A combinatorial procedure for finding isolating neighbourhoods and index pairs

1997 ◽  
Vol 127 (5) ◽  
pp. 1075-1088 ◽  
Author(s):  
Andrzej Szymczak
Keyword(s):  
Periodic Orbits ◽  
Conley Index ◽  
Finite Union ◽  
Computer Assisted ◽  
Invariant Subset ◽  
Index Pair ◽  
Index Pairs

SynopsisWe present a purely combinatorial procedure for finding an isolating neighbourhood and an index pair contained in a given set, being a finite union of cubes in Rs. It is applied for a computer-assisted computation of the Conley index of an isolated invariant subset of the Hénon attractor. As a corollary, it is shown that the Hénon attractor contains periodic orbits of all principal periods except for 3 and 5.

scholarly journals On Computer-Assisted Proving The Existence Of Periodic And Bounded Orbits

2015 ◽  
Vol 29 (1) ◽  
pp. 7-17
Author(s):  
Roman Srzednicki
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Conley Index ◽  
Computer Assisted ◽  
Small Step ◽  
Index Pair ◽  
Time Periodic ◽  
Autonomous Ordinary Differential Equation

AbstractWe announce a new result on determining the Conley index of the Poincaré map for a time-periodic non-autonomous ordinary differential equation. The index is computed using some singular cycles related to an index pair of a small-step discretization of the equation. We indicate how the result can be applied to computer-assisted proofs of the existence of bounded and periodic solutions. We provide also some comments on computer-assisted proving in dynamics.

Finding Unstable Fixed Points From Time Series Data

2003 ◽  
Author(s):  
Takaaki Maehara ◽  
Mikio Nakai
Keyword(s):  
Time Series ◽  
Fixed Points ◽  
Time Series Data ◽  
Piecewise Linear ◽  
Duffing Oscillator ◽  
Conley Index ◽  
Series Data ◽  
Experimental Time ◽  
Index Pairs ◽  
Experimental Time Series

This study employs topological methods to extract unstable fixed points in phase space from both numerical and experimental time series data. Conley index of an isolated invariant subset and the R-B method can determine unstable fixed points contained in strange attractor from numerical time series data. For experimental time series data, the theorem for the relationship between index pairs and Conley index enables one to predict them with acceptable accuracy. As a corollary, some results for Duffing oscillator and piecewise linear system are shown.

Computer-assisted proof of skeletons of periodic orbits

2012 ◽  
Vol 183 (1) ◽  
pp. 80-85 ◽  
Author(s):  
Roberto Barrio ◽  
Marcos Rodríguez ◽  
Fernando Blesa
Keyword(s):  
Periodic Orbits ◽  
Computer Assisted ◽  
Computer Assisted Proof

THE THIRD POWER OF THE SMALE HORSESHOE INDUCES ALL LINK TYPES

2000 ◽  
Vol 09 (07) ◽  
pp. 939-953 ◽  
Author(s):  
EIKO KIN
Keyword(s):  
Periodic Orbits ◽  
Solid Torus ◽  
Finite Union ◽  
Smale Horseshoe ◽  
The Third

Let φ:D2→D2 be an orientation preserving homeomorphism of the disk into itself, and Φ= {φt}0≤t≤1 an isotopy with φ0=idD2 and φ1=φ. Then for a finite union of periodic orbits P of φ, the set [Formula: see text] is a link in D2×S1. We say that φ induces all link types (for Φ) if there exists a homeomorphism h of D2×S1 into a standardly embedded solid torus in the 3-sphere S3 such that any link L in S3 can be realized by a finite union of periodic orbits PL of φ so that L and [Formula: see text] are equivalent. We will show that the Smale horseshoe and its second power do not induce all link types, but its third power does induce all link types.

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