Bifurcation Diagrams for Discrete-Time Maps

This repository provides Python implementations for generating interactive bifurcation diagrams and analysing chaotic behavior in discrete-time dynamical systems.

The following maps are currently included:

• Logistic Map
• Tent Map
• Sine Map

Overview

Each map is implemented to analyse its bifurcation diagram and Lyapunov exponent, which are key to understanding the dynamics and chaos within these systems.

Installation

1. Create and source virtual environment:

python -m venv env
source env/bin/activate  # On Windows use `env\Scripts\activate`

2. Install the dependencies:

pip install -r requirements.txt

Example Usage

To analyse a specific map, run the corresponding script in the Maps directory. You can tweak the map parameters as needed.

Interactive plots will be generated and saved as offline HTML files within each map's directory.

Note

Some plots have been uploaded to the Maps directories for reference.

Scripts

python Maps/LogisticMap/main.py

python Maps/TentMap/main.py

python Maps/SineMap/main.py

Example (Logistic Map):

from logistic import LogisticMap
from utils.plotting import saveInteractivePlot
import datetime

# Parameters for logistic map
params = {
    'paramMin': 3.57,  # bifurcation parameter
    'paramMax': 4.0,   
    'stepSize': 1e-3,
    'numTransient': 300,
    'numPlotPoints': 300,
    'numIterationsLyapunov': 200
}

logisticMap = LogisticMap(**params)

# Generate and save bifurcation diagram
rhoValues, xValues, hoverText = logisticMap.generateBifurcationData()

timestamp = datetime.datetime.now().strftime("%Y-%m-%d_%H-%M-%S")

plotParams = {
    'title': 'Logistic Map Bifurcation Diagram',
    'xAxisTitle': 'Bifurcation Parameter (rho)',
    'yAxisTitle': 'System States x(n)',
    'fileName': f'logisticMap_bifurcation_{timestamp}.html',
    'markerSize': 0.05,
    'opacity': 0.6,
}

saveInteractivePlot(rhoValues, xValues, hoverText, **plotParams)

# Generate and save Lyapunov exponent plot
rhoValuesLyapunov, lyapunovExponents, lyapunovHoverText = logisticMap.generateLyapunovData()

plotParams = {
    'title': 'Logistic Map Lyapunov Exponent',
    'xAxisTitle': 'Bifurcation Parameter (rho)',
    'yAxisTitle': 'Lyapunov Exponent',
    'fileName': f'logisticMap_lyapunov_exponent_{timestamp}.html',
    'mode': 'lines'
}

saveInteractivePlot(rhoValuesLyapunov, lyapunovExponents, lyapunovHoverText, **plotParams)