Applied Math Lab

Syllabus (Program Overview)

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.

--- 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.