Mathematical Simulations
Dynamical Systems • Geometry • Chaos • Visualization
Numerical explorations of complex mathematical phenomena: differential equations, geometric transformations, and chaotic behaviors.
Copula Laboratory
Probability • Dependence Structures • Financial & Climate Risk
A copula separates the dependence between variables from their marginal distributions — this is Sklar's theorem. This laboratory simulates five families (Gaussian, Student-t, Clayton, Gumbel, Frank) in real time: vary the parameter θ and watch the dependence structure morph while the marginals stay fixed. Every empirical measure (Kendall's τ, Spearman's ρ, tail dependence λL, λU) is compared against its closed-form theoretical value.
Sklar's Theorem
F(x, y) = C( F_X(x) , F_Y(y) )
λ_L = lim_{u→0⁺} P( U₂ ≤ u | U₁ ≤ u ) (lower-tail dependence)
λ_U = lim_{u→1⁻} P( U₂ > u | U₁ > u ) (upper-tail dependence)
τ = 4 ∬ C(u,v) dC(u,v) − 1 (Kendall's tau)
Applied Scenarios
Technique: simulation via conditional / inversion methods (Marshall-Olkin for Archimedean families, Cholesky decomposition for elliptical ones), numerically computed inverse CDFs for the marginals, additive-blending Canvas rendering. No external dependencies, single file. Empirical measures vs. closed forms from Cherubini, Durante & Mulinacci, Principles of Copula Theory.
GPS Trajectory — Cinematic Visualization
Trajectories • Kinematics • Data Visualization
Cinematic visualization of a GPS trajectory using time-based interpolation, smoothing techniques, and dynamic camera motion to reveal the underlying kinematic structure of the path.
Coming next: velocity-based coloring, temporal annotations, and interactive exploration of the trajectory.
Point Cloud Lab
3D • Projections & Spherical • LiDAR / Vision
Interactive exploration of point clouds: 3D visualization, orthographic (top-down) and spherical projection type panorama. Demonstration of a 3D → 2D pipeline used in computer vision, photogrammetry, and LiDAR.
Polyhedra Morphing
Geometry • Transformations • Interpolation
A visual exploration of geometric transitions between regular and semi-regular polyhedra. The simulation demonstrates continuous interpolations between different polyhedral structures, revealing underlying geometric relationships.
Included Polyhedra
Technique: Linear interpolation of vertex coordinates with normalization to maintain scale. Continuous rotation highlights the symmetry of each shape.
Lorenz Attractor
Deterministic Chaos • Sensitivity to Initial Conditions
The Lorenz system is a classic example of deterministic chaos. Despite perfectly deterministic evolution rules, the system exhibits extreme sensitivity to initial conditions. The video shows the 3D trajectory and its projections onto canonical planes.
System Equations
dx/dt = σ (y − x) dy/dt = x (ρ − z) − y dz/dt = x y − β z
Coming soon: Visualization of exponential divergence of nearby trajectories, fractal dimension of the attractor, and coloring by time or local velocity.
Nonlinear Damped Pendulum
Differential Equation • Energy Dissipation
Simulation of a rigid pendulum under gravity with linear damping.
The sin(θ) term introduces a nonlinearity that distinguishes
the real behavior from the small-angle harmonic approximation.
Equation of Motion
d²θ/dt² + a dθ/dt + (g/ℓ) sin(θ) = 0
Where θ is the angle from vertical, a is the damping coefficient, g is gravitational acceleration, and ℓ is the pendulum length.
Coming soon: Energy graphs E(t), phase portrait (θ, dθ/dt), and comparison with the linearized solution to highlight nonlinear effects.
Methodology
Each simulation combines rigorous mathematical modeling with optimized visual rendering to reveal the underlying structure of the studied phenomena.
Modeling
Precise mathematical formulation (ODE, PDE, geometric transformations). Phase space study and stability analysis.
Numerical Computation
Integration with adaptive methods (RK4, RK45) and optimized time step. Validation through conservation of invariants.
Visualization
3D rendering with dynamic camera, fluid trajectories, and multiple projections. High-quality export for web and presentation.
Analysis
Geometric and physical interpretation of results. Complete documentation of parameters and assumptions.
Next step: Development of an interactive interface allowing real-time parameter adjustment and guided exploration of the solution space.
Contact
📍 Québec, Canada