Construction of Lyapunov functions using Helmholtz–Hodge decomposition

Tomoharu Suda

doi:10.3934/dcds.2019103

Article Contents

2019, Volume 39, Issue 5: 2437-2454. Doi: 10.3934/dcds.2019103

This issue Previous Article Self-excited vibrations for damped and delayed higher dimensional wave equations Next Article Diophantine approximation of the orbits in topological dynamical systems

Construction of Lyapunov functions using Helmholtz–Hodge decomposition

Tomoharu Suda

Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto, 606-8501, Japan

Received: November 2017

Revised: May 2018

Published: January 2019

The first author is supported by Grant-in-Aid for JSPS Fellows (17J03931)

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Abstract

The Helmholtz–Hodge decomposition (HHD) is applied to the construction of Lyapunov functions. It is shown that if a stability condition is satisfied, such a decomposition can be chosen so that its potential function is a Lyapunov function. In connection with the Lyapunov function, vector fields with strictly orthogonal HHD are analyzed. It is shown that they are a generalization of gradient vector fields and have similar properties. Finally, to examine the limitations of the proposed method, planar vector fields are analyzed.

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Mathematics Subject Classification: Primary: 37B25; Secondary: 37C10, 37B35.

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Figure 1. Left: Contours of $ V_1 $ and the sign of $ \dot{V_1} $. In the shaded domain, $ \dot{V_1} $ is positive. Right: Solution curves of Equation (2). A contour of $ V_1 $ is given for comparison with the left panel

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Figure 2. Contours of $ V_2 $ and the sign of $ \dot{V_2} $. In the shaded domain, $ \dot{V_2} $ is positive

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Figure 3. Solution curves of the vector field (4)

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Figure 4. Strictly orthogonal HHD of the vector field (4). Left: solution curves of $ -\nabla V $. Right: solution curves of $ {\bf u} $

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