Surface of section (or Poincaré map) for a frictionless double pendulum cart with low energy.
The Poincaré map is acquired by mapping angle and angle momentum of the first pendulum each time the second pendulum crosses a certain angle. This is done simultaneously for around 50 double pendulum carts that all have the same energy and are initialized with zero cart momentum. The resulting map shows regular trajectories (closed orbits) and some seemingly random and chaotic motion (irregular orbits). The orbits never intersect on the map and the chaotic orbits will therefore reveal modes hidden within the chaotic regions of the double pendulum cart. The large red area depicting chaotic motion is constructed from five simulations each which fully maps the area red if the simulation is run for long enough.

The video shows a survey of the Poincaré map by depicting the double pendulum cart motion at selected positions of the map.

The simulations were done using high order explicit symplectic integrators and was rendered in real time. This type of simulation requires exact time evolution in phase space and can not be performed properly using regular integration methods.

The mass of each pendulum bob is 1 and the mass of the cart is 2. All other parts have no mass.
A low energy was selected to reveal a large number of different periodic and quasiperiodic configurations.

The simulation was made using high order explicit symplectic integrators. This type of simulation requires exact time evolution in phase space and can not be accurately performed properly using regular integration methods.

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