Iterative Construction of Complete Lyapunov Functions.
Carlos Argáez,Peter Giesl,Sigurdur F. Hafstein +2 more
- 01 Jan 2018
- pp 211-222
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TL;DR: A new iterative algorithm that avoids obtaining trivial solutions when constructing complete Lyapunov functions is presented, based on mesh-free numerical approximation and analyzes the failure of convergence in certain areas to determine the chain-recurrent set.
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Abstract: Dynamical systems describe the evolution of quantities governed by differential equations. Hence, they represent a very powerful prediction tool in many disciplines such as physics and engineering, chemistry and biology and even in economics, among others. Their importance relies on their capability of predicting, as a function of time, future states of the corresponding system under consideration by means of the current, known state. Many difficulties arise when trying to solve such systems. Complete Lyapunov functions allow for the systematic study of complicated dynamical systems. In this paper, we present a new iterative algorithm that avoids obtaining trivial solutions when constructing complete Lyapunov functions. This algorithm is based on mesh-free numerical approximation and analyzes the failure of convergence in certain areas to determine the chain-recurrent set.
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Citations
Fast basin stability estimation for dynamic systems under large perturbations with sequential support vector machine
TL;DR: This work proposes a sequential support vector machine (SVM) framework to efficiently locate the stability boundaries and delineate the basin of attraction and reduces over 90% of the computational cost in the conventional time-domain simulation.
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Iterative Construction of Complete Lyapunov Functions: Analysis of Algorithm Efficiency
Carlos Argáez,Peter Giesl,Sigurdur F. Hafstein +2 more
- 29 Jul 2018
TL;DR: The efficiency of the algorithm is studied, important sections of the code are discussed and the systematic study of the qualitative behaviour of complicated systems is studied.
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Improved estimation of the chain-recurrent set
Carlos Argáez,Peter Giesl,Sigurdur F. Hafstein +2 more
- 25 Jun 2019
TL;DR: This work introduces a heuristic algorithm that reduces the overestimation of the chain-recurrent set in a simple and efficient way and a new and improved grid to evaluate the complete Lyapunov function is introduced to avoid unevaluated regions in the domain of the function.
7
LyapXool – a program to compute complete Lyapunov functions
TL;DR: How the code is organized, how it can be used and an interesting example of its application are provided are described.
5
Efficient Construction of Neural Networks Lyapunov Functions with Domain Of Attraction Maximization
Benjamin Bocquillon,Philippe Feyel,Guillaume Sandou,Pedro Rodriguez-Ayerbe +3 more
- 07 Jul 2020
TL;DR: This work proposes an optimization method for determining a Lyapunov function modelled by a neural network while maximizing the domain of attraction.
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References
Efficient computation of Lyapunov functions for Morse decompositions
TL;DR: It is proved that if the diameters of the grid elements go to zero, then the sequence of piecewise constant Lyapunov functions generated by the algorithm converge to a continuous LyAPunov function for the dynamics generated the nonlinear map.
Computational methods for Lyapunov functions
Peter Giesl,Sigurdur F. Hafstein +1 more
TL;DR: Lyapunov functions are functions which decrease along solution trajectories of systems, and they can be used to show stability of an invariant set, such as an equilibrium, as well as to determine its basin of attraction.
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Computational Approach for Complete Lyapunov Functions
Carlos Argáez,Peter Giesl,Sigurdur F. Hafstein +2 more
- 11 Dec 2017
TL;DR: This work presents significant improvements of an algorithm recently suggested by the authors to compute complete Lyapunov functions, incapable to fully detect chain-recurrent sets in dynamical systems with high differences in speed.
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Stability domains computation and stabilization of nonlinear systems : implications for biological systems
AI Alina Doban
- 04 Oct 2016
TL;DR: The final author version and the galley proof are versions of the publication after peer review that features the final layout of the paper including the volume, issue and page numbers.
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Computation of Complete Lyapunov Functions for Three-Dimensional Systems
Carlos Argáez,Peter Giesl,Sigurdur F. Hafstein +2 more
- 01 Dec 2018
TL;DR: This paper discusses the extension of the methods to compute complete Lyapunov functions in the plane to the three-dimensional case, which is directly applicable to higher dimensions, too.
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