Josephson junction at the onset of chaos: A complete devil’s staircase

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Abstract
By analog computer calculations of the resistively and capacitively shunted Josephson junction model, I-V characteristics are measured for several choices of the parameters in the Josephson equation. The points, where hysteresis sets in, are related to cubic inflection points in the return map. For different values of the amplitude and the frequency of the imposed ac field the critical line is determined in the (I,G) space, where I is the dc current and G is the damping factor. Furthermore, the subharmonic steps along the critical line form a complete devil’s staircase with a fractal dimension D∼0.87 and a decay exponent for the (1/Q)-steps δ∼3. Besides the hysteresis which gives occasion for a chaotic behavior everywhere below a certain critical voltage, hysteresis also turns up locally. It is suggested that the critical points where local hysteresis occurs can be found by use of a local approximation.