Transversality for Cyclic Negative Feedback Systems

TL;DR

This paper proves the transversality of stable and unstable manifolds in monotone cyclic systems with negative feedback, extending understanding beyond traditional monotone dynamical systems using cone of high rank.

Contribution

It establishes transversality results for hyperbolic trajectories in systems not classified as monotone, employing the cone of high rank as a key analytical tool.

Findings

Transversality of manifolds between hyperbolic periodic trajectories and equilibria.

Extension of transversality results to systems with different types of hyperbolic trajectories.

Use of cone of high rank to analyze non-monotone cyclic systems.

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.