# American Institute of Mathematical Sciences

May  2019, 39(5): 2437-2454. doi: 10.3934/dcds.2019103

## Construction of Lyapunov functions using Helmholtz–Hodge decomposition

 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

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

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.

Citation: Tomoharu Suda. Construction of Lyapunov functions using Helmholtz–Hodge decomposition. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2437-2454. doi: 10.3934/dcds.2019103
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##### References:
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
Contours of $V_2$ and the sign of $\dot{V_2}$. In the shaded domain, $\dot{V_2}$ is positive
Solution curves of the vector field (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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