Visualization of Set-valued Dynamical System

Iteractive web-based visualization tool for exploring set-valued dynamical systems with bounded noise, developed as part of the Advanced Computing Project (ACP2) research course at the University of Oulu.

Mathematical Bacground

In classical analysis, a single-valued function (or simply a function) $f: X \to Y$ assigns each point $x \in X$ to exactly one point $y \in Y$, written $y = f(x)$. Traditional dynamical systems using single-valued maps to describe deterministic evolution: given initial state $x_0$, the trajectory is uniquuely determined as $x_1 = f(x_0), x_2 = f(x_1), x_3= f(x_2)$ and so forth.

In contrast, a set-valued function (or multivalued map) $F: X \to \mathcal{(Y)}$ assigns to each point $x \in X$ a subset $F(x) \subseteq Y$ where $\mathcal{P}(Y)$ denotes the power set of $Y$. Rather than producing a single output, set-valued functions produce a set of possible outputs:

$F(A) = \bigcup_{x \in A} F(x)$ In our setting, we model bounded additive noise through set-valued map: $F(x) = B_\epsilon(f(x)) = {f(x) + \xi : |\xi| \leq \epsilon}$ where $f: \mathbb{R}^n \to \mathbb{R}^n$ is the underlying single-valued deterministic map (the Hénon map in our case), and $B_\epsilon(f(x))$ represent all possible perturbed states within distance $\epsilon$ of the deterministic image.

Rather than tracking every possible point within the noise ball $B_\epsilon(f(x))$ which would too computationally expensive to compute as the noise balls grow, we instea adopt a worse-case boundary approach. Since the maximum uncertainty occurs at the boundary $\partial B_\epsilon(f(x))$ (points at distance exactly $\epsilon$ from the deterministic image), we focus exclusively on tracking how these boundary points evolve.

Visualization tool

Features

Feature	Status	Description
Algorithm 1 Implementation	✅ Done	Core boundary tracking with normal vector transformation
Interactive 2D Visualization	✅ Done	Real-time rendering with Three.js (mapped points, noise circles, boundaries)
Step-by-Step Execution	✅ Done	Detailed iteration-by-iteration inspection mode
Batch Execution	✅ Done	Automatic iteration until convergence ("Run Full" mode)
Divergence Detection	✅ Done	Monitors escaping points and halts when >50% diverge
Real-Time Parameter Control	✅ Done	Interactive sliders for $a$, $b$, $\epsilon$, $x_0$, $y_0$
Multi-Mode Visualization	✅ Done	Toggle between all elements, mapped points only, noise circles only, etc.
Hausdorff Distance	🔄 Planned	Rigorous convergence assessment using $d_H(M_n, M_{n-1})$
Bifurcation Visualization	🔄 Planned	Detect and visualize topological and boundary bifurcations
Color-Coded Orbits	🔄 Planned	Distinguish fixed points, periodic points, and singular points
Export Functionality	🔄 Planned	PNG screenshots, CSV data export, MP4 animations
Parameter Space Exploration	🔄 Planned	Visual indicators for bifurcation regions in $(a, b, \epsilon)$ space

An example of a 4-periodic point found with A = 1.4 and B = 0.3

Getting Started

1. Clone the Repository

git clone <repository-url>
cd set-valued-viz

2. Build WebAssembly Module

# Build the Rust code to WebAssembly
wasm-pack build --target web --out-dir pkg

This creates the WebAssembly module in the pkg/ directory.

3. Install Frontend Dependencies

cd frontend
npm install

4. Copy WASM Files to Frontend

# Copy WebAssembly files to the frontend source
npm run build-wasm

Alternative: Manual copy if the npm script doesn't work:

cp -r ../pkg/* src/pkg/

5. Start Development Server

npm run dev