Collisions · Gravity · Patterns · Recurrence
Turn the MPIPKS visualization collection into experiments. Change an initial condition, follow the evolution, and compare the image with the mathematical model.
Six laboratories covering all project families on the supplied page, including its Sinai download and standard-map example. Standalone HTML · generated locally · no installation. Numeric edits apply on Enter or when you leave the field.
Click either table to choose the common starting position. Compare nearby trajectories at the same physical time.
At a smooth wall, the tangential component stays unchanged and the normal component reverses. Speed is conserved. Corners are singular, so a corner hit stops the experiment.
Without a field, flight is linear. With a perpendicular magnetic field, v̇=ΩJv and the flight radius is |v|/|Ω|. Circular arcs and their intersections with the walls are computed analytically. Ω=0 recovers straight flight.
In the radius-one semicircular cap, trajectories whose angular-momentum magnitude exceeds the opening half-width cannot enter the foot. Other seeds can visit it. The boundary preset starts close to this geometric threshold; long visits do not by themselves prove a statistical power law.
The Sinai heatmap samples positions at uniform physical-time intervals. It is an empirical occupation measure, not a proof of ergodicity. A finite separation curve is likewise not an asymptotic Lyapunov exponent.
Source: Billiards, magnetic motion, mixed phase space, and intermittency. New ACT implementation; source links open online.
Source: Sinai comparison and occupation histogram. New ACT implementation; source links open online.
Click the square to launch an orbit. The gray background is a fixed collection of reference trajectories.
Change K or move a seed close to an island boundary. Look for long intervals of apparently regular motion followed by excursions. Use the return gaps to formulate a hypothesis, then increase the sample size. A short orbit cannot establish a tail exponent or distinguish a very long transient from an invariant curve.
From (q′,p′), recover q=q′−p′ and then p=p′−K sin q, both modulo 2π. The kick and drift are individually invertible shears.
Source: The standard-map illustration under “Intermittent Chaos”. New ACT implementation; source links open online.
Switch between a reciprocal three-body model and a massless asteroid moving among circular primaries.
Unit Sun mass and Earth–Sun distance; G=1. Earth mass 3.003×10⁻⁶; Moon mass 3.694×10⁻⁸; lunar separation 0.00257. All three bodies affect each other.
The inset uses approximate physical Sun and Moon radii. Orbital inclination, Earth rotation, observer latitude, and ephemerides are absent, so it does not predict real eclipses.
The Sun–Earth–Moon simulation uses velocity Verlet. Total energy should have small bounded numerical error over resolved orbits. Try halving the time step. An Earth-centered plot subtracts the Earth position at each recorded instant.
Primary separation and angular speed are one; their masses are 1−μ and μ. RK4 integrates the rotating coordinates. L₄ is linearly stable only if μ(1−μ)<1/27. The six presets are newly specified initial states, not recovered parameters from the old desktop program.
Source: Sun–Earth–Moon and Sun–Jupiter–asteroid demonstrations. New ACT implementation; source links open online.
Each map pixel launches a ray into the chamber between four spheres. Click a pixel to inspect its path, then zoom around it.
Exit 0: charcoal · exit 1: blue · exit 2: teal · exit 3: rust. Light gray is unresolved at the reflection limit, not an exit basin.
Sphere surfaces are shown as transparent wireframes so interior reflections remain visible.
The four unit spheres have centers at alternating corners of a cube, scaled to form a regular tetrahedron. At separation 2 they touch. The central chamber is bounded by the four face planes; a ray terminates at its first outward face crossing.
Ray–sphere quadratic equations determine the next collision. The reflection preserves the ray’s speed. This is the same local law as in the two-dimensional billiards.
Increase the reflection limit and zoom into a boundary. Long-lived rays may reveal additional structure. Unresolved rays retain a separate color; they are not silently assigned to an exit.
A finite image can suggest complicated basin boundaries but cannot establish their dimension or the Wada property. The view uses parallel incident rays; its geometry and sampling are explicit rather than reproducing an undocumented historical camera.
Source: Four-sphere billiard and chaotic scattering. New ACT implementation; source links open online.
Compare fourfold and sixfold anisotropy with weakly directional growth. Switch between the solid fraction and the released heat.
Let φ≈1 denote solid and φ≈0 liquid. With θ=arg ∇φ and ε(θ)=W[1+δ cos(mθ)], define
Here J(a,b)=(−b,a), η is seeded interface noise, W=0.5, D=0.4, L=1.8. Anisotropy enters the interfacial energy, not a decorative color filter.
Face fluxes and explicit time steps advance the 144² grid with zero-flux boundaries. Therefore the mean of T−Lφ is conserved up to roundoff. The displayed drift tests this heat balance, not convergence of the spatial approximation.
The interface spans only a few cells. Grid bias and this coarse interface width affect morphology. The low-anisotropy phase-field model may remain compact. The separate aggregation preset illustrates irregular diffusion-limited branching without solving a temperature field. Side branches depend on perturbations and available growth time. Continue growing or vary the seed.
The time step is 0.1. Sixfold anisotropy is capped at 0.027 to keep ε+ε″ positive; fourfold is capped at 0.06.
In diffusion-limited aggregation, seeded random walkers approach a cluster and stick on first nearest-neighbor contact. Exposed tips intercept more walkers than shielded interior sites. Launch and kill circles keep the finite computation tractable; the run stops before the cluster reaches the boundary.
This stochastic lattice model provides an irregular branching comparison to the deterministic transport equations. It is not a quantitative reconstruction of the source’s seaweed movie. Update count is not physical time, and no heat-balance diagnostic applies.
T. A. Witten & L. M. Sander (1981), Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon.
Source: Crystal growth: dendrites, symmetry, and seaweed. New ACT implementation; source links open online.
