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PreprintCC BY 4.0arXiv

Basins of attraction and escape in the Lozi map

  • Kristijan Kilassa Kvaternik
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Abstract

For the Lozi map La,b, we consider parameter pairs for which the fixed point X in the first quadrant has no homoclinic points and the period-two orbit {P,P'} is attracting. For such parameters, let ℓ denote the set of accumulation points of the unstable manifold WXu that do not belong to WXu. We completely classify the forward asymptotic behavior of points in the phase space. The forward orbit of every point in the plane either converges to X, to the other fixed point Y in the third quadrant, or to ℓ, or it escapes to infinity. The global phase space is organized by the stable manifolds of the fixed points: WYs separates the basin of ℓ from the region of escaping orbits, while WXs is the exceptional set of points whose orbits converge to X. In particular, if 𝒜1 denotes the component of ℝ2 ∖ WYs containing X, then 𝒜1 ∖ WXs is precisely the basin of attraction of ℓ.

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Cite

BibTeXRIS
BibTeX
@misc{kvaternik2026basins,
  title = {{Basins of attraction and escape in the Lozi map}},
  author = {Kristijan Kilassa Kvaternik},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11540},
  eprint = {2609.11540},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11540},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Basins of attraction and escape in the Lozi map
AU  - Kristijan Kilassa Kvaternik
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11540
UR  - https://arxiv.org/abs/2609.11540
AB  - For the Lozi map L_a,b, we consider parameter pairs for which the fixed point X in the first quadrant has no homoclinic points and the period-two orbit {P,P'} is attracting. For such parameters, let ℓ denote the set of accumulation points of the unstable manifold W_X^u that do not belong to W_X^u. We completely classify the forward asymptotic behavior of points in the phase space. The forward orbit of every point in the plane either converges to X, to the other fixed point Y in the third quadrant, or to ℓ, or it escapes to infinity. The global phase space is organized by the stable manifolds of the fixed points: W_Y^s separates the basin of ℓ from the region of escaping orbits, while W_X^s is the exceptional set of points whose orbits converge to X. In particular, if 𝒜_1 denotes the component of ℝ^2 ∖ W_Y^s containing X, then 𝒜_1 ∖ W_X^s is precisely the basin of attraction of ℓ.
ER  -