Symmetry structure of the hyperbolic bifurcation without reflection of periodic orbits in the standard map
type: Item
instance of: scholarly article
title: Symmetry structure of the hyperbolic bifurcation without reflection of periodic orbits in the standard map
zbMATH Open document ID: 1023.37026
DOI: 10.1016/S0960-0779(00)00189-2
author: Yanyan Li
published in: Chaos, Solitons and Fractals
publication date: 2003-03-19T00:00:00Z
review text: For two-dimensional area preserving maps, the linearized tangent map determines the stability of a fixed point. When the trace of the tangent map is less than \(-2\), a fixed point is inversion hyperbolic, thus subsequent points of a map alternate across the destabilized fixed point (periodic doubling bifurcation). While for the trace of the tangent map is larger than \(+2\), a fixed point undergoes the hyperbolic bifurcation without reflection. In the paper, the processes of the hyperbolic bifurcation without reflection in the standard map \((q,p)\to (q+p+A\sin 2\pi q,p+A\sin 2\pi q)\), where \(A\) is a stochastic parameter, are examined in terms of the higher-order symmetry in the momentum inversion. The higher-order symmetry lines are shown to approach asymptotically to the separatrix of the hyperbolic fixed point. It is also shown that the existing symmetry lines cannot determine the structure of periodic islands born after the hyperbolic bifurcation without reflection. The analysis is undertaken for a period-4 orbit and a period-3 accelerator mode in the standard map.
Mathematics Subject Classification ID: 37E30
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 1884660
zbMATH Keywords: standard map
zbMATH Keywords: two-dimensional area preserving maps
zbMATH Keywords: hyperbolic bifurcation of periodic orbits
zbMATH Keywords: accelerator modes
zbMATH Keywords: symmetry lines
reviewed by: Eugene Ershov
MaRDI profile type: Publication
cites work: Q4023618
cites work: Q4220588
full work available at URL: s0960-0779(00)00189-2
OpenAlex ID: W2043832230
description: scientific article; zbMATH DE number 1884660