Dynamical System Simulator

This project simulates a dynamical system, given by a set of 1st-order ordinary differential equations, with C++. It generates a .csv file with the data, which is analysed and plotted with Python. This project was made purely out of interest, which was sparked from a Differential Equations and Dynamical Systems module in my 2nd year of University as a Theoretical Physics and Applied Mathematics student. Below are some graphics generated from a simulation of a Lorenz system.

Animation	3D Plot

How It's Made

Languages, Tech and Packages:

• C++ (Visual C++ 14.50)
• Python 3.11 (Jupyter Notebook)
  • Matplotlib for graphics
    • Pyplot
    • Animation
  • Numpy for calculation
  • Pandas for handling data

Theory: An $n$-dimensional dynamical system can be written in the fundamental form $\frac{d\mathbf{x}}{dt} = \mathbf{f}(\mathbf{x}; t)$, where $\mathbf{x}$ is the $n$-dimensional vector of all the dynamical variables. In order to simulate this, we can use the Euler method, which simply treats $dt$ as a finite and small constant $\Delta t$ rather than infinitecimal; thus naturally, the approximation gets better as $\Delta t \to 0^+$. So for each of our $n$ dynamical variables $x_i$, $0 \leq i < n$, we can approximate the change in $x_i$ as $\Delta x_i = f_i(x_i; t) \Delta t$ Then, the value of $x_i$ at time $t + \Delta t$ is $x_i(t+\Delta t) = x_i(t) + \Delta x_i$ This process is repeated in each timestep.

Simulation: All of the methodology for simulating the system is entirely contained in the DynamicalSystem class. It manages the evolution of the system in time, from a set of initial conditions (ICs), equations (disguised as functions) and definitions, which are passed upon instantiation. The DynamicalSystem::step() function updates the state of the system according to the Euler method described above when called.

Output: There is a simulation loop that stops once a time limit (set when the dynamical system is defined) is reached. In each iteration, the state of the system is entered as a row in a .csv file, with a column for each variable along with time.

Analysis and Graphics: Pandas is used to process and analyse the .csv file, which in turn is plotted with Matplotlib with standard methods.

Instructions to Build/Run

This project is a Visual Studio 2026 Solution, so can be cloned directly into VS2026. Otherwise, you can clone the repo and compile with any C++ standard, as there are no libraries beyond C++98 used. The Jupyter Notebooks should work with any version of Jupyter with Python3.

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Changelog

All notable changes to this project will be documented in this file.

The format is based on Keep a Changelog, and this project adheres to Semantic Versioning.

[Unreleased]

• Other dynamical system examples such as the Rössler system, or simple cases like the pendulum or pulley
• Analysis of chaotic characteristics such as the Lyapunov exponent or bifurcation
• More generalised classes and functions for either creating systems or producing examples
• Better recording of .csv files and graphics, with names that accurately describe the system
• Better output organisation
• Automated analysis, plotting and animation with Python (hopefully without Jupyter)
• Executable for simulation with arguments for parameters

[v0.0.0] - 2026-01-31

Added

• DynamicalSystem Class
• Lorenz system example
• Plot and animation generation with Python