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# MODULE: Dynamical Systems.

## Part 4: Linear and Nonlinear Functions

Linear functions are in general easier to study than nonlinear
ones. However, there is a relationship between nonlinear and linear
functions. Namely, iteration of a nonlinear function near a fixed
point is similar to iteration of a linear function with same
slope. The following section explains this statement and shows how it
can be used to predict whether a fixed point is attracting, repelling,
or neutral.

*A function in the quadratic family, along with a line of the same
slope through the fixed point.*

Study the long term behavior of linear maps. Namely, what is the
dynamics of g(x)=*a*x, a linear map with slope *a*? How does
the stability of the fixed point 0 change as parameter *a*
changes? Make sure to think about the case of *a* negative?

As mentioned before, near a fixed point iteration of a nonlinear map
is similar to the iteration of a specific linear map. Using this, you
can find a relationship between the slope of the graph at the fixed
point and whether the fixed point is attracting, repelling, or
neutral. First do so for the quadratic family:

- For a map in the quadratic family, what is the slope of its
graph at the fixed point?

- Iteration of a linear map with this slope is similar to
iteration of the quadratic function near the fixed point. For what
values of c should fixed points for the quadratic family be
attracting? Repelling? Neutral? Test your values using the Chaos Lab
#2.

- State the relationship between slope of the graph at the
fixed point and whether the fixed point is attracting, repelling, or
neutral.

- Recall that in order to find period two points, you graphed the
second iterate of the map. Can you find a relationship between the
slope of the graph of the second iterate of the function and whether the
period two points are attracting, repelling, or neutral?

- Describe geometrically how your relationship might generalize
to period n points.

Note: The procedure you found only works for periodic points with
small periods. Although it is nice to have the condition above for
whether a period n point is attracting, it becomes computationally
difficult to accurately graph the nth iterate of a map as n becomes
large.

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**Prev:** *Iteration*

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Created: Jun 09 1996 ---
Last modified: Jul 31 1996