Tangential homoclinic points for Lozi maps

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.

Figures (12)

• Figure 1: The stable (red) and unstable (blue) manifold of $X$ for parameter values $a=1$, $b=0.95$, together with some iterates of $Z$ and $V$.
• Figure 2: The figure illustrates the proof of Lemma \ref{['lem:zigzag']}. The zigzag part of the stable manifold $W_X^s$ is represented in red.
• Figure 3: Intersections of two polygonal lines $\eta$ and $\xi$ in the plane: tangential (left), transverse (middle), intersection along a segment (right).
• Figure 4: Sketch of the general structure of $W_X^u$. Here, we denote $\gamma_n = [Z^{2n-1},Z^{2n+1}]^u$ and $\delta_n = [Z^{2n},Z^{2n+2}]^u$ for all $n \in \mathbb{N}_0$.
• Figure 5: The figure illustrates the proof of Lemma \ref{['lemma:unstable_yaxis_first']}. Further iterations of $\gamma_i$ under $L_{a,b}^2$ do not intersect $L_{a,b}^{-1}(\varphi)$ outside the shaded polygon $\mathcal{F}$.

Theorems & Definitions (28)

• Theorem 1.1
• Lemma 3.1: Some geometric properties of $L_{a,b}$ and $L_{a,b}^{-1}$
• Lemma 3.2
• proof
• Lemma 3.3: Zigzag structure of $W_X^{s}$ in the third quadrant
• proof
• Definition 3.4
• Lemma 3.5
• proof
• Lemma 3.6