Chaotic-Attractors

Simulating Rössler attractor and the Lorenz attractor

Lorenz Attractor Simulation

The Lorenz system is chaotic dynamical system described by a set of three coupled ordinary differential equations, and it exhibits unpredictable behavior over time.

Prerequisites

Make sure you have the required dependencies installed. You can install them using:

pip install numpy scipy matplotlib

Code Overview

The code is organized into three main functions:

lorenz_system(t, xyz, sigma, rho, beta):

Defines the Lorenz system of differential equations. Parameters: t: Time variable. xyz: Current state vector [x, y, z]. sigma, rho, beta: System parameters. Returns the rate of change of each variable.

simulate_lorenz(sigma, rho, beta, initial_conditions, t_span, num_points):

Uses SciPy's solve_ivp to numerically solve the Lorenz system. Parameters: sigma, rho, beta: System parameters. initial_conditions: Initial values for [x, y, z]. t_span: Time span for simulation. num_points: Number of points for time discretization. Returns time values and the state trajectory.

plot_lorenz_attractor(t, xyz):

Plots the Lorenz attractor in 3D using Matplotlib. Uses a rainbow color map to add visual appeal. Parameters: t: Time values. xyz: State trajectory.

Main Block

if __name__ == "__main__":
    # Set parameters and initial conditions
    sigma = 10
    rho = 28
    beta = 8/3
    initial_conditions = [0, 1, 20]

    # Simulate Lorenz attractor
    t, xyz = simulate_lorenz(sigma, rho, beta, initial_conditions)

    # Plot colorful Lorenz attractor
    plot_lorenz_attractor(t, xyz)

Usage

1. Run the script by executing the following command:

   python lorenz_attractor.py

2. The script will simulate the Lorenz attractor and display a 3D plot with rainbow-colored trajectories.

Feel free to experiment with different parameters and initial conditions to observe the impact on the Lorenz attractor's behavior.

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Rossler Attractor

Rössler Attractor Simulation

The Rössler system is described by three coupled nonlinear differential equations that exhibit chaotic behavior characterized by interesting trajectories in three-dimensional space.

Code Overview

The code consists of three main functions:

1. rossler_attractor(t, xyz, a, b, c):

   • Defines the Rössler system of differential equations.
   • Parameters:
     • t: Time variable.
     • xyz: Current state vector [x, y, z].
     • a, b, c: System parameters.
   • Returns the rate of change of each variable.

2. simulate_rossler_attractor(a, b, c, initial_conditions, t_span, num_points):

   • Uses SciPy's solve_ivp to numerically solve the Rössler system.
   • Parameters:
     • a, b, c: System parameters.
     • initial_conditions: Initial values for [x, y, z].
     • t_span: Time span for simulation.
     • num_points: Number of points for time discretization.
   • Returns time values and the state trajectory.

3. plot_rossler_attractor(t, xyz):

   • Plots the Rössler attractor in 3D using Matplotlib.
   • Uses a rainbow color map to add visual appeal.
   • Parameters:
     • t: Time values.
     • xyz: State trajectory.

Usage

1. Set the parameters and initial conditions in the __main__ block.
2. Run the script.
3. The simulation results are displayed in a 3D plot showing the evolution of the Rössler attractor over time.

Dependencies

• numpy: For numerical operations.
• scipy: For solving differential equations using solve_ivp.
• matplotlib: For creating 3D visualizations.

Example

if __name__ == "__main__":
    # Set parameters for the Rössler attractor
    a = 0.2
    b = 0.2
    c = 5.7

    # Set initial conditions
    initial_conditions = [0.1, 0.0, 0.0]

    # Simulate Rössler attractor
    t, xyz = simulate_rossler_attractor(a, b, c, initial_conditions)

    # Plot colorful Rössler attractor
    plot_rossler_attractor(t, xyz)

Feel free to experiment with different parameter values and initial conditions to observe the impact on the Rössler attractor's behavior. Contributions and improvements are welcome!