On the singularity analysis of intersecting separatrices in near- integrable dynamical systems
type: Item
instance of: scholarly article
title: On the singularity analysis of intersecting separatrices in near- integrable dynamical systems
zbMATH Open document ID: 0613.70023
DOI: 10.1016/0167-2789(87)90081-9
published in: Physica D
publication date: 1987-01-01T00:00:00Z
review text: It has been recently proved by \textit{S.L. Ziglin} [Tr. Mosk. Mat. 0.-va 41, 287-302 (1980; Zbl 0466.70009)] that transversal intersections of separatrices (invariant manifolds) in near-integrable Hamiltonian systems of two degrees of freedom imply the existence of multi-valued solutions with infinitely many Riemann sheets. Ziglin's theorem is illustrated here on a simple example and then extended and applied to non-Hamiltonian, analytic flows \(\dot x=f(x)+\epsilon g(x,t)\), with \(x\equiv (x,y)\) and \(g(x,t)=g(x,t+2\pi)\), which for \(\epsilon =0\) possess the Painlevé property. On the other hand, the theoretically expected logarithmic singularities for \(\epsilon\neq 0\) are obtained explicitly in solutions near the intersecting separatrices. Thus, we conjecture that dynamical systems with the Painlevé property, can have no separatrix intersections and hence no strange attractors, etc. These singularities are then numerically located and found to form certain very interesting ''chimney'' patterns in the complex t-plane, on which they accumulate densely. The upper part of the chimneys (away from the Re t axis) is asymptotically quite insensitive to changes in parameter values or initial conditions. The singularity pattern itself, however, becomes ''denser'' and each chimney is seen to gradually ''collapse'' towards the Re t axis, as the amplitude of the driving term increases and the motion becomes more chaotic.
Mathematics Subject Classification ID: 70K40
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 3992976
zbMATH Keywords: transversal intersections
zbMATH Keywords: separatrices
zbMATH Keywords: invariant manifolds
zbMATH Keywords: near- integrable Hamiltonian systems
zbMATH Keywords: two degrees of freedom
zbMATH Keywords: existence
zbMATH Keywords: multi- valued solutions
zbMATH Keywords: infinitely many Riemann sheets
zbMATH Keywords: logarithmic singularities
zbMATH Keywords: Painlevé property
zbMATH Keywords: separatrix intersections
zbMATH Keywords: strange attractors
author: Tassos Bountis
author: V. G. Papageorgiou
author: Martin Bier
MaRDI profile type: Publication
full work available at URL: 0167-2789(87)90081-9
OpenAlex ID: W1976431015
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description: scientific article; zbMATH DE number 3992976