Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 5 min ago
PreprintarXiv licencearXiv

Computation of Attractor Lattices for a Class of Hybrid Systems

  • William Kalies
  • Bernardo Rivas
  • Tony Wehbe
Open at sourcePDFDOICite

Abstract

We study the attractor lattice of a class of hybrid dynamical systems on compact metric spaces. Under the trapping guard condition, we prove that hybrid attractors form a bounded distributive lattice isomorphic to the attractor lattice of an associated semiflow. The regularized preimages define a lattice epimorphism from a sublattice of the regular closed, attracting neighborhoods of the semiflow onto those of the hybrid system, and from the whole lattice when the guard set and its reset image have empty interior. Then, using grids on the state space, we construct a convergent cofiltration of outer approximations of a fixed-time map of this semiflow. This construction realizes every finite sublattice of hybrid attractors by a lattice of attracting neighborhoods that are unions of grid elements at sufficiently fine resolutions, and provides index pairs for computing Conley indices. We illustrate these constructions computationally by extracting the Conley--Morse graphs of the bouncing ball, the rimless wheel, the spiking neuron, and exhibiting the attractor lattice for an impacting oscillator.

Related

  1. On statistical inference for non-linear dynamical systems evolving in their global attractor
    Dimitri Konen, Richard Nickl
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsDynamical systemsarXiv
  2. Separation Conditions for Iterated Function Systems with a Common Attractor
    Cai-Yun Ma
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  3. Class numbers and finiteness of norm Euclidean quadratic lattices over algebraic number fields
    Zilong He
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Iterated Graph Systems (III): Alexander-Orbach conjecture on hierarchical lattices
    Ziyu Neroli
    arXivPreprintCC BY 4.0Full textMathematical physicsProbabilityarXiv
  5. On covering systems over finite nilpotent groups and lattices
    Lian Duan, Zhenchao Ge, Biao Wang +2
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{kalies2026computation,
  title = {{Computation of Attractor Lattices for a Class of Hybrid Systems}},
  author = {William Kalies and Bernardo Rivas and Tony Wehbe},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15123},
  eprint = {2610.15123},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15123},
}
RIS
TY  - UNPB
TI  - Computation of Attractor Lattices for a Class of Hybrid Systems
AU  - William Kalies
AU  - Bernardo Rivas
AU  - Tony Wehbe
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15123
UR  - https://arxiv.org/abs/2610.15123
AB  - We study the attractor lattice of a class of hybrid dynamical systems on compact metric spaces. Under the trapping guard condition, we prove that hybrid attractors form a bounded distributive lattice isomorphic to the attractor lattice of an associated semiflow. The regularized preimages define a lattice epimorphism from a sublattice of the regular closed, attracting neighborhoods of the semiflow onto those of the hybrid system, and from the whole lattice when the guard set and its reset image have empty interior. Then, using grids on the state space, we construct a convergent cofiltration of outer approximations of a fixed-time map of this semiflow. This construction realizes every finite sublattice of hybrid attractors by a lattice of attracting neighborhoods that are unions of grid elements at sufficiently fine resolutions, and provides index pairs for computing Conley indices. We illustrate these constructions computationally by extracting the Conley--Morse graphs of the bouncing ball, the rimless wheel, the spiking neuron, and exhibiting the attractor lattice for an impacting oscillator.
ER  -