Transversality for Cyclic Negative Feedback Systems

2014/12/04 by Yi Wang, Wang, Yi, Dun Zhou +1 · 1 citation
Engineering · Physics and Astronomy · #Control and Stability of Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1412.1540

openalex publication_date 2014/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Transversality of stable and unstable manifolds of hyperbolic periodic trajectories is proved for monotone cyclic systems with negative feedback. Such systems in general are not in the category of monotone dynamical systems in the sense of Hirsch. Our main tool utilized in the proofs is the so-called cone of high rank. We further show that stable and unstable manifolds between a hyperbolic equilibrium and a hyperbolic periodic trajectory, or between two hyperbolic equilibria with different dimensional unstable manifolds also intersect transversely.

Cited by

• Structural stability for the bidirectional cyclic negative feedback systems

Related

• Structural stability for the bidirectional cyclic negative feedback systems
• The strong unstable manifold and periodic solutions in differential delay equations with cyclic monotone negative feedbck
• Averaging theory and catastrophes: The persistence of bifurcations under time-varying perturbations
• On the shifts of stable and unstable manifolds of a hyperbolic cycle under perturbation
• A geometric criterion for the existence of chaos based on periodic orbits in continuous-time autonomous systems