Louis-Finegan/Lorenz-Attractors-Pygame-Python3

Chaos Equations (Lorenz Attractors) in python3 using the pygame, scipy and numpy libaries.

Lorenz Attractors

Lorenz Attractors

The Lorenz Attractors are governed by the following System of Ordinary Differential Equations:

$$\frac{dx}{dt}=\sigma(y-x)$$

$$\frac{dy}{dt}=x(\rho-z)-y$$

$$\frac{dz}{dt}=xy-\beta z$$

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Purpose

Demonstrate the chaotic nature of the Lorenz Attractors with slight variation in the initial conditions.

Libaries Used:

1. numpy

2. scipy using the integrate module

3. pygame

Approach

3 instances of the lorenz class were created with slight variation in their initial conditions. these conditions were random using numpy's uniform random number generator:

    [np.random.uniform(0.99, 1.01),np.random.uniform(0.99, 1.01),np.random.uniform(0.99, 1.01)]

The initial condition was centered around [1, 1, 1]. This is to stop the points in the solution from displaying off the pygame display.

NOTE: Initial conditions will be printed in the terminal.

The System of Ordinary Differential Equation were solved by using, solve_ivp function, Then the plot.game method is called which generates the interactive pygame display with the 3 solutions: white red and/or blue, appearing point by point at 30 fps.

How to use

Run the python file main.py.

1. Press s to start then select one of the following options:

• press a to generate all 3 solution on the display at the same time.

• press w to generate the white solution.

• press g to generate the green solution.

• press b to generate the green solution.

2. Press q or close the window to quit.

3. Press r to reset the display.

4. follow on from step 1 to generate a new plot.

Conclusion

It is clear, by observing all 3 solutions after a sufficent amount of time, there nature is hugely different. Then it can be said with a slight change in the Lorenz Attractors initial conditions, these models will have different solutions after a long period of time.