DOUBLE PENDULUM / first-flip time

θ₁₀ → horizontal · θ₂₀ ↑ vertical

One fixed plot: θ₁₀, θ₂₀ ∈ [−π, π]. Click to inspect a trajectory. Dragging and scrolling do not change the plot.

Double pendulum / translation

Each pixel is a pair of starting angles. Both angular velocities begin at zero.

Physics

flip target time horizon series order max time step local tolerance detail supersampling

Optional final anti-aliasing pass: simulates each subpixel separately and averages colors. Native pixels render first, then supersampling refines them; 3×3 and 4×4 can be very slow.

Colors

palette cycle (time)

early fliplater / repeating

show angle axes

Initializing…

Selected pendulum

Click a point in the fractal to inspect its motion.

Units: equal masses and lengths, with g = L = 1. Time is measured in √(L/g).

Method: each motion uses algebraic Taylor-coefficient recurrences, then shifts the reference time and repeats. No RK4 is used to render pixels.

Sharpness: Ultra renders each display pixel (up to 8.5 megapixels) following fast previews. Choose 2×2, 3×3, or 4×4 supersampling for an additional pass that averages independently calculated subpixel trajectories. It smooths pixel-scale jagged boundaries; it does not remove the sensitivity of chaotic trajectories. A supersampling pass can take many times longer than native rendering. Changing the palette during supersampling restarts rendering.

Flip: first time the selected angle reaches ±π (an upright crossing). Black indicates no flip within the horizon. Fine boundaries can shift with the numerical settings.

Domain: one bounded plot only: θ₁₀ and θ₂₀ each range from −π to π. Drag and wheel navigation are disabled; the frame does not repeat.

Stand-alone version inspired by the earlier Fractals / zuv viewer. No external scripts or color files required.