Periodic Solutions of the Double Pendulum

Double pendulum is famous for its chaotic behaviour, but within that chaos lay pieces of order, structure, and crystal-like beauty: periodic solutions. Repetition of pendulum movement is a very rare occurance, and trying to collect them feels like an exploratory and taxonomic effort performed with code in the space of methematical physics.

Alas, solutions shown here do not stand up to the mathematical rigor required to be a true periodic solution¹ of the differential equation describing the double pendulum. These are just close enough² to create a seemless loop showing the repeated trajectory as a gif animation.

For each of the cyclic solutions there is the pendulum animation, trajectory of the second pendulum in the cartesian space, and its phase portrait. Results are split into two groups: one where the phase plot is continuous and the second where it is not³.

Group 1

• Duration: 9.28 seconds
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• Duration: 13.13 seconds
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• Duration: 18.83 seconds
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• Duration: 12.85 seconds
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• Duration: 16.13 seconds
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• Duration: 13.52 seconds
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• Duration: 19.19 seconds
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• Duration: 15.81 seconds
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• Duration: 19.49 seconds
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• Duration: 13.75 seconds
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• Duration: 22.40 seconds
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• Duration: 22.41 seconds
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• Duration: 16.85 seconds
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• Duration: 10.29 seconds
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• Duration: 19.50 seconds
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Group 2

• Duration: 16.82 seconds
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• Duration: 13.88 seconds
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• Duration: 19.29 seconds
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• Duration: 9.90 seconds
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• Duration: 11.07 seconds
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• Duration: 9.89 seconds
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• Duration: 19.32 seconds
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• Duration: 8.48 seconds
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• Duration: 17.04 seconds
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• Duration: 16.68 seconds
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• Duration: 16.68 seconds
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• Duration: 16.68 seconds
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• Duration: 14.68 seconds
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• Duration: 6.22 seconds
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• Duration: 7.24 seconds
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• Duration: 6.43 seconds
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• Duration: 8.49 seconds
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• Duration: 17.12 seconds
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• Duration: 11.81 seconds
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• Duration: 16.84 seconds
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• Duration: 9.78 seconds
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1. See this and this for a in-depth discussion about periodicity in case of the double pendulum. ↩︎

2. The distance (cartesian and in the phase space) between two ends of each loop is within range of the standard distribution of distances in the pendulum movement. ↩︎

3. Technically, the phase space should be represented as a toroidal surface with circumference equal 2π (full rotation of a pendulum). In that case the phase plot would by cyclical and continuous just like the cartesian trajectory. ↩︎