Bifurcations in a class of polycycles involving two saddle-nodes on a Mobius band

TL;DR

This paper analyzes bifurcations of a specific class of polycycles called lips on a Mobius band, determining the maximum number of bifurcating limit cycles and describing the bifurcation diagram under generic conditions.

Contribution

It provides a detailed study of bifurcations involving two saddle-nodes on a Mobius band, including maximum limit cycles and bifurcation structure, which was previously unexplored.

Findings

Maximum number of bifurcating limit cycles is established.

Bifurcation diagram for lips polycycles is described.

Conditions for generic bifurcations are identified.

Abstract

In this paper we study the bifurcations of a class of polycycles, called lips, occurring in generic three-parameter smooth families of vector fields on a M\"obius band. The lips consists of a set of polycycles formed by two saddle-nodes, one attracting and the other repelling, connected by the hyperbolic separatrices of the saddle-nodes and by orbits interior to both nodal sectors. We determine, under certain genericity hypotheses, the maximum number of limits cycles that may bifurcate from a graphic belonging to the lips and we describe its bifurcation diagram.