A topological classification of plane polynomial systems having a globally attracting singular point

TL;DR

This paper classifies plane polynomial systems with a globally attracting singular point using a combinatorial invariant, providing polynomial representatives and an explicit example, thus advancing understanding of such dynamical systems.

Contribution

It introduces a topological classification method for these systems via feasible sets and supplies polynomial examples for each class, including a novel explicit example.

Findings

Classification achieved through feasible sets

Polynomial representatives for each class found

Explicit example of a non-trivial globally attracting singular point provided

Abstract

In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible set" consisting of finitely many vectors with components in the set $\{n/3: n=0,1,2,\ldots\}$), so that two such systems are equivalent if and only if (after appropriately fixing an orientation in $\mathbb{R}^2$ and a heteroclinic separatrix) they have the same feasible set. In fact, this classification is achieved in the more general setting of continuous flows having finitely many separatrices. Polynomial representatives for each equivalence class are found, although in a non-constructive way. Since, to the best of our knowledge, the literature does not provide any concrete polynomial system having a non-trivial globally attracting singular point, an explicit example is given as well.