kalimusada: a Python library for solving the Ma-Chen financial chaotic system

A Python-based solver for demonstrating sensitivity to initial conditions in economic dynamics.

Model

The Ma-Chen system describes financial dynamics through three coupled ordinary differential equations:

$$ \dot{x} = z + (y - a)x, \quad \dot{y} = 1 - by - x^2, \quad \dot{z} = -x - cz $$

where the state variables are:

and the parameters are:

The solver simulates two trajectories with infinitesimal initial separation $\delta_0 \sim \mathcal{O}(10^{-5})$ to visualize exponential divergence characteristic of deterministic chaos.

Installation

From PyPI:

pip install kalimusada

From source:

git clone https://github.com/sandyherho/kalimusada.git
cd kalimusada
pip install -e .

Quick start

CLI:

kalimusada case1          # run standard chaos scenario
kalimusada --all          # run all test cases

Python API:

from kalimusada import MaChenSolver, MaChenSystem

system = MaChenSystem(a=0.9, b=0.2, c=1.2)
solver = MaChenSolver()

result = solver.solve(
    system=system,
    init_A=[1.0, 2.0, 0.5],
    init_B=[1.00001, 2.0, 0.5],
    t_span=(0, 250),
    n_points=100000
)

print(f"Max divergence: {result['max_euclidean_distance']:.6f}")

Features

• High-precision ODE integration (LSODA)
• Dual trajectory sensitivity analysis
• Error metrics: Euclidean distance, RMSE, log divergence
• Output formats: CSV, NetCDF, PNG, GIF

License

MIT © Sandy H. S. Herho

Citation

@article{herho2026butterfly,
  title = {{T}he butterfly effect in economics: {E}xploring chaos with a simple financial model},
  author = {Herho, Sandy H. S.},
  journal = {CODEE Journal},
  volume = {20},
  number = {1},
  pages = {1},
  year = {2026},
  note = {https://scholarship.claremont.edu/codee/vol20/iss1/1}
}