Complexity reduction of C-algorithm

TL;DR

This paper introduces ReCA, an efficient implementation of the C-Algorithm for determining isochronous centers in differential systems, significantly reducing computational complexity and enabling analysis of higher-degree cases.

Contribution

It presents ReCA, a reduced complexity version of the C-Algorithm, and an adapted rational case version, facilitating practical computations in isochronous center detection.

Findings

ReCA significantly reduces computational complexity.

ReCA enables analysis of quartic and higher-degree polynomial systems.

RCA extends the approach to rational systems, broadening applicability.

Abstract

The C-Algorithm introduced in [Chouikha2007] is designed to determine isochronous centers for Lienard-type differential systems, in the general real analytic case. However, it has a large complexity that prevents computations, even in the quartic polynomial case. The main result of this paper is an efficient algorithmic implementation of C-Algorithm, called ReCA (Reduced C-Algorithm). Moreover, an adapted version of it is proposed in the rational case. It is called RCA (Rational C-Algorithm) and is widely used in [BardetBoussaadaChouikhaStrelcyn2010] and [BoussaadaChouikhaStrelcyn2010] to find many new examples of isochronous centers for the Li\'enard type equation.