Authors:
Carlos Argáez
1
;
Sigurdur Hafstein
1
and
Peter Giesl
2
Affiliations:
1
University of Iceland, Iceland
;
2
University of Sussex, United Kingdom
Keyword(s):
Dynamical System, Complete Lyapunov Function, Meshless Collocation, Radial Basis Functions.
Related
Ontology
Subjects/Areas/Topics:
Dynamical Systems Models and Methods
;
Formal Methods
;
Mathematical Simulation
;
Non-Linear Systems
;
Simulation and Modeling
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.