A Necessary Stability Condition for Differentiable Autonomous Systems

Created on December 06, 2025 by Your Name

2025   ·   control   dynamical-systems   math

(Under Construction)

Consider the differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Let $x^* \in \mathbb{R}^n$ be an isolated equilibrium point. Because $x^*$ is isolated, there exists $\varepsilon > 0$ such that for the ball $ B_{\varepsilon}(x^*) $ centered at $x^*$ there are no other equilibrium points. Let $ S_{\varepsilon}(x^*) $ be the boundary of $ B_{\varepsilon}(x^*) $. Then we can define the well-defined map

\[f_{\varepsilon} : S_{\varepsilon}(x^*) \rightarrow S^{n-1}, \quad x \mapsto \frac{f(x)}{\Vert f(x)\Vert}.\]

We define the index of the isolated equilibrium point $x^*$ to be the topological degree of $f_{\varepsilon}$ around $x^*$. Since $S_{\varepsilon}(x^*)$ and $S^{n-1}$ are both compact smooth manifolds and $f_{\varepsilon}$ is differentiable on its domain, we can use a particular definition of degree with respect to some regular value $y \in S^{n-1}$:

\[\text{Ind}_{x^{*}}(f) = \deg (f_{\varepsilon}) = \sum_{x\in f_{\varepsilon}^{-1}(y)} \text{sign} \left( \det \left( \frac{\partial f_{\varepsilon}}{\partial x}\Big\rvert_{x} \right) \right).\]

Note that because $S_{\varepsilon}(x^*)$ is compact and $y$ is a regular value, the cardinality of $f_{\varepsilon}^{-1}(y)$ is finite. The index of a map around an equilibrium point can be useful since it provides a necessary condition for local asymptotic stability with the following statement.

Theorem.

Consider the autonomous system \[ \dot x = f(x) \] in dimension \( n \neq 4 \) with isolated equilibrium point \( x^* \). Then \( x^* \) is locally asymptotically stable only if the index of \( f \) at \( x^* \) is \( (-1)^n \), that is \[ \text{Ind}_{x^*}(f) = (-1)^n. \]

This can be a useful condition to check for nonlinear systems as opposed to the sufficient Lyapunov test. However, the index is preserved under homotopy, which is a rather weak notion of equivalence. There are plenty of linear systems which are homotopic to one another but have different stability properties.

Note that the requirement of $n \neq 4$ is because we do not know yet if sublevel sets of a smooth Lyapunov function are diffeomorphic to a 3-dimensional disk. It seems likely due to Perelman’s proof of the Poincaré conjecture, Wilson’s theorem says that superlevel sets of smooth Lyapunov functions are diffeomorphic to $S^{n-1}$ for all dimensions. This condition can actually be removed by considering homeomorphism of trajectories …