Read the Notes and Credits. Click on the Green Flag, Enter a value for x, record the value of x1 or x2 that is to be iterated. Press Space to start the iteration. The x-values will iterate to one of the three roots of the equation -1, 0, and 1.
This project is for students in the Fractals and Chaos class I am teaching. It is used to explore a question in the Midterm exam. One of the characteristics of a chaotic system (like the Lorenz Attractor) is 'sensitivity to initial conditions'. Consider turbulence in a fluid. Two particles that are very close together at the top of a waterfall are widely separated 25 feet downstream. This project uses Newton's Method to iterate y = 4x^4 – 4x^2. By selecting particles (x-values) that are closer and closer together, one can show that they can become widely separated. The program then adds or subtracts √(2)/2 to x (Newton's method is undefined at √(2)/2 and –√(2)/2 because the denominator of Newton's method, f'(x(n), evaluates to 0). Using this program one can demonstrate mathematically that two points (x-values) arbitrarily close together can iterate to widely separate points. In short, the project is designed to simulate mathematical turbulence.
