Structural stability for the bidirectional cyclic negative feedback systems

Abstract

We establish the generic Kupka--Smale property for bidirectional cyclic negative feedback systems on open convex domains in the compact-open $C^1$ topology. Under an inward-pointing dissipativity condition, we also prove the generic Morse--Smale property in the relative topology of the dissipative class. A key ingredient is automatic transversality between the stable and unstable manifolds of hyperbolic critical elements, except possibly when both elements are equilibria with the same odd Morse index. The proofs use an integer-valued Lyapunov function, a Floquet-type spectral decomposition, and perturbations that preserve the feedback structure. Under a stronger dissipativity condition, we further show that the equilibrium-free Poincaré--Bendixson property persists under sufficiently small multiples of bounded $C^1$ perturbations, even when these perturbations do not preserve the feedback structure.

57 pages