State, unfolding. A view through four dimensions.

This preset is a pair of harmonic oscillator modes, not a driven pendulum. Each complex mode obeys z′ = iΩz; its amplitude stays constant. For odd ratio k and cycle duration T, Ω₁=(k+1)π/T and Ω₂=(1−k)π/T. The default k=5 and T=180 s produce a five-petal rose in U–V.

Raw state is [x₁,y₁,x₂,y₂]. The explicit map is U=x₁+x₂, V=y₁+y₂, W=0.18(x₁−x₂), S=0.18(y₁−y₂). The 0.18 factor is display depth scaling. Eight seeded nearby mode states surround the central identity. Every strand contains only its own history.

Collection tests the wrapped relative phase of the two modes against the chosen center and full width. Width opens at 0, 45, 90, and 135 seconds: 12°, 24°, 48°, 72°. Amber nodes are exact accepted samples; there is no noisy sensor or estimator in this preset. Full trajectories remain visible outside the aperture.

Changing cycle duration or petal ratio preserves current phases and history; the combined path may then differ from a closed rose. Seed or initial spread restarts while retaining other selected settings. The run pauses for inspection at 180 s; Resume continues. U–V, U–W and V–W show direct coordinate slices. Dragging returns to the spatial view. Export retains raw samples, the map, and identities.

Each sample is q = [θ, ω, θ − estimated θ, time since last accepted observation]. Before acquisition, error and age are undefined, so the projected trajectory waits. A small physical pendulum remains visible for reference.

For display only, coordinates are normalized as tanh(q / [1 rad, 2 rad/s, 0.25 rad, 3 s]). This bounds long observation ages without hiding the raw values in exported data. Six ordered plane rotations transform R⁴, followed by R⁴→R³ and R³→R² projection. Perspective distances are 4 and 6; orthographic mode removes both perspective divisions.

Cyan shows history outside the collection window; amber shows collection-open history. Each accepted observation adds a permanent amber node; a cyan endpoint preserves the immediately preceding state. Dashed amber connectors mark correction jumps, not a physical trajectory. Nodes remain visible even when a shorter trail is selected. Reset clears the archive. All historical samples are reprojected using the current viewing orientation. The geometry is a view of the state—not the bob's physical path.

Preset 6, Evolving driver / narrow glimpse, holds collection at 1°, center at 0°, and sampling at 10 Hz while the physical drive passes through six deterministic stages over 210 simulation seconds. Smooth ramps change amplitude, frequency, and damping; forcing phase is the integral of frequency. The final driver is held after 210 s. Center and sensor edits no longer interrupt physical driver evolution. A manual edit to drive amplitude, frequency, or damping pins only that selected driver value; other scheduled values keep evolving. The state clock and retained geometry continue. Reset during automatic evolution restarts its schedule and seeded observations.

The underlying 2 m pendulum uses RK4 at 120 Hz. The Driven pendulum / chaos preset adds an external periodic drive using the classical nonlinear driven-damped pendulum equation θ¨ + cθ˙ + (g/L)sinθ = A cos(Ωt). Its parameters map to the classic dimensionless chaotic regime (c/ω₀ = 0.5, A/ω₀² = 1.2, Ω/ω₀ = 2/3), where ω₀ = √(g/L). Drive amplitude and frequency are visible live parameters; set drive amplitude to zero to return to the unforced pendulum. The EKF uses the same known drive, while observations still depend on sensor coverage, window, and noise. A nonlinear predictor and extended Kalman filter estimate motion from intermittent noisy angle measurements. Collection requires the physical angle to fall inside the window and the sensor to be uncovered. Measurements beyond the innovation threshold are rejected. After 3 s without acceptance, the next available measurement reinitializes angle and resets estimated velocity to zero.

Pause state freezes physical and estimation time while viewing rotation can continue. Auto rotate freezes only the view. Reset clears state history but preserves viewing orientation. Full run retains every recorded sample in memory (30 per simulation second); very long sessions can slow rendering. Kill the plot suppresses drawing without changing either model or projection clocks. Export preserves raw state samples, normalization, view settings, and recent sensor events; it is an inspection record, not a resumable checkpoint.