syllabus.qmd

---
title: "Applied Math Lab"
subtitle: "Syllabus (Program Overview)"
format: html
---

## Program

This site is organized around **nine core modules** plus one extra Lorenz attractor module.

The sidebar navigation is the ground-truth order for the material, and this page follows the same structure.

### ODEs in 1D

Build the first simulation workflow with one-dimensional ODEs. You solve IVPs with `scipy.integrate.solve_ivp`, compare trajectories, and interpret parameter changes through the SIR epidemic model, Michaelis-Menten kinetics, and the spruce budworm model.

### ODEs in 2D

Move from scalar models to planar dynamics. You work with the CDIMA reaction, the Van der Pol oscillator, and the FitzHugh-Nagumo model, then animate trajectories and phase portraits to study oscillations and excitability.

### Coupled ODEs

Study synchronization through the Kuramoto model. You simulate many interacting oscillators, track order parameters, and connect time-domain behavior with summary diagrams.

### Collective Motion

Model flocking and alignment in continuous space. The main case studies use Vicsek and Couzin-style rules, then extend them with interactive animation and predator avoidance.

### Networks

Represent systems as graphs with `networkx`. You measure connectivity and centrality, generate standard graph models, and simulate SIS and SIR spreading on synthetic and real networks.

### PDEs in 1D

Discretize reaction-diffusion systems on an interval. You build finite-difference Laplacians, apply boundary conditions, study Turing instability, and use the linearized mode picture to predict which spatial patterns should grow.

### PDEs in 2D

Extend the same workflow to rectangular grids. You reuse the vectorized solver structure in two spatial dimensions, compare boundary conditions, and simulate Gierer-Meinhardt and Gray-Scott pattern formation.

### Cellular Automata

Switch from continuous fields to fully discrete local rules. You implement one-dimensional cellular automata, build space-time diagrams, and compare how rule choice and initial conditions affect long-run behavior.

### Agent-Based Modeling

Use traffic as the main agent-based case study. You implement the Nagel-Schreckenberg model, visualize stop-and-go waves, and measure density, flow, and congestion.

### Extra: Lorenz Attractor

Extend the ODE material into deterministic chaos. You simulate the Lorenz system, plot the attractor in three dimensions, and measure sensitivity to initial conditions.