Nonlinear oscillation modes in the 3rd order Josephson junction circuits

Autonomous third-order Josephson junction (JJ) circuits containing an angular variable are analyzed. Using piecewise linearizing approximation, the Poincare map, bifurcation diagram, attractor dimension, and Lyapunov spectrum have been efficiently obtained, especially for the chaos in this system. The authors have also obtained the almost one-dimensional feature of the chaos orbit and the fine structure of the chaos oscillation. This chaos has a low attractor dimension nearly equal to that of the quasi-periodic oscillation in nonautonomous second-order JJ circuits. The conditions for the generation and extinction of chaos are provided, since they can be useful in designing a Josephson heterodyne detector, Josephson parametric amplifiers, etc. All the unstable periodic oscillation modes are found to be embedded into the chaos orbit, and these expanding directions are tangential to the chaos orbit. Lyapunov exponents of the chaos orbit are equal to those of unstable periodic oscillation modes. The C/N of the JJ oscillators falls below -20 dB when the chaos occurs.