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

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Tangential Homoclinic Points for Lozi Maps

  • 期刊论文

  • 发表日期：

    2026年4月9日

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  • Kristijan Kilassa Kvaternik

摘要

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.