Nonintegrability of dynamical systems with homo- and heteroclinic orbits
type: Item
instance of: scholarly article
title: Nonintegrability of dynamical systems with homo- and heteroclinic orbits
zbMATH Open document ID: 1379.37113
author: Kazuyuki Yagasaki
author: Shogo Yamanaka
published in: Journal of Differential Equations
publication date: 2017-05-08T00:00:00Z
review text: The present paper concerns the integrability of a general differential equation \(\dot{x}=f(x)\) where \(f:\mathbb{R}^n\rightarrow \mathbb{R}^n\) is analytic. This equation is assumed to have equilibria connected by heteroclinic orbits on an \((n-2)\)-dimensional locally invariant manifold \(\mathcal{N}\), but it may be non-Hamiltonian and the equilibria do not have to be saddle centers. The local invariance means that the trajectory \(x(t)\) may pass through \(\partial \mathcal{N}\) and escape \(\mathcal{N}\) even if \(x(0)\in \mathcal{N}\). The equilibria are also allowed to be the same and connected by a homoclinic orbit. The main result is that, under some additional assumptions, the monodromy group of the \textit{normal variational equation} associated to the given differential equation is diagonalizable or infinitely cyclic. The used tools belong to differential Galois theory for linear differential equations.
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37J30
Mathematics Subject Classification ID: 37J05
zbMATH DE Number: 6712369
zbMATH Keywords: systems of differential equations
zbMATH Keywords: homoclinic orbit
zbMATH Keywords: Riemann surface
zbMATH Keywords: variational equation
reviewed by: Mircea Crasmareanu
MaRDI profile type: Publication
full work available at URL: j.jde.2017.03.006
OpenAlex ID: W2595758916
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DOI: 10.1016/J.JDE.2017.03.006
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description: scientific article; zbMATH DE number 6712369