Extend results to vector fields and generic attractors

Develop a version of the stability-preserving homotopy results that applies directly to continuous vector fields and to generic compact attractors (not only equilibria), thus extending the semiflow-level results that preserve global asymptotic stability to vector-field-level statements for general attractors.

Background

The paper proves that if two vector fields (under suitable regularity) render an equilibrium globally asymptotically stable (GAS), then their associated semiflows are homotopic through semiflows that preserve GAS. This addresses Conley’s converse question at the level of semiflows and mainly for equilibria.

The authors highlight that pushing this equivalence down to vector fields themselves, and beyond equilibria to generic attractors, would be a major strengthening and is the central remaining challenge.