitude of the perturbation is inversely proportional to some algebraic order of the frequency, and it depends only on the asymptotic form of the solutions near the complex time singularities.