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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)=ax, 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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