Existence of complete Lyapunov functions with prescribed orbital derivative

Peter Giesl; Sigurdur Freyr Hafstein; Stefan Suhr

doi:10.3934/dcdsb.2022027

Article Contents

2022, Volume 27, Issue 11: 6927-6941. Doi: 10.3934/dcdsb.2022027

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Existence of complete Lyapunov functions with prescribed orbital derivative

1.
Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom
2.
Faculty of Physical Sciences, Dunhagi 5,107 Reykjavík, Iceland; Current address: Faculty of Physical Sciences, University of Iceland, Dunhagi 5,107 Reykjavík, Iceland
3.
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany

^*Corresponding author: Stefan Suhr
^*Corresponding author: Stefan Suhr

Received: September 2021

Revised: January 2022

Early access: May 20, 2022

Published online: May 20, 2022

Suhr is partially supported by the SFB/TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics", funded by the Deutsche Forschungsgemeinschaft.

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Abstract

Complete Lyapunov functions for a dynamical system, given by an autonomous ordinary differential equation, are scalar-valued functions that are strictly decreasing along orbits outside the chain-recurrent set. In this paper we show that we can prescribe the (negative) values of the derivative along orbits in any compact set, which is contained in the complement of the chain-recurrent set. Further, the complete Lyapunov function is as smooth as the vector field defining the dynamics. This delivers a theoretical foundation for numerical methods to construct complete Lyapunov functions and renders them accessible for further theoretical analysis and development.

Addendum: “Current address: Faculty of Physical Sciences, University of Iceland, Dunhagi 5,107 Reykjavík, Iceland” is added for the second author Sigurdur Freyr Hafstein. We apologize for any inconvenience this may cause.

Keywords:

Smooth complete Lyapunov function,
chain recurrent set.

Mathematics Subject Classification: Primary: 34D05, 93D30; Secondary: 37C10.

Citation:

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Figure 1. Schematic figure of a flow box $ \mathcal{V}_{\tau', r, T} $

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Figure 2. Schematic presentation of the sets $ \mathcal{V}_{s_i, 1} $ in the construction of the functions $ \tilde{\tau}_i $

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Figure 3. The first step. Note that $ M $ can intersect the boundary of $ [-(k+1), k+1]\times W_s $ at $ \{-(k+1)\}\times W_s $ and $ [-(k+1), k+1)\times \partial W_s $, but not at $ \{k+1\}\times W_s $ (right side)

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Figure 4. The second step

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Figure 5. The third step

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Figure 6. The fourth step

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