One-dimensional global attractor for the damped and driven sine-Gordon equation
type: Item
instance of: scholarly article
title: One-dimensional global attractor for the damped and driven sine-Gordon equation
zbMATH Open document ID: 0906.35015
DOI: 10.1007/BF02897436
author: Min Qian
author: Shengfan Zhou
author: Shu Zhu
published in: Science in China. Series A
publication date: 1999-02-16T00:00:00Z
review text: It is known that the dynamical behavior of a continuous Josephson junction is described by the damped and driven sine-Gordon equation, \[ u_{tt} +\alpha u_t- d\Delta u+ \sin u=I+ b\sin \omega t. \tag{1} \] In this paper, we consider system (1) with homogeneous Neumann boundary condition. We show that if \(d\) and \(\alpha\) satisfy \(\sigma= d\lambda_1-1>0\) and \(\alpha> \max\{2,1/ \sqrt \sigma\}\), the global attractor of the system is a one-dimensional periodic horizontal curve which attracts orbits exponentially. Therefore it is an inertial manifold when \(b=0\). Moreover, on this curve the restriction of system (1) is exactly governed by the equation of single point junction: \[ \ddot \varphi+ a\dot\varphi+ \sin \varphi= I+ b\sin \omega t. \] Therefore, the long term behavior of (1) is exactly like that of a one-dimensional system. By virtue of the periodicity, it can be regarded as a closed curve. Then the dynamics of the system could be reduced to some one-dimensional maps on a circle. This explains why physicists sometimes use circle maps to describe Josephson junctions.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37C70
zbMATH DE Number: 1180418
zbMATH Keywords: continuous Josephson junction
zbMATH Keywords: homogeneous Neumann boundary condition
zbMATH Keywords: inertial manifold
zbMATH Keywords: single point junction
MaRDI profile type: Publication
cites work: Infinite-dimensional dynamical systems in mechanics and physics
cites work: Invariant manifolds for flows in Banach spaces
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Trend to spatial homogeneity for solutions to semilinear damped wave equations
cites work: Q3240942
cites work: Q3123062
cites work: Global behavior in the dynamical equation
full work available at URL: https://doi.org/10.1007/bf02897436
OpenAlex ID: W2075321402
description: scientific article; zbMATH DE number 1180418