Coupled pendulum modes chaos fractals

############################# Video Source: www.youtube.com/watch?v=YZ2ENWinLLA

Surface of section (or Poincaré map) for the spring-coupled pendulums with low energy. • The Poincaré map is acquired by mapping angle and angle momentum of the green pendulum each time the red pendulum crosses a certain angle as illustrated on the section plane. This is done simultaneously for 100 coupled pendulum systems with varying initial conditions but the the same energy. The resulting map shows regular trajectories (closed orbits) and some seemingly random and chaotic motion (irregular orbits). The orbits never intersect and the chaotic orbits will therefore reveal modes hidden within the chaotic regions of the coupled pendulum. • The video shows a survey of the Poincaré map, where various location on the map are illustrated with a corresponding simulation and a depiction of the motion in phase space. The motion in phase space typically span the same area for both pendulums, but is otherwise illustrated separately. Note that the phase space in the selected generalized coordinates may appear mirrored for the green pendulum. • 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. • 0:00 intro (chaotic 🔥) • 0:15 Poincaré map (100 coupled pendulums) • 0:28 regular • 0:41 regular • 0:54 regular (different pendulum phase space) • 1:12 irregular (chaotic 🔥) • 1:33 regular (different pendulum phase space) • Physical parameters are all set to 1 except the Hamiltonian which is -0.75 and equilibrium length and spring constant which are 2.

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