Let us consider the linear dynamical system

as it was introduced in the
*Introduction*. For now, do not
think in terms of squirrels too much, both *x* and
can be negative
numbers, and we accept non-integer values for them as well.

- The graph of
*f*is simply a straight line. Draw the graphs for = 0.1, 0.3, 1, 2, and 5. Explain for each what the long term behavior of the system is for arbitrary*x*. What happens for*x*= 0? What happens for = 1?

Now look at the slope of each graph. Do you see a relation between the long term behavior of the system and the slope of its graph?

- For negative the
graph of
*f*is still a straight line. Draw the graphs for = -3, -1.8, -1, -0.5, and -0.2. For each describe the long term behavior, and explain the difference in behavior compared to positive .

Formulate the relation between the long term behavior of the system and the slope of its graph, for both positive and negative slopes.

- In the above two exercises you have found two special lines,
corresponding to two special linear systems, namely
*f*(*x*) =*x*and*f*(*x*) = -*x*. Draw the graphs of these two functions. In the same picture sketch the graph of a nonlinear function which is such that iterates of this function go to infinity for all*x*0.

Make another picture with the graph of a nonlinear function whose iterates go to 0 for all*x*.

To summarize the results:**The iterates of a linear system either go to 0 or to infinity, for all***x*0.More complicated dynamics can only be obtained with nonlinear dynamical systems. You should already be able to guess what the graph of such a function needs to look like, in order to get different dynamics. Let us consider nonlinear dynamical stystems that can be found on your calculator.

- Suppose we have a dynamical system given by the square root
function on your calculator. Type in any number and calculate its
square root. Take the square root of the result. Take the square root
again and so on. Give a full description of the dynamics, similar to
what was done for the squirrel model. In particular, look for values
of
*x*that do not change when taking the square root. Only consider*x*0.

Draw the graph of the square root function, together with the two special lines*y = x*and*y = -x*. What is special about the points where the graph of the square root function intersects one of the two lines?

- Give a description of the dynamics for a dynamical system given by
the cosine button, where
*x*is given in radians.

Draw a graph of the cosine function for*x*given in radians. Also draw the special lines*y = x*and*y = -x*. Describe what is special about the points on the intersection of the graph of the cosine function and these two lines.

Written by Hinke Osinga

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Created: Mar 31 1998 ---
Last modified: Fri Apr 10 15:11:11 1998