DYNAMICAL INTEGRATION

Abstract

In this letter we show how to use a new form of integration, called dynamical integration, that utilizes the dynamics of a system defined by an ODE to construct a map that is in effect a one-step integrator. This method contrasts sharply with classical numerical methods that utilize polynomial or rational function approximations to construct integrators. The advantages of this integrator is that it uses only one step while preserving important dynamical properties of the solution of the ODE: First, if the ODE is conservative, then the one-step integrator is measure preserving. This is significant for a system having a highly nonlinear component. Second, the one-step integrator is actually a one-parameter family of one-step maps and is derived from a continuous transformation group as is the set of solutions of the ODE. If each element of the continuous transformation group of the ODE is topologically conjugate to its inverse, then so is each member of the one-parameter family of one-step integrators. If the solutions of the ODE are elliptic, then for sufficiently small values of the parameter, the one-step integrator is also elliptic. In the limit as the parameter of the one-step family of maps goes to zero, the one-step integrator satisfies the ODE exactly. Further, it can be experimentally verified that if the ODE is chaotic, then so is the one-step integrator. In effect, the one-step integrator retains the dynamical characteristics of the solutions of the ODE, even with relatively large step sizes, while in the limit as the parameter goes to zero, it solves the ODE exactly. We illustrate the dynamical, in contrast to numerical, accuracy of this integrator with two distinctly different examples: First we use it to integrate the unforced Van der Pol equation for large ∊, ∊≥10 which corresponds to an almost continuous square-wave solution. Second, we use it to obtain the Poincaré map for two different versions of the periodically forced Duffing equation for parameter values where the solutions are chaotic. The dynamical accuracy of the integrator is illustrated by the reproduction of well-known strange attractors. The production of these attractors is eleven times longer when using a conventional fourth-order predictor-corrector method. The theory presented here extends to higher dimensions and will be discussed in detail in a forthcoming paper. However, we caution that the theory we present here is not intended as a line of research in numerical methods for ODEs.