Complexity

Volume 2018, Article ID 8931525, 10 pages

https://doi.org/10.1155/2018/8931525

## Influence of Bifurcation Structures Revealed by Refinement of a Nonlinear Conductance in JosephsonJunction Element

^{1}Graduate School of Advanced Technology and Science, Tokushima University, Tokushima, Japan^{2}Center for Administration of Information Technology, Tokushima University, Tokushima, Japan

Correspondence should be addressed to Yuu Miino; moc.liamg@uuy.oniim

Received 30 June 2018; Revised 10 October 2018; Accepted 21 October 2018; Published 2 December 2018

Academic Editor: Yongping Pan

Copyright © 2018 Yuu Miino and Tetsushi Ueta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

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. [1] discovered chaotic responses in a JJ circuit and a forced JJ circuit. 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.

On the other hand, hybrid dynamical systems (HDSs) have been studied intensively by many researchers [5–7]. An HDS combines a continuous-time dynamical system and a discrete-time dynamical system, e.g., a circuit including a switch like a DC/DC converter [8], a neuron model including a firing scheme like the Izhikevich neuron model [9], or a conflict system like a bell [10]. HDSs embody a rich variety of bifurcation phenomena reflecting their interrupting properties and the discontinuity of their derivatives, and many bifurcation analysis tools and applications have been developed for them. For example, Kousaka et al. [11] studied the periodic orbits in an autonomous HDS and proposed the method to compute the local stability of their orbits and local bifurcation sets. Our previous study [12] suggested a scheme to apply Kousaka’s method to the nonautonomous HDS. Piiroinen et al. [13] discovered chaotic behavior and grazing bifurcations occurring in an HDS. Ito et al. [14] suggested a method to control chaos in HDS by perturbing the threshold value of the system.

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 [12] to analyze the PWL system. In what follows, we define the system and its mathematical characteristics (Sections 2.1-2.2). We then describe the single-junction superconducting quantum interferometer with a vibrating external flux [15, 16]; we derive its circuit equation and normalized equation. We define the PWL function from the current-voltage characteristics of the JJ. We also consider a linear conductance, because there are some studies that use this approximation [17]. Next, we explain the local stability and local bifurcation phenomena of the periodic orbit (Section 2.3). We introduce the Poincaré map and variational equation and present criteria under which local bifurcations arise. We describe the criteria for a grazing bifurcation as well (Section 2.4). As the main result (Section 3), we reveal the bifurcation structure of the system in a two-dimensional parameter space. We discuss the bifurcations observed in the system by using one-dimensional bifurcation diagrams and trajectories. We make a comparison between the case of a linear function and the case of a PWL function, identifying the similarities and differences between them. Finally, we conclude this study (Section 4).

#### 2. Materials and Methods

##### 2.1. Single-Junction Superconducting Quantum Interferometer with the External Flux

Let us consider a single-junction superconducting quantum interferometer (SQUID) with an external flux [15, 16], as shown in Figure 1(a). This is also called an RF-SQUID core [18]. Taking as the sum of DC and AC components [16], we get the equivalent circuit shown in Figure 1(b).