7 - Period doubling bifurcations: what good are they?

Summary

Prologue

The statement that systems near the onset of instabilities are sensitive to fluctuations may be a haggard cliche, but is true nevertheless. The very notion of ‘instability’ means that a small fluctuation tends to grow with time rather than decay away. This has some distinct disadvantages, as was driven home to me during an experiment performed by Bob Miracky on an electrical circuit. Our goal was to determine the precise parameter value at which a period doubling bifurcation occurred, in order to compare this with analytic results derived from the circuit equation. As an experimental matter, this turned out to be tricky due to some curious interactions of the circuit with unwanted electrical noise.

Figure 7.1 reproduces a typical sequence of power spectra generated by sweeping a control parameter. If one has a good machine, the power spectrum is a very accurate means for determining the onset of period doubling bifurcations: the signature is the birth of a sharp spectral line at one-half the fundamental oscillation frequency ω₀, this line initially growing from zero height. Figure 7.1(a) shows the system well below the bifurcation point; however, in addition to the expected lines at frequencies ω = 0 and ω = ω₀ there appear two bumps in the broadband noise. Though mysterious, these bumps are not a problem since they are far away from the crucial frequency ω₀/2. Unfortunately, as the bifurcation point is neared these bumps move together, eventually colliding at ω₀/2 and obscuring the onset of the sharp line heralding the bifurcation.