Interrogate parameter space
Reveal how boundary conditions, coupled variables, and uncertainty reshape the feasible region.
Computational visualization
Make complex systems visible.
SKAR translates mathematics, physics-informed models, and simulation data into visual structures that expose geometry, dynamics, uncertainty, stability, and constraint.
Parametric geometryNonlinear dynamicsSimulation & uncertainty
From governing structure to physical intuition
See the mechanism, not only the result.
A simulation can produce an answer while concealing the mechanism that produced it. We construct visual representations of state space, phase evolution, boundary conditions, coupled modes, and stochastic convergence so technical behavior can be inspected rather than merely reported.
Visualization becomes part of the analytical method: a way to interrogate sensitivity, compare parameter regimes, expose instability, and communicate why a conclusion remains physically and mathematically defensible.
Identify governing dynamics
Distinguish transient behavior from stable modes, periodic structure, and the variables controlling system response.
Communicate simulation evidence
Preserve the chain from governing assumptions and equations through observed behavior and a defensible decision.
Analytical environments
Structure the mathematics around the decision.
High-dimensional systems are governed by interacting constraints, nonlinear response, uncertainty, and path dependence. We build computational views that preserve those relationships while making the governing behavior possible to examine.
Parameter-space exploration
Vary initial conditions, boundary conditions, and coupled parameters to locate sensitivity, stability, and feasible operating regions.
- Response surfaces
- Sensitivity gradients
- Feasibility boundaries
Dynamical-system views
Track state evolution, modal behavior, constraint interaction, and the propagation of disturbances through a system.
- State trajectories
- Transient response
- Stability indicators
Simulation evidence
Connect model assumptions, numerical experiments, uncertainty, and observed behavior in one inspectable analytical record.
- Model provenance
- Uncertainty structure
- Physical interpretation
Featured simulation · Stochastic convergence
Ensemble convergence
Resolve stable structure from a cloud of possible states.
Uncertainty quantification begins with an ensemble, not a single deterministic path. Each particle represents a sampled state drawn from a broader distribution of possible conditions.
As the ensemble evolves, individual trajectories remain distinct while their occupancy converges toward a coherent manifold. The distribution—not one privileged sample—becomes the object of analysis.
The resulting structure exposes range, concentration, outliers, and the assumptions capable of moving the system into a different regime.
Computational studies
Parameterized systems in motion.
These are live equations evaluated in the browser at every frame—not prerecorded animations. Together they render cyclic symmetry, driven ensembles, constrained surfaces, coupled phase modes, stochastic transport, and topology through continuous state space.
STUDY 01 · CYCLIC SYMMETRY
Phase-locked rotation
Seven parameterized branches form a cyclically symmetric field. A traveling phase perturbation modulates curvature and radius while the global mode remains coherent—an analogue for synchronized oscillators, rotating machinery, and periodic loading.
C₇ symmetry · θₚ(t) = 2πp/7 + ωt
STUDY 02 · DRIVEN ENSEMBLE
Collective field response
Twenty thousand sample points evolve under a shared nonlinear phase law. Local trajectories remain distinct while macroscopic structure emerges, illustrating coherent response in many-body systems, network dynamics, and distributed control.
N = 20,000 · xᵢ(t) = Φ(i,t)
STUDY 03 · CONSTRAINED GEOMETRY
Boundary-defined state space
Stacked parametric sections reconstruct a deforming surface of revolution. Changes in waist, curvature, and modal amplitude show how boundary conditions restrict the admissible geometry of a design or physical system.
r = r(z,θ,t) · ∂Ω(t)
STUDY 04 · COUPLED PHASE SPACE
Reciprocal trajectories
Two phase-shifted parametric bodies move within one nonlinear field. Their shared governing law preserves coherence while relative position changes—an analogue for coupled oscillators, interacting states, and second-order dependency.
Two branches · Δφ = 3 rad · xᵢ(t) = Φ(i,t)
STUDY 05 · NONLINEAR MODES
Mode exchange
Nested trigonometric forcing produces two related structures that exchange phase and orientation. Continuous deformation reveals mode coupling and adaptive response without breaking the topology of the system.
m ∈ {0,3} · q = q(k,e,d,t)
STUDY 06 · PERIODIC TRANSPORT
Transport on a closed orbit
Two ribbon manifolds evolve along a 1:2 Lissajous trajectory. Normal-direction perturbations generate local deformation while the orbit remains closed, clarifying periodic transport, handoffs, and stability through a complete state cycle.
x = sin θ · y = 0.78 sin 2θ
STUDY 07 · STOCHASTIC TRANSPORT
Advection through uncertainty
Lagrangian tracers evolve through a time-varying, divergence-free velocity field while stochastic diffusion separates neighboring paths. The resulting density reveals coherent transport barriers, mixing regions, and the rate at which initial certainty is lost.
dXₜ = u(Xₜ,t)dt + √(2κ)dWₜ · ∇·u = 0
STUDY 08 · KNOT TOPOLOGY
Closed phase manifold
A dense ribbon is transported around a (3,8) torus knot. Slow orbital motion exposes periodic closure, winding number, and the continuous local frame that turns a one-dimensional trajectory into an oriented surface without changing its topology.
γ(t) = ((R+r cos 8t)cos 3t, (R+r cos 8t)sin 3t, r sin 8t)
Interactive visualization
Explore an interactive field.
Move your pointer across the canvas to guide the flowing strands.
This interactive field requires motion effects and WebGL to be available.