Phase-field basis: R. Kobayashi (1993), Modeling and numerical simulations of dendritic crystal growth. Related primary implementation documentation: MOOSE anisotropic Allen–Cahn kernels.
The two panels use the same cat map and image. Only the within-pixel sample coordinates differ.
The matrix A has determinant one and eigenvalues (3±√5)/2. On an N×N rational grid, A acts as a permutation. Its order is the smallest positive T with Aᵀ=I modulo N. For the power-of-two resolutions here, T=3N/4; N=256 gives 192.
Each output pixel is colored by following A⁻ⁿ back into the original image. Exact modular matrix powers allow a jump directly to the grid return.
The second panel fixes one pseudorandom point inside each pixel on a lattice with denominator 65,536N. These coordinates are still rational. The lattice has a much longer exact map period, and individual points can have shorter periods.
Pixel-color matches can occur without coordinate recurrence; both are reported. No independent randomness is added during the evolution. This avoids presenting finite computer arithmetic as genuinely irrational initial data.
Poincaré recurrence requires an invariant finite measure. It gives an almost-everywhere return statement, not the short global image period of a finite permutation or a universal numerical waiting-time estimate.
Source: Poincaré recurrences: rational versus randomized pixel coordinates. New ACT implementation; source links open online.
Each distinct demonstration on the supplied MPIPKS page has a working laboratory here. Duplicate download links refer to the same systems.
| Source demonstration | Interactive recreation | Model choice |
|---|---|---|
| Mushroom billiards | Billiards: free flight, close seeds, regular and boundary presets | Semicircular cap with a triangular foot; exact collisions |
| Magnetic billiards | Billiards: field strength and signed curvature | Exact circular flight arcs and specular impacts |
| Sinai download | Two adjacent tables and occupation heatmap | Square with reflecting central disk |
| Intermittency / standard map | Standard map: phase portrait and target return gaps | Area-preserving kick–drift map on the torus |
| Sun–Earth–Moon | Three bodies: whole-system, Earth zoom, solar-overlap inset | Planar reciprocal Newtonian gravity; approximate ratios |
| Sun–Jupiter–asteroid | Six editable initial states and nearby-orbit comparison | Circular restricted three-body problem |
| 3D billiard / chaotic scattering | Exit basins, ray paths, zoom, reflection limit | Four equal spheres on a regular tetrahedron |
| Crystal growth | Fourfold, sixfold, basal-arm, and irregular-growth presets | Phase-field model plus stochastic aggregation limit |
| Poincaré recurrence | Image scrambling on grid and jittered coordinates | Exact modular arithmetic; no floating-point recurrence artifact |
Use a visualization to identify a pattern, formulate a conjecture, and design a discriminating comparison. Conservation errors, resolution changes, event handling, and arithmetic choices determine whether the image supports that interpretation.
Chaos, ergodicity, fractal dimension, and asymptotic recurrence are mathematical properties. They are not established by one long-looking trajectory. Here the conservation diagnostics and explicit models make those questions visible.
Begin with a parameter or initial-condition change. State what should remain invariant, run both cases, and explain the disagreement if the numerical picture differs from the prediction.
Every laboratory exports a PNG with parameters. The cat-map panel can transform an uploaded image locally. Crystal and scattering experiments can generate compositions while retaining an explanation of how the structure arose.
The app has no external scripts or downloaded models. Paper links need internet; all experiments run locally. Simulations pause when you change tabs.
MPIPKS: Visualization of Dynamical Systems, Nonlinear Dynamics and Time Series Analysis Group. The model-specific pages are linked in each laboratory.
L. Bunimovich (2001), Mushrooms and other billiards with divided phase space, Chaos 11, 802. E. G. Altmann, A. E. Motter & H. Kantz (2005), Stickiness in mushroom billiards.
Altmann, Motter & Kantz (2006), Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space. Phys. Rev. E 73, 026207.
D. Sweet, E. Ott & J. A. Yorke, Topology in Chaotic Scattering, the reference named by the MPIPKS four-sphere page. This app traces the specified geometry; finite escape images do not prove topological basin properties.
T. Ihle & H. Müller-Krumbhaar (1994), Fractal and compact growth morphologies in phase transitions with diffusion transport. R. Kobayashi (1993), anisotropic phase-field growth.
F. J. Dyson & H. Falk (1992), Period of a discrete cat mapping, American Mathematical Monthly 99, 603–614. The app reports a modular matrix order and distinguishes it from a generic continuum recurrence time.
Billiards: exact line/arc intersections, unit speed, finite stored trail, equal-time comparison. Singular corners stop the orbit. Magnetic tangent collisions may be ill-conditioned.
Standard map: deterministic double-precision iteration, 60 reference seeds × 500 iterations, at most 20,000 selected points displayed. Three-body models: velocity Verlet or RK4 with a selectable step; close encounters stop before the chosen resolution becomes inadequate.
Scattering: rays stop on exit, at the reflection cap, or outside the entrance. Rendering is divided into rows so controls remain responsive. The image resolution and reflection cap are independent parameters.
Crystal: explicit face-flux phase and heat updates on 144² cells, Δt=0.1, finite interface width, pseudorandom noise, no-flux boundaries. The heat-balance diagnostic does not remove grid effects. The aggregation option uses seeded four-neighbor random walks with finite launch/kill circles and a boundary stop. No sharp-interface convergence claim is made.
Recurrence: selected resolutions are powers of two. All modular coordinates and matrix products stay below JavaScript’s exact-integer limit. Image colors may recur sooner than sample coordinates. Uploaded images are processed in the browser.
The source’s Xmgrace and image-conversion pages describe animation workflows rather than additional dynamical systems. External resource directories are not recursively copied into this collection.