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.