Application of Helmholtz-Hodge decomposition to the study of certain vector fields

TL;DR

This paper explores the Helmholtz-Hodge decomposition for specific vector fields, providing methods to find orthogonal decompositions, construct Lyapunov functions, and analyze planar quadratic fields using complex potentials.

Contribution

It introduces new techniques for obtaining orthogonal Helmholtz-Hodge decompositions for linear and planar vector fields, including solving algebraic Riccati equations and applying complex potential formalism.

Findings

Orthogonal HHD for linear fields via Riccati equations

Explicit Lyapunov function construction for certain vector fields

Application of complex potentials to quadratic planar fields

Abstract

Smooth vector fields on $\mathbb{R}^n$ can be decomposed into the sum of a gradient vector field and divergence-free (solenoidal) vector field under suitable hypotheses. This is called the Helmholtz-Hodge decomposition (HHD), which has been applied to analyze the topological features of vector fields. In this study, we apply the HHD to study certain types of vector fields. In particular, we investigate the existence of strictly orthogonal HHDs, which assure an effective analysis. The first object of the study is linear vector fields. We demonstrate that a strictly orthogonal HHD for a vector field of the form $\textbf{F}(\textbf{x}) = A \textbf{x}$ can be obtained by solving an algebraic Riccati equation. Subsequently, a method to explicitly construct a Lyapunov function is established. In particular, if A is normal, there exists an easy solution to this equation. Next, we study planar vector fields. In this case, the HHD yields a complex potential, which is a generalization of the notion in hydrodynamics with the same name. We demonstrate the convenience of the complex potential formalism by analyzing vector fields given by homogeneous quadratic polynomials.