A Farey staircase from the two-extremum return map of a Josephson junction

Abstract

We report the presence of a Farey staircase in the simulated current–voltage characteristics between the second and third harmonic steps of an underdamped Josephson junction under external electromagnetic radiation. The steps constituting the staircase are interrupted by chaotic intervals. The dynamics is due to a two-extremum return map and is not associated with phase locking on an invariant torus. On decreasing the current, the third harmonic step ends in the bifurcation known as blue-sky catastrophe.

Appendix: Scaling law for Farey steps

In the limit as I approaches $I_{\mathrm{BSC}}$, the scaling law $T\propto \left( I_{\mathrm{BSC}}-I\right) ^{-1/2}$ leads to an expression for the step widths, in the CVC, in terms of the denominator in the p / q ratio. Consider two currents $I_q$ and $I_{q+1}$, corresponding to q and $q+1$. Since $T = q\tau$, the scaling law implies

$$\begin{aligned} I_{\mathrm{BSC}}-I_q \propto \frac{1}{T_q^2} \propto \frac{1}{q^2} , \end{aligned}$$

(4)

and

$$\begin{aligned} I_{\mathrm{BSC}}-I_{q+1} \propto \frac{1}{T_{q+1}^2} \propto \frac{1}{(q+1)^2} . \end{aligned}$$

(5)

By subtracting Eq. (5) from (4), we find that the step width is given by

$$\begin{aligned} I_{q+1}-I_{q} = l_0\left( \frac{1}{q^2}-\frac{1}{(q+1)^2}\right) , \end{aligned}$$

(6)

where the proportionality constant is found to be $l_0=0.32159$. It is interesting to note that the derived scaling law is the same as Eq. (4) of Ref. [20]. For comparison, the constant of proportionality for the case discussed in Ref. [20] is $l_0=2.720$.