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https://github.com/shreyasmehta05/chaos-and-bifurcations
Exploring chaos and recursion through logistic maps and bifurcation diagrams using Python.
https://github.com/shreyasmehta05/chaos-and-bifurcations
Last synced: about 1 month ago
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Exploring chaos and recursion through logistic maps and bifurcation diagrams using Python.
- Host: GitHub
- URL: https://github.com/shreyasmehta05/chaos-and-bifurcations
- Owner: shreyasMehta05
- Created: 2024-12-08T08:20:34.000Z (about 2 months ago)
- Default Branch: main
- Last Pushed: 2024-12-08T08:34:39.000Z (about 2 months ago)
- Last Synced: 2024-12-08T09:24:33.894Z (about 2 months ago)
- Language: Python
- Size: 813 KB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# **🌌 Chaos and Bifurcations**
This project explores **chaos theory** and **bifurcations** through visualizations of the **logistic map** and **bifurcation diagrams**. Using Python, it implements mathematical models to demonstrate how simple nonlinear dynamical systems exhibit **complex behaviors** as system parameters vary.
✨ _This project was part of the_ `Science-1` _course at IIIT-Hyderabad._
---
## **👨💻 Author**
- **Shreyas Mehta**
---
## **📂 Directory Structure**
```plaintext
Chaos-and-Bifurcations/
├── bifurcation_diagram_CosineCurve.png
├── bifurcation_diagram_SinosudalCurve.png
├── logistic_map_2023101059_last1.png
├── README.md # 📖 This file
├── bifurcation_diagram_sine.py # 🐍 Code for bifurcation diagrams (sinusoidal map)
├── logistic_map_visualization.py # 🐍 Code for visualizing logistic maps
└── Results-Img/ # 📁 Output images folder
├── bifurcation_diagram_2023101059.png
├── bifurcation_diagram_sineCurve1.png
├── logistic_map_2023101059_last1.png
├── logistic_map_2023101059_last2.png
├── logistic_map_2023101059_last3.png
├── logistic_map_2023101059_start1.png
├── logistic_map_2023101059_start2.png
└── logistic_map_2023101059_start3.png
```
---
## **✨ Features**
- 🔍 **Generate Bifurcation Diagrams** for **logistic** and **sinusoidal maps**.
- 📈 **Visualize Logistic Map Dynamics** with parameters `r` and `x0`.
- 🌟 **Demonstrate Chaotic Behavior** and **Periodic Orbits** in dynamic systems.
- 💾 **Save Results** as high-quality images for further analysis.
---
## **🐍 Python Files**
### 1. **`bifurcation_diagram_sine.py`**
Generates a **bifurcation diagram** by iterating a sinusoidal variant of the logistic map:
\[
x(t+1) = r \cdot \sin(x(t))
\]
**🔑 Key Features**:
- Iterates through a range of system parameter `r`.
- Plots **bifurcation points** to visualize transitions to chaos.
- Highlights how sinusoidal maps exhibit **chaotic behavior** at certain thresholds.
**📁 Output**:
- `bifurcation_diagram_sineCurve1.png`
---
### 2. **`logistic_map_visualization.py`**
Visualizes the behavior of the classic **logistic map**:
\[
x(t+1) = r \cdot x(t) \cdot (1 - x(t))
\]
**🔑 Key Features**:
- Simulates system behavior for different values of `r` and initial state `x0`.
- Visualizes **periodic orbits** and **chaotic regimes** over specified iterations.
- Allows comparison of behaviors for different parameters.
**📁 Outputs**:
- `logistic_map_2023101059_last1.png`
- `logistic_map_2023101059_start1.png`
- Other visualizations in the `Results-Img/` folder.
---
## **🚀 How to Run the Project**
1. Clone the repository:
```bash
git clone https://github.com/shreyasMehta05/Chaos-and-Bifurcations.git
cd Chaos-and-Bifurcations
```
2. Install dependencies:
```bash
pip install matplotlib numpy
```
3. Run the scripts:
- **To generate the bifurcation diagram**:
```bash
python3 bifurcation_diagram_sine.py
```
- **To visualize logistic map dynamics**:
```bash
python3 logistic_map_visualization.py
```
4. View the output images in the **`Results-Img/` folder**.
---
## **📊 Results**
### 1. **Bifurcation Diagram for Logistic Map**
### 2. **Logistic Map Dynamics**
### 3. **Bifurcation Diagram for Sinusoidal Map**
---
Enjoy exploring the **chaos**! 🌀