Homoclinic chaos in a pair of parametrically-driven coupled SQUIDs

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.

Figures (2)

• Figure 1: Orbits homoclinic to $\Pi _{0}=\{\, (x, y, I, \phi):x=y=0,1<\delta<3, I>\frac{\Omega_{1}}{3-\delta}\,\}$ . For $I=I_{0}$ (the orbit in the middle),$d\phi/dT =0$ on $\Pi _{0}$, and the orbit is heteroclinic, connecting fixed points on $\Pi _{0}$ that are $\Delta\phi$ apart. For $I \lessgtr I_{0}, d\phi/ dT \lessgtr 0$ on $\Pi _{0}$. The parameters are $\delta =1.49704$, $\Omega =-110$, $\Omega _{1} =27.6085$.
• Figure 2: a) The heteroclinic orbit given by Eqs. \ref{['18a']} and \ref{['18d']} with $I=I_{0}$, superimposed with the phase portrait of the unperturbed scaled system on $\Pi_{\varepsilon}$ near resonance. The parameters are $\delta=1.55, \Omega=-150, \Omega_{1}=35, \zeta= 0.45, h=1$ ($\xi=\gamma/h$ and $\zeta=\xi/ \delta$). This value of $\zeta$ sets $\phi_{c}=0.233383$ and we can see from the figure that $\phi_{s}<\phi_{c}+\Delta \phi <{\phi}_{m}$. b) The values of $\phi _{s}, \phi _{c} ,\phi_{c} +\Delta \phi$ and $\phi _{m}$ as functions of $\delta$ , for $\Omega=-150$. For $\delta>1.524$ the condition is satisfied. c) In gray are indicated the parameter values for which the condition \ref{['eq40']} is satisfied. In figures b and c: $\varepsilon =0.01$