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Structural stability for the bidirectional cyclic negative feedback systems

Xu Cheng, Yi Wang, Dun Zhou

TL;DR

This work develops a robust framework for structural stability of bidirectional cyclic negative feedback systems under $C^1$ perturbations. By integrating a generalized Floquet theory, nested invariant cones, an integer-valued Lyapunov function, and Sard–Smale arguments, the authors prove automatic transversality, generic hyperbolicity of critical elements, and the generic Kupka–Smale property, followed by generic Morse–Smale behavior under dissipativity. The results show that asymptotic dynamics are preserved under small perturbations and that, in a broad class of models, generic vector fields exhibit structurally stable dynamics with transversal invariant-manifold intersections. This yields a rigorous, topological notion of stability for high- and possibly infinite-dimensional cyclic feedback networks with wide applicability to biological and cybernetic systems.

Abstract

The present paper investigates the structural stability of bidirectional cyclic negative feedback systems. To address this, we develop a generalized Floquet theory and construct nested invariant cones for the systems. Subsequently, we demonstrate that the Poincaré-Bendixson property for the limit set persists under $C^1$-perturbations. By applying the generalized Floquet theory and the nested invariant cones, we establish that the stable and unstable manifolds of any two connecting hyperbolic critical elements intersect transversally, with the exception of two hyperbolic equilibria possessing the same odd Morse index. Next, we prove the generic hyperbolicity of critical elements using the Sard-Smale theorem. Furthermore, by formulating the transversality condition as a functional constraint, we generically preclude connecting orbits between hyperbolic equilibrium points with the same odd Morse index, thereby establishing the generic Kupka-Smale property. Finally, under dissipative assumptions, we show that the system generically exhibits the Morse-Smale property. Our findings characterize the stability of bidirectional cyclic negative feedback systems in two distinct manners: $C^1$- small perturbations of the system do not alter the asymptotic behavior of orbits; from a topological perspective, generic vector fields in bidirectional cyclic negative feedback systems are structurally stable.

Structural stability for the bidirectional cyclic negative feedback systems

TL;DR

This work develops a robust framework for structural stability of bidirectional cyclic negative feedback systems under perturbations. By integrating a generalized Floquet theory, nested invariant cones, an integer-valued Lyapunov function, and Sard–Smale arguments, the authors prove automatic transversality, generic hyperbolicity of critical elements, and the generic Kupka–Smale property, followed by generic Morse–Smale behavior under dissipativity. The results show that asymptotic dynamics are preserved under small perturbations and that, in a broad class of models, generic vector fields exhibit structurally stable dynamics with transversal invariant-manifold intersections. This yields a rigorous, topological notion of stability for high- and possibly infinite-dimensional cyclic feedback networks with wide applicability to biological and cybernetic systems.

Abstract

The present paper investigates the structural stability of bidirectional cyclic negative feedback systems. To address this, we develop a generalized Floquet theory and construct nested invariant cones for the systems. Subsequently, we demonstrate that the Poincaré-Bendixson property for the limit set persists under -perturbations. By applying the generalized Floquet theory and the nested invariant cones, we establish that the stable and unstable manifolds of any two connecting hyperbolic critical elements intersect transversally, with the exception of two hyperbolic equilibria possessing the same odd Morse index. Next, we prove the generic hyperbolicity of critical elements using the Sard-Smale theorem. Furthermore, by formulating the transversality condition as a functional constraint, we generically preclude connecting orbits between hyperbolic equilibrium points with the same odd Morse index, thereby establishing the generic Kupka-Smale property. Finally, under dissipative assumptions, we show that the system generically exhibits the Morse-Smale property. Our findings characterize the stability of bidirectional cyclic negative feedback systems in two distinct manners: - small perturbations of the system do not alter the asymptotic behavior of orbits; from a topological perspective, generic vector fields in bidirectional cyclic negative feedback systems are structurally stable.

Paper Structure

This paper contains 16 sections, 47 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Background for cyclic feedback system
  3. Main results
  4. Remarks
  5. Integer-valued Lyapunov function
  6. Spectral theory, invariant spaces and nested invariant cones
  7. Poincaré-Bendixson property and robustness
  8. Automatic Transversality
  9. Generic hyperbolicity of critical elements
  10. Proof of Proposition \ref{['G-H-P']}
  11. Proof of Proposition \ref{['G-H-O-M']}
  12. Generic Kupka-Smale properties
  13. The main idea of proving Theorem \ref{['Generic non-existence of homoindexed connecting orbits']}
  14. $\mathcal{G}^{l,p}\left(f_0,m\right)$ is an open set in $\mathcal{U}\left(f_0,m\right)$
  15. $\mathcal{G}^{l,p}\left(f_0,m\right)$ is dense in $\mathcal{U}\left(f_0,m\right)$
  16. ...and 1 more sections

Key Result

Theorem 1.1

(Automatic Transversality) Let $\gamma ^-$, $\gamma ^+$ be hyperbolic critical elements of negative feedback system. Then if one of the following conditions is true:

Theorems & Definitions (94)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 84 more