A motion that returns to the same initial condition after one period, then repeats.
Phase space
The space of all possible system states, represented here by angles and angular velocities: (θ₁, θ₂, θ̇₁, θ̇₂).
Poincaré section
A lower-dimensional slice through phase space that records where trajectories cross it.
Reading the visualization
The animated pendulum shows one periodic orbit; the white trace is the path of the lower mass.
The mini map shows the complete orbit map. Its black dot marks the current orbit's initial condition.
The map uses a Poincaré section at θ̇₁ = 0, when the upper arm is momentarily at rest. The visualization plots the stored initial condition for each orbit entry.
Each colored sequence samples a continuous family of periodic orbits. Family crossings reveal bifurcations.
Conventions
Angles: 0° means the pendulum arm is hanging straight down; positive angles are counterclockwise.
Energy: Zero is the stable fixed point, with both masses hanging at rest. Masses, arm lengths, and gravitational acceleration are all set to 1.
Phase Space Overlay
Click the mini map to open the full, interactive Poincaré section.
Hover the large plot to see initial conditions and properties of orbits (initial θ₁, θ₂, |θ̇₂|, Energy, Period).
Click (or tap) a point to load a new orbit.
Pinch or Ctrl/⌘-scroll to zoom. Two-finger scroll, mouse-wheel scroll, or drag the plot to pan.
Hold the arrow keys to pan smoothly (hold Shift to move faster), and press 0 or Reset to restore the full view.
Use the buttons to recolor points by Initial θ̇₂, Energy, Period, or orbit Family. Press 1 / 2 / 3 / 4, respectively, as shortcuts.
Hover a point and press Enter (or choose Navigate to Point) to load that orbit. You can also click or tap a point directly.
Playback Controls
Play / Pause: Toggles the animation. When paused, additional buttons appear to step the simulation forward or backward.
Speed + / Speed −: Changes playback speed and the size of simulation time steps.
Next / Prev: Moves to the next or previous orbit in the same family.
Editable values: Click an underlined Time, Period, Speed, Energy, or Orbit value, type a number, and press Enter.
Keyboard Shortcuts
Press Esc to close any overlay. On this Info overlay, Enter also closes it.
Press Space from the main view, or K anywhere outside an overlay, to play / pause. When a control is focused, Space activates that control.
On the main animation, press ↑ / ↓ to step through the orbit family.
On the main animation, press ← / → to step one frame backward / forward when paused.
On the main animation, press Shift+← / Shift+→ to step 10 frames backward / forward when paused.
On the main animation, press + / - to increase / decrease playback speed.
On the main animation, press 0 to reset speed to default.