Countable and Uncountable Boundaries in Chaotic Scattering

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17 Citations (Scopus)

Abstract

We study the topological structure of basin boundaries of open chaotic Hamiltonian systems in general. We show that basin boundaries can be classified as either type I or type II, according to their topology. Let B be the intersection of the boundary with a one-dimensional curve. In type I boundaries, B is a Cantor set, whereas in type II boundaries B is a Cantor set plus a countably infinite set of isolated points. We show that the occurrence of one or the other type of boundary is determined by the topology of the accessible configuration space, and also by the chosen definition of escapes. We show that the basin boundary may undergo a transition from type I to type II, as the system's energy crosses a critical value. We illustrate our results with a two-dimensional scattering system.

Original languageEnglish
Article number046214
Number of pages4
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Volume66
Issue number4
DOIs
Publication statusPublished - Oct 2002

Fingerprint

Uncountable
Countable
Scattering
scattering
Cantor set
topology
Topology
Topological Structure
Configuration Space
Chaotic System
intersections
Hamiltonian Systems
escape
Critical value
Intersection
occurrences
Curve
curves
configurations
Energy

Keywords

  • open hydrodynamical flows
  • nonlinear dynamical-systems
  • fractal basin boundaries
  • WADA
  • metamorphoses
  • bifurcation
  • transients

Cite this

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title = "Countable and Uncountable Boundaries in Chaotic Scattering",
abstract = "We study the topological structure of basin boundaries of open chaotic Hamiltonian systems in general. We show that basin boundaries can be classified as either type I or type II, according to their topology. Let B be the intersection of the boundary with a one-dimensional curve. In type I boundaries, B is a Cantor set, whereas in type II boundaries B is a Cantor set plus a countably infinite set of isolated points. We show that the occurrence of one or the other type of boundary is determined by the topology of the accessible configuration space, and also by the chosen definition of escapes. We show that the basin boundary may undergo a transition from type I to type II, as the system's energy crosses a critical value. We illustrate our results with a two-dimensional scattering system.",
keywords = "open hydrodynamical flows, nonlinear dynamical-systems, fractal basin boundaries, WADA, metamorphoses, bifurcation, transients",
author = "{de Moura}, {A P S} and C Grebogi",
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doi = "10.1103/PhysRevE.66.046214",
language = "English",
volume = "66",
journal = "Physical Review. E, Statistical, Nonlinear and Soft Matter Physics",
issn = "1539-3755",
publisher = "AMER PHYSICAL SOC",
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T1 - Countable and Uncountable Boundaries in Chaotic Scattering

AU - de Moura, A P S

AU - Grebogi, C

PY - 2002/10

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N2 - We study the topological structure of basin boundaries of open chaotic Hamiltonian systems in general. We show that basin boundaries can be classified as either type I or type II, according to their topology. Let B be the intersection of the boundary with a one-dimensional curve. In type I boundaries, B is a Cantor set, whereas in type II boundaries B is a Cantor set plus a countably infinite set of isolated points. We show that the occurrence of one or the other type of boundary is determined by the topology of the accessible configuration space, and also by the chosen definition of escapes. We show that the basin boundary may undergo a transition from type I to type II, as the system's energy crosses a critical value. We illustrate our results with a two-dimensional scattering system.

AB - We study the topological structure of basin boundaries of open chaotic Hamiltonian systems in general. We show that basin boundaries can be classified as either type I or type II, according to their topology. Let B be the intersection of the boundary with a one-dimensional curve. In type I boundaries, B is a Cantor set, whereas in type II boundaries B is a Cantor set plus a countably infinite set of isolated points. We show that the occurrence of one or the other type of boundary is determined by the topology of the accessible configuration space, and also by the chosen definition of escapes. We show that the basin boundary may undergo a transition from type I to type II, as the system's energy crosses a critical value. We illustrate our results with a two-dimensional scattering system.

KW - open hydrodynamical flows

KW - nonlinear dynamical-systems

KW - fractal basin boundaries

KW - WADA

KW - metamorphoses

KW - bifurcation

KW - transients

U2 - 10.1103/PhysRevE.66.046214

DO - 10.1103/PhysRevE.66.046214

M3 - Article

VL - 66

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 4

M1 - 046214

ER -