# Middle Point Reduction of the Chain-recurrent Set

### Carlos Argáez, Peter Giesl, Sigurdur Hafstein

#### Abstract

Describing dynamical systems requires capability to isolate periodic behaviour. In Lyapunov’s theory, the qualitative behaviour of a dynamical system given by a differential equation can be described by a scalar function that decreases along solutions: the Complete Lyapunov Function. The chain-recurrent set will produce constant values of an associated complete Lyapunov function and zero values of its orbital derivative. Recently, we have managed to isolate the chain-recurrent set of different dynamical systems in 2- and 3-di- mensions. An overestimation, however, is always obtained. In this paper, we present a method to reduce such overestimation based on geometrical middle points for 2-dimensional systems.

Download#### Paper Citation

#### in Harvard Style

Argáez C., Giesl P. and Hafstein S. (2019). **Middle Point Reduction of the Chain-recurrent Set**.In *Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,* ISBN 978-989-758-381-0, pages 141-152. DOI: 10.5220/0007920601410152

#### in Bibtex Style

@conference{simultech19,

author={Carlos Argáez and Peter Giesl and Sigurdur Hafstein},

title={Middle Point Reduction of the Chain-recurrent Set},

booktitle={Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,},

year={2019},

pages={141-152},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0007920601410152},

isbn={978-989-758-381-0},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,

TI - Middle Point Reduction of the Chain-recurrent Set

SN - 978-989-758-381-0

AU - Argáez C.

AU - Giesl P.

AU - Hafstein S.

PY - 2019

SP - 141

EP - 152

DO - 10.5220/0007920601410152