On the finiteness of uniform sinks

TL;DR

This paper proves that under certain conditions, a flow can only have finitely many uniform sinks, extending previous results and providing new insights into the structure of dynamical systems with hyperbolic and dissipative singularities.

Contribution

It generalizes Liao's theorem by showing finiteness of uniform sinks for vector fields with specific types of singularities.

Findings

Finiteness of uniform sinks under hyperbolic or sectionally dissipative singularities

Extension of Liao's theorem to broader class of vector fields

Conditions ensuring only finitely many uniform sinks

Abstract

We study the finiteness of uniform sinks for flow. Precisely, we prove that, for $\alpha>0$ $T>0$, if a vector field $X$ has only hyperbolic singularities or sectionally dissipative singularities, then $X$ can have only finitely many $(\alpha,T)$-uniform sinks. This is a generalized version of a theorem of Liao