Spatially Homogeneous and Inhomogeneous Oscillations and Chaotic Motion in the Active Josephson Junction Line

Abstract

We consider an active Josephson junction line which forms a ring of length l. Its dynamical behavior is described by the “damped Sine-Gordon equation” $\phi _{tt} + \varepsilon \phi _t + \sin \phi = \phi _{xx} + I,\,x \in [ 0,l ]$, subject to the periodicity boundary conditions. When the parameters $\varepsilon > 0$ and $I > 0$ are taken in a suitable range, this equation has a spatially homogeneous periodic solution (of the second kind) $\phi = \phi ( {t;\varepsilon ,I} )$, which corresponds to an oscillating state of the Josephson junction. It is shown that $\phi ( {t;\varepsilon ,I} )$ is stable if $l > 0$ is small and it becomes unstable when l exceeds a certain critical length $l = l^ * ( {\varepsilon ,I} )$. A spatially inhomogeneous periodic solution (of the second kind) $\phi = \phi ( {x,t;l} )$ bifurcates from $\phi ( {t;\varepsilon ,I} )$ at this critical length. It is shown that the bifurcating periodic solution is stable [unstable] if the bifurcation at $l = l^ * $ is supercritical [subcritical]. Results of numerical solutions show that the bifurcation at $l = l^ * $ is supercritical, i.e., $\phi ( {x,t;l} )$ is defined for $l > l^ * $. Numerical results show further that there is another (secondary) bifurcation point $l = l_2 ( {\varepsilon ,I} )( { > l^ * } )$ at which another inhomogeneous periodic solution (with more complicated temporal and spatial structure) bifurcates from $\phi ( {x,t;l} )$. There is a sequence of subsequent bifurcation points $l_i ( {\varepsilon ,I} )$, $i = 3,4, \cdots ( {l_2 < l_3 < \cdots } )$, at which further complicated periodic solutions bifurcate, and a chaotic motion is observed when the length l is larger than a certain threshold length $l = l_\infty ( {\varepsilon ,I} )$.