Homoclinic chaos in a pair of parametrically-driven coupled SQUIDs

M Agaoglou,H Susanto,V M Rothos

doi:10.1088/1742-6596/574/1/012027

Abstract

An rf superconducting quantum interference device (SQUID) consists of a superconducting ring interrupted by a Josephson junction (JJ). When driven by an alternating magnetic field, the induced supercurrents around the ring are determined by the JJ through the celebrated Josephson relations. This system exhibits rich nonlinear behavior, including chaotic effects. We study the dynamics of a pair of parametrically-driven coupled SQUIDs arranged in series. We take advantage of the weak damping that characterizes these systems to perform a multiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture allows us to expose the existence of homoclinic orbits in the dynamics of the integrable part of the slow equations of motion. Using high-dimensional Melnikov theory, we are able to obtain explicit parameter values for which these orbits persist in the full system, consisting of both Hamiltonian and non-Hamiltonian perturbations, to form so-called Silnikov orbits, indicating a loss of integrability and the existence of chaos.

Highlights

We consider a series of electrical circuits in a line and we could derive the following nonlinear lattice equation [1], fn + γfn + fn + βsin(2πfn) − λ(fn+1 − 2fn + fn−1) = 0 (1.1)The coefficient of the nonlinearity, which corresponds to the so-called Josephson critical current, is modulated in time, which can be realized experimentally by modulating the surrounding temperature
It is in the slow dynamics that the homoclinic orbits are found
In particular we calculate the Energy-difference function using the Melnikov integral evaluated on the homoclinic solutions and applying the singular perturbation theory we study the dynamics near resonances

Summary

Introduction

We wish to study the possibility of homoclinic chaos near resonances in typical SQUID lattice, which is described by eq (2.1). We consider a specific dimer for the lattice equation (2.1) and show that the unperturbed system has a homoclinic orbit in their collective dynamics. We state the theorem for the exisence of multi-homoclinic orbits near

Results

Conclusion