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Let

$$\dot{y} = Ay + \cdots \, ,$$

where the dots represent higher order terms in $y$. Make the change of variables $y \mapsto x - h(y)$, where $h$ is a vector valued polynomial of order $r$ in $y$. If the eigenvalues of $A$ are non-resonant, that is, if none of them can be expressed as a linear combination of the others, then the system above can be brought to the form

$$ \dot{x} = Ax + v(x) + \cdots \, ,$$

where the dots express terms of order higher than $r$ and

$$v(x) = \frac{\partial h(x)}{\partial x} Ax - Ah(x) := L_A h(x) \, .$$

Then equation $L_A h = v$ is called the homological equation associated to $h$ ($L_A$ can be seen as a Lie derivative). The possibility for such a linearisation is called the Poincaré theorem. The context is the one of normal forms for dynamical systems.

My question is: I know some objects in mathematics are named sort of randomly, but homological is a very, very specific name. I simply cannot associate the homological equation with homology (the one with classes, exact sequences, cohomology and Betti numbers). Is there a connection or is this name simply unfortunate?

QuantumBrick
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1 Answers1

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Almost ten years after asking this question, I ended up stumbling upon an attempted answer sort of by chance. It comes from the book Normal Forms and Unfoldings for Local Dynamical Systems, by James Murdock.

What I reproduce here is his Remark 1.1.1 on page 5. Start by defining $\mathcal{V}_j$ as the space of homogeneous polynomial vector fields of degree $j{+}1$ on $\mathbb{R}^n$ (well known to those who have computed normal forms). Consider the exact sequence

$$ 0 \longrightarrow \mathcal{V}_j \stackrel{L}{\longrightarrow} \mathcal{V}_j \longrightarrow 0 \quad, $$

where $L$ is the homological operator as defined in the question. The homology space $\mathcal{V}_j/ \mathrm{im} (L)$ is precisely the space where normal form terms live.

Although I appreciate Murdock's attempt at providing an explanation for the terminology, I find this remark quite trivial (he says that himself). It does not help in understanding why it would be advantageous to interpret normal forms in a homological way. If every field in mathematics named the study of an operator's kernel as "homological something" we would be swimming in a sea of homology.

Anyway, since this is the closest to an answer that I have found, I thought it would be worthwhile to let others know.

QuantumBrick
  • 516