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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.

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. For example, Salam et al. [

On the other hand, hybrid dynamical systems (HDSs) have been studied intensively by many researchers [

In this study, we solve the bifurcation problem of a single-junction superconducting quantum interferometer with an external flux. We assume two types of conductance in the equivalent circuit of the JJ: a linear conductance and a PWL conductance. By defining a PWL function that models the actual response of the JJ, we expect we can determine its properties. We use the HDS approach [

Let us consider a single-junction superconducting quantum interferometer (SQUID) with an external flux [

(a) Single-junction superconducting quantum interferometer with external flux

The circuit equations for the circuit in Figure

(a) Modeled conductance characteristic in the Josephson junction equivalent circuit. (b) Approximations of (a) by a piecewise linear function.

An

Since one of the state variables in (

Equation (

When discussing the asymptotic stability of the periodic orbit, we generally use

The variational equation from the fixed point

A fixed point is

A fixed point is

A fixed point is

A fixed point is

The index at the bottom left of the symbol indicates the number of unstable dimensions for the fixed point. Generally,

We numerically obtain the point

By perturbing some of the parameters, we find the parameter sets where the stability of

Let us consider the case that a periodic orbit passes near the border

The condition for a grazing bifurcation to occur is that a periodic orbit and the border of the system are tangent to each other at the time

Let us choose the parameter values as follows:

In the case of a linear JJ with

Bifurcation diagram of system (

1-dimensional bifurcation diagram of system (

By increasing

By decreasing

Trajectories of system (

By the way, there is a discontinuity in the curve of

In the case of a PWL JJ with

Bifurcation diagram of system (

1-dimensional bifurcation diagram of system (

By increasing

By decreasing

Another phenomenon in Figure

Focusing on these grazing bifurcations, let us discuss the change in stability of periodic orbits. Figure

Enlargement of Figure

Trajectories of system (

We conducted a bifurcation analysis of a single-junction superconducting quantum interferometer with an external flux. We used two types of conductance: a linear conductance and a PWL conductance, to approximate the current-voltage characteristic of the JJ. We described a method to compute the local stability of the solution orbit and to solve the bifurcation problem of the system. For the case of the PWL function, we applied the hybrid dynamical system approach to the system.

The main results of the analysis of system (

We exactly revealed the bifurcation structure of the system in two-dimensional parameter space. From the one-dimensional bifurcation diagram, we plainly explained what kind of bifurcation arises.

The comparison of the linear function and PWL function cases indicated a difference in the shapes of the bifurcation sets in two-dimensional parameter space, despite that there were no differences in the one-dimensional bifurcation diagrams or trajectories.

We discovered that grazing bifurcations terminate the calculation of the local bifurcations because they drastically change the stability of the periodic orbit.

As future work, we would like to explore cases with other values of

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.