Measurements off(α)spectra of attractors at transitions to chaos in driven diode resonator systems

Measurements of the $f\left(\alpha \right)$ spectrum, which is a measure of the dimension of the set of singularities of strength $\alpha$ on the attractor, are reported for the diode resonator system at the onset of chaos from both the quasiperiodic route and the period-doubling route. For the period-doubling route a driven single diode resonator is used and quasiperiodicity is attained by driving two resistively coupled diode resonators. In the driven coupled diode resonator system, a second frequency occurs naturally, but a winding number equal to the golden mean cannot be reached. However, the system can be made to respond at a winding number approximated by the continued fraction $〈4111\dots 〉$ to an accuracy of one part in ${10}^6$. The measured $f\left(\alpha \right)$ functions are compared with those calculated for the sine-circle map (quasiperiodic route) and the logistic map (period-doubling route). We find good agreement in both cases, providing experimental evidence for the universality of $f\left(\alpha \right)$ within classes of situations representing particular routes to chaos.