Tangential homoclinic points for Lozi maps

TL;DR

This paper investigates the boundary of parameters in Lozi maps where homoclinic points for a saddle fixed point are tangent, characterizing all such points and describing boundary curves explicitly.

Contribution

It characterizes the boundary of homoclinic points in Lozi maps, showing all such points are iterates of two special points or points on a segment, with explicit boundary equations.

Findings

All homoclinic points on the boundary are iterates of points Z and V or on a segment between them.

The boundary of the homoclinic region is composed of explicit parameter curves.

Intersections of stable and unstable manifolds are tangential or along segments on the boundary.

Abstract

For the family of Lozi maps, we study homoclinic points for the saddle fixed point $X$ in the first quadrant. Specifically, in the parameter space, we examine the boundary of the region in which homoclinic points for $X$ exist. For all parameters on that boundary, all intersections of the stable and unstable manifold of $X$, apart from $X$, are tangential, or these manifolds intersect along a segment. We ultimately prove that for such parameters, all possible homoclinic points for $X$ are iterates of two special points $Z$ and $V$, or iterates of points on a segment joining $V$ with an iterate of $Z$. Additionally, we describe the parameter curves that form the boundary and provide explicit equations for several of them.