Chaos

Chaos Visualization

Chaos theory is a mathematical theory dealing with dynamic systems that are highly sensitive to initial conditions.
The following visualization is called a logistic map. We take the function x_n+1 = rx_n(1-x_n) and perform iterations starting with some x₀ between 0 and 1.
After performing enough iterations, several behaviors may be observed:

1. The value of the function will be 0.
2. The value of the function will diverge.
3. The value of the function will converge to a single value.
4. The value of the function will oscillate between a set of possible values.
5. The value of the function will be chaotic - no oscillations can be detected.

Different values of 'r' have different behaviors.
The interesting region in which the function converges is between r=-2 and r=4.

To use the applet, enter the start and end values for 'r' and click "Draw".


The logistic map shows the convergence values x_n for different values of 'r'.