Sketch the graph of a continuous function which has an attracting fixed point and a repelling period-2 orbit.
I am having a hard time trying to come up with a graph with the above conditions. I know that to have an attracting fixed point, the magnitude of the slope of the tangent line has to be between 0 and 1. But how do you draw a repelling period-2 orbit?
Suppose you want your orbit to be $x_1\to x_2\to x_1$. Draw the graph of your function $f$ so that it goes through the points $(x_1,x_2)$ and $(x_2,x_1)$. In addition, make sure that the absolute value of the slope of the graph as it goes through those points is larger than $1$. For, if $F=f \circ