• Derivation of the model
  
• Eventual behaviour
  
• Visualising an orbit

One-Dimensional Dynamical Systems

Definition (Fixed Point): The point p is a fixed point of the function f if f(p) = p.

• What are the fixed points of f(x) = x²?
  
  What is the relation between the fixed points of the square function and the intersection points of the graph of the square function with the special lines y = x and y = -x?
  
  
• Consider the square function on your calculator. Pick a point "near" (distance less than 0.1 is near enough) a fixed point. Keep iterating (pushing the squaring button) until you see a pattern for the long term behaviour. Try again with other numbers near each of the fixed points. Write down what happens for each choice. Does it matter if the number you pick is less than or greater than the fixed point?

Notice that for the square function, some orbits converge to a point, a fixed point, and some orbits diverge (get arbitrarily large). Are there orbits which always stay bounded, but do not converge to just one point? In general, the answer is yes. You will observe more complicated long term behaviour of orbits in your computer investigations.

Definition (Attracting and Repelling Fixed Point): A fixed point p is easy to detect computationally when orbits of nearby points converge to p. In this case, p is called attracting. If the orbits of nearby points move away from p, p is called repelling. If p is neither attracting nor repelling, p is called neutral.

• Which of the fixed points for the square function are attracting? Which are repelling?

• Derivation of the model
  
• Eventual behaviour
  
• Visualising an orbit