Abstract
We conduct a bifurcation analysis of a single-junction superconducting quantum interferometer with an external flux. We approximate the current-voltage characteristics of the conductance in the equivalent circuit of the JJ by using two types of functions: a linear function and a piecewise linear (PWL) function. We describe a method to compute the local stability of the solution orbit and to solve the bifurcation problem. As a result, we reveal the bifurcation structure of the systems in a two-dimensional parameter plane. By making a comparison between the linear and PWL cases, we find a difference in the shapes of their bifurcation sets in the two-dimensional parameter plane even though there are no differences in the one-dimensional bifurcation diagrams or the trajectories. As for the influence of piecewise linearization, we discovered that grazing bifurcations terminate the calculation of the local bifurcations, because they drastically change the stability of the periodic orbit.
Highlights
Josephson junctions (JJs) are devices composed of two superconductors coupled with a weak link
We describe a method to compute the local stability of the solution orbit and to solve the bifurcation problem
As for the influence of piecewise linearization, we discovered that grazing bifurcations terminate the calculation of the local bifurcations, because they drastically change the stability of the periodic orbit
Summary
Josephson junctions (JJs) are devices composed of two superconductors coupled with a weak link. JJs have an extraordinary current-voltage characteristic, and circuits incorporating them show a plentiful variety of nonlinear phenomena. Hadley et al [2] found phase locking of JJ series arrays, while Cirillo and Pedersen [3] studied bifurcation phenomena and chaos in the response of JJs. Solving the bifurcation problem is important for comprehending the properties of the system, but most of the previous studies failed to solve it or did so imprecisely, e.g., Dana et al [4] suggested a simulation of JJ circuits defined by the piecewise linear conductance but did not solve its bifurcation problem. Our previous study [12] suggested a scheme to apply Kousaka’s method to the nonautonomous HDS. Ito et al [14] suggested a method to control chaos in HDS by perturbing the threshold value of the system
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