Recurrence Relations (RR) and Cellular Automata (CA) - Mathematics Assignment Help

Assignment Task

1. Find a RR for the number of bit strings of length n, an, that contain three consecutive zeros. More zeros are allowed. Somewhere in the string there must be a 000 sequence. What is the initial condition? Solve the problem on the computer. How many such strings of length 30 are there? Plot with DiscretePlot the first 10 values of an. Hint: split the total number of strings of length n into strings with and without 000. OP

2. Plot a stable 3-periodic orbit in the logistic map. Give the three x values for the orbit. Try to find the a value for which it goes through a period doubling and loose stability.

3. Consider rule 30 and start with one black cell in the middle. Your string can be 1000 cells long. Run it for 300,400,500 iterations. Plot the value in the middle cell, b(n), where n is generation n. b(0) = 1 since you start with one black cell. Black is equal to 1 and white is 0 in the list. Does it look random? Equally many black and white cells?

4. Investigate the CA where birth occurs when the cell has exactly three living neighbours and there is survival when the living cell has not more than four living cells as neighbours. What is the rule number? Try random seeds of various size in the middle of the grid. What happens if you start with one-dimensional seeds with 3,4 or 5 black cells in a row? Take a sufficiently large grids so you don’t hit the boundary. Do at least 1000 iterations.

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