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Construction of Finsler-Lyapunov functions with meshless collocation
We study the stability of invariant sets such as equilibria or periodic orbits of a Dynamical System given by a general autonomous nonlinear ordinary differential equation (ODE). A classical tool to analyse the stability are Lyapunov functions, i.e. scalar-valued functions, which decrease along solutions of the ODE. An alternative to Lyapunov functions is contraction analysis. Here, stability (or incremental stability) is a consequence of the contraction property between two adjacent solutions, formulated as the local property of a Finsler-Lyapunov function. This has the advantage that the invariant set plays no special role and does not need to be known a priori. In this paper, we propose a method to numerically construct a Finsler-Lyapunov function by solving a first-order partial differential equation using meshless collocation. Depending on the expected attractor, the contraction only takes place in certain directions, which can easily be implemented within the method. In the basin of attraction of an exponentially stable equilibrium or periodic orbit, we show that the PDE problem has a solution, which provides error estimates for the numerical method. Furthermore, we show how the method can also be applied outside the basin of attraction and can detect the stability as well as the stable/unstable directions of equilibria. The method is illustrated with several examples.
History
Publication status
- Published
File Version
- Accepted version
Journal
ZAMMISSN
0044-2267Publisher
WileyExternal DOI
Issue
4Volume
99Page range
1-18Article number
e201800141Department affiliated with
- Mathematics Publications
Research groups affiliated with
- Analysis and Partial Differential Equations Research Group Publications
Notes
Article published in Special Issue: 14th International conference on “Dynamical Systems — Theory and Applications” 2017Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2018-12-14First Open Access (FOA) Date
2020-01-09First Compliant Deposit (FCD) Date
2018-12-13Usage metrics
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