Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions

TL;DR

The paper provides a comprehensive topological analysis of spaces of smooth asymptotically stable vector fields $\mathcal{S}^r(\mathbb{R}^n)$ and Lyapunov-like functions $\mathcal{L}^r(\mathbb{R}^n)$, showing they are path-connected, simply connected (for $n\neq 4,5$), and weakly contractible (for $n\le 3$), with a weak homotopy type equal to the nonlinear Grassmannian $\textnormal{Gr}(D^n,\mathbb{R}^n)$. The core strategy combines Lyapunov theory with diffeomorphism-group topology, proving that the sublevel-set map $p:\mathcal{L}_0^\infty(\mathbb{R}^n)\to \textnormal{Gr}_0(D^n,\mathbb{R}^n)$ is a fiber bundle with weakly contractible fibers, and then leveraging Smale–Cerf–Hatcher-type results to identify the base's topology. These insights transfer to the vector-field spaces through smoothing-to-gradient reductions, yielding weak homotopy equivalences across the several related spaces and providing a unified description via $\textnormal{Gr}(D^n,\mathbb{R}^n)$. The work also delivers notable applications: partial answers to Conley’s question, parametric forms of the Hartman-Grobman theorem for nonhyperbolic equilibria, and a parametric Morse-type lemma for degenerate minima, along with a detailed treatment of unstable components in parametric families. In addition, the paper discusses obstructions to parametric asymptotic stabilization and reveals deep connections to four-dimensional topology, including an implication that the $4$-dimensional smooth Poincaré conjecture would follow from certain path-connectedness statements in dimension five. Collectively, these results provide a robust global-topological framework for understanding asymptotic stability and Lyapunov structures in high dimensions with broad dynamical-system implications.

Abstract

We study the topology of the space of all smooth asymptotically stable vector fields on $\mathbb{R}^n$, as well as the space of all proper smooth Lyapunov functions for such vector fields. We prove that both spaces are path-connected and simply connected when $n\neq 4,5$ and weakly contractible when $n\leq 3$. Moreover, both spaces have the weak homotopy type of the nonlinear Grassmannian of submanifolds of $\mathbb{R}^n$ diffeomorphic to the $n$-disc. The proofs rely on Lyapunov theory and differential topology, such as the work of Smale and Perelman on the generalized Poincaré conjecture and results of Smale, Cerf, and Hatcher on the topology of diffeomorphism groups of discs. Applications include a partial answer to a question of Conley, a parametric Hartman-Grobman theorem for nonyperbolic but asymptotically stable equilibria, and a parametric Morse lemma for degenerate minima. We also study the related topics of hyperbolic equilibria, Morse minima, and relative homotopy groups of the space of asymptotically stable vector fields inside the space of those vanishing at a single point.

Theorems & Definitions (101)

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• Corollary 1