The Pendulum - Demo 5

Driven Damped Pendulum - period doubling

At a driving strength of 1.07 the limit cycle takes two orbits to exactly repeat itself. This is period doubling. (The transient, if you study it, is quite complicated, but we are interested really in only the asymptotic behavior i.e. the behavior of the attractors.

The "power spectrum"

shows a new, small peak at one half of the driving frequency (and its multiples) i.e. at subharmonics. We also refer to this phenomenon as the orbit having undergone a "subharmonic bifurcation". Note that harmonics grow in smoothly as the nonlinearity increases, whereas the subharmonic first appears at a particular driving or nonlinearity strength.

Increasing the driving very slightly to a=1.083 leads to a second subharmonic bifurcation with an even smaller peak at one quarter of the fundamental (and its harmonics), the orbit:

and its power spectrum:

(There actually exists a second attracting orbit, which you can get to starting with and initial condition of X0=0.1, Y=0.)

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