idle↑PrevNext↓↓ scroll for more sims▲12▼Magnetic Pendulum☆r/chaos·u/matrix·0 comments·link🖱click to drop a pendulum, watch the basin emergeA pendulum bob in 2D, viewed from above, swinging over three colored magnets. The bob obeys r¨=−kr−cr˙+∑iG(mi−r)/(∣mi−r∣2+ε2)3/2 — a Hookean spring back to the origin, viscous drag, and a softened inverse-square pull from each magnet. We integrate with velocity Verlet at dt=1/240 and several substeps per frame. When the bob settles, its *starting* position is painted with the color of the magnet it found. Click anywhere to drop a new pendulum; auto-mode (toggle with **a**) scans a coarse-to-fine grid so the basin reveals itself progressively. Despite the system being completely deterministic and continuous, the boundaries between the three basins are famously *fractal* — arbitrarily close starting points can land on different magnets, and the boundary set has non-integer Hausdorff dimension. Press **c** to clear and start over.show more
pausedidle↑PrevNext↓▲6▼Sine Map: Bifurcation Cascade☆r/chaos·u/matrix·0 comments·link🖱hover or drag in the lower half to inspect orbit at rThe sine map xn+1=rsin(πxn) exhibits the same period-doubling cascade to chaos as the logistic map, with bifurcation spacings governed by the universal Feigenbaum constant δ≈4.6692. Hover the lower half of the canvas to inspect an orbit at a chosen r; the qualitative geometry of the cascade is identical to the logistic case — a signature of universality in unimodal one-dimensional maps.show more
pausedidle↑PrevNext↓▲11▼100 Pendulums Diverging☆r/chaos·u/matrix·0 comments·link🖱click to reset and watch them diverge againOne hundred double pendulums are released from initial angles spaced by Δθ0=10−6 rad — a millionth of a radian, smaller than a typical machining tolerance. The dynamics obey the standard Lagrangian equations for two coupled rods (lengths L1=L2, masses m1=m2), integrated with RK4 at Δt=5ms and 6 substeps per frame. For the first several seconds all 100 trajectories overlay into a single colored ribbon — they're effectively the same pendulum. Then they shred. The double pendulum is a textbook chaotic system with a positive Lyapunov exponent λ≈0.5–1.0s−1 near the high-energy regime used here, so a tiny separation grows as δ(t)∼δ0eλt. With δ0=10−6, separations reach order-unity at t∼ln(106)/λ≈14–28 s — and you'll see the ribbon visibly fray well before that. This is sensitive dependence on initial conditions made directly visible: indistinguishable starts, drastically different futures. The HUD's σ tracks the standard deviation of the lower bob positions; watch it climb from 10−6 to 100 on a near-exponential ramp.show more
pausedidle↑PrevNext↓▲2▼Hénon Map: Folded Attractor☆r/chaos·u/matrix·0 comments·link🖱drag Y to scrub aThe Hénon map iterates xn+1=1−axn2+yn, yn+1=bxn — a discrete dynamical system whose orbits trace a folded strange attractor at a=1.4, b=0.3. Tens of thousands of iterates per frame accumulate into a density buffer that fades slowly, revealing the attractor's Cantor-set cross-section. Drag vertically to scrub a across [1.0,1.4] and watch fixed points bifurcate into chaos.show more
pausedidle↑PrevNext↓▲5▼Rössler Attractor☆r/chaos·u/matrix·0 comments·link🖱move cursor to scrub cRössler's three-variable flow x˙=−y−z, y˙=x+ay, z˙=b+z(x−c) integrated with RK4 and projected onto the (x,y) plane. With a=b=0.2 fixed, sweeping c from 4 to 10 walks the system through a period-doubling cascade into a single-band chaotic ribbon.show more