Dirichlet series  for dynamical systems of first-order ordinary differential equations

Bin Wang; Arieh Iserles

doi:10.3934/dcdsb.2014.19.281

Article Contents

2014, Volume 19, Issue 1: 281-298. Doi: 10.3934/dcdsb.2014.19.281

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Dirichlet series for dynamical systems of first-order ordinary differential equations

Bin Wang^1, and
Arieh Iserles^2,

1.
School of Mathematics & Physics, Qingdao University of Science & Technology, Qingdao 266061
2.
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA

Received: July 31, 2012

Revised: September 30, 2013

Published: December 2013

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Abstract

In this paper, inspired by the work by A. Iserles and G. Söderlind [Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), pp. 97-112], we present comprehensive discussion on Dirichlet series for dynamical systems of first-order ordinary differential equations (ODEs). We first derive the scheme of Dirichlet approximation for scalar dynamical systems and present the bounds on the terms of Dirichlet series. The global error and the right choice of a term in Dirichlet series are analysed and two numerical experiments are carried out to demonstrate the efficiency of Dirichlet approximation. Then we consider applying Dirichlet series to multivariate dynamical systems and present a new scheme of Dirichlet approximation for such systems. Some discussion and a numerical experiment are accordingly carried out for the new Dirichlet approximation. Compared with routine time-stepping algorithms, Dirichlet series does not need time stepping and yields a continuous solution that is equally valid along an interval, which is significant for obtaining long-time numerical solution. As a result of the special nature of Dirichlet series, the Dirichlet approximation delivers considerable information on dynamical systems of first-order ODEs and provides a novel and effective approach to numerical solutions of these dynamical systems.

Keywords:

Dirichlet series,
dynamical systems,
first-order ordinary differential equations,
stationary points,
global bounds.

Mathematics Subject Classification: Primary: 37M10, 37N30; Secondary: 65L05, 65L70.

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