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Construction of Lyapunov functions for the estimation of basins of attraction

Spelsberg-Korspeter, G. and Hochlenert, D. and Heffel, Eduard and Wagner, A. and Hagedorn, P. and Sampaio, R. (2012):
Construction of Lyapunov functions for the estimation of basins of attraction.
In: Journal of the Brazilian Society of Mechanical Sciences and Engineering, pp. 633-639, 34, (2), ISSN 1678-5878,
[Article]

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Abstract

Technical systems are often modeled through systems of differential equations in which the parameters and initial conditions are subject to uncertainties. Usually, special solutions of the differential equations like equilibrium positions and periodic orbits are of importance and frequently the corresponding equations are only set up with the intent to describe the behavior in the vicinity of a limit cycle or an equilibrium position. For the validity of the analysis it must therefore be assumed that the initial conditions lie indeed in the basins of attraction of the corresponding attractors. In order to estimate basins of attraction, Lyapunov functions can be used. However, there are no systematic approaches available for the construction of Lyapunov functions with the goal to achieve a good approximation of the basin of attraction. The present paper suggests a method for defining appropriate Lyapunov functions using insight from center manifold theory. With this approach, not only variations in the initial conditions, but also in the parameters can be studied. The results are used to calculate the likelihood for the system to reach a certain attractor assuming different random distributions for the initial conditions.

Item Type: Article
Erschienen: 2012
Creators: Spelsberg-Korspeter, G. and Hochlenert, D. and Heffel, Eduard and Wagner, A. and Hagedorn, P. and Sampaio, R.
Title: Construction of Lyapunov functions for the estimation of basins of attraction
Language: English
Abstract:

Technical systems are often modeled through systems of differential equations in which the parameters and initial conditions are subject to uncertainties. Usually, special solutions of the differential equations like equilibrium positions and periodic orbits are of importance and frequently the corresponding equations are only set up with the intent to describe the behavior in the vicinity of a limit cycle or an equilibrium position. For the validity of the analysis it must therefore be assumed that the initial conditions lie indeed in the basins of attraction of the corresponding attractors. In order to estimate basins of attraction, Lyapunov functions can be used. However, there are no systematic approaches available for the construction of Lyapunov functions with the goal to achieve a good approximation of the basin of attraction. The present paper suggests a method for defining appropriate Lyapunov functions using insight from center manifold theory. With this approach, not only variations in the initial conditions, but also in the parameters can be studied. The results are used to calculate the likelihood for the system to reach a certain attractor assuming different random distributions for the initial conditions.

Journal or Publication Title: Journal of the Brazilian Society of Mechanical Sciences and Engineering
Volume: 34
Number: 2
Uncontrolled Keywords: Lyapunov functions, basins of attraction, center manifold theory
Divisions: 16 Department of Mechanical Engineering
16 Department of Mechanical Engineering > Dynamics and Vibrations
Exzellenzinitiative
Exzellenzinitiative > Graduate Schools
Exzellenzinitiative > Graduate Schools > Graduate School of Computational Engineering (CE)
Zentrale Einrichtungen
Date Deposited: 28 May 2014 12:50
Identification Number: doi:10.1590/S1678-58782012000600012
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