# Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

### Carlos Argáez, Sigurdur Hafstein, Peter Giesl

#### Abstract

Ordinary differential equations arise in a variety of applications, including e.g. climate systems, and can exhibit complicated dynamical behaviour. Complete Lyapunov functions can capture this behaviour by dividing the phase space into the chain-recurrent set, determining the long-time behaviour, and the transient part, where solutions pass through. In this paper, we present an algorithm to construct complete Lyapunov functions. It is based on mesh-free numerical approximation and uses the failure of convergence in certain areas to determine the chain-recurrent set. The algorithm is applied to three examples and is able to determine attractors and repellers, including periodic orbits and homoclinic orbits.

Download#### Paper Citation

#### in Harvard Style

Argáez C., Hafstein S. and Giesl P. (2017). **Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions** . In *Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,* ISBN 978-989-758-265-3, pages 134-144. DOI: 10.5220/0006440601340144

#### in Bibtex Style

@conference{simultech17,

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

title={Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions},

booktitle={Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,},

year={2017},

pages={134-144},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006440601340144},

isbn={978-989-758-265-3},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,

TI - Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

SN - 978-989-758-265-3

AU - Argáez C.

AU - Hafstein S.

AU - Giesl P.

PY - 2017

SP - 134

EP - 144

DO - 10.5220/0006440601340144