Phase portrait of Hamiltonian systems with homogeneous nonlinearities
type: Item
instance of: scholarly article
title: Phase portrait of Hamiltonian systems with homogeneous nonlinearities
zbMATH Open document ID: 0965.34015
DOI: 10.1016/S0362-546X(99)00131-5
author: Armengol Gasull
author: Antoni Guillamon
author: V. Mañosa
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-07-17T00:00:00Z
review text: This nice paper is concerned with the Hamiltonian system of the form \[ \dot{x}=-H_y(x,y), \qquad \dot{y}=H_x(x,y), \tag{1} \] with \(H(x,y)=(x^2+y^2)/2+H_{n+1}(x,y),\) and \(H_{n+1}\) is a homogeneous polynomial of degree \(n+1.\) An algorithm which allows to depict the phase portrait of any system of type (1) in terms of the shape of \(g(\Theta)=H_{n+1}(\cos\Theta, \sin\Theta)\) is presented. This approach can be also used for obtaining the full bifurcation diagram of (1) for the case when some parameters \(\Lambda\) are involved in terms of the associated function \(g_\Lambda(\Theta).\) In order to get a complete description of the topological behaviour of the flow in the neighbourhood of a finite or infinite critical point, the authors use a variation of the Poincaré compactification and a corresponding classification of finite and infinite critical points in this compactification. Then the new algorithm is applied to obtain all phase portraits of (1) for \(n=2\) and \(n=3,\) as well as two bifurcation diagrams: the complete one for \(n=2\) and the one corresponding to reversible systems of type (1) for \(n=3.\) The results obtained are in complete correspondence with those known in the literature.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37G10
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 37G20
Mathematics Subject Classification ID: 37J20
Mathematics Subject Classification ID: 37J35
zbMATH DE Number: 1513368
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: phase portrait
zbMATH Keywords: bifurcations
zbMATH Keywords: Poincaré compactification
zbMATH Keywords: isochronous center
reviewed by: Yuriy V. Rogovchenko
MaRDI profile type: Publication
full work available at URL: s0362-546x(99)00131-5
OpenAlex ID: W2028532048
Wikidata QID: Q127720113
description: scientific article; zbMATH DE number 1513368