One-dimensional global attractor for the damped and driven sine-Gordon equation
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.
scientific article; zbMATH DE number 1180418