We now turn our attention to resonance in the Duffing oscillator. The notion of a resonance is a physical concept with no exact mathematical definition. Physically, a resonance is a large-amplitude response, or output, of a system that is subject to a fixed-amplitude input. The concept of a resonance is best described experimentally, and resonances are easy to see in the string apparatus described in section 3.2 by constructing a sort of experimental bifurcation diagram for forced string vibrations.
Imagine that the string apparatus is running with a small excitation amplitude (the amount of current in the wire is small) and a low forcing frequency (the frequency of the alternating current in the wire is much less than the natural frequency of free wire vibrations). To construct a resonance diagram we need to measure the response of the system, by measuring the maximum amplitude of the string vibrations as a function of the forcing frequency. To do this we slowly increase (scan through) the forcing frequency while recording the response of the string with the optical detectors. The results of this experiment depend on the forcing amplitude as well as where the frequency scan begins and ends. Decreasing frequency scans can produce different results from increasing frequency scans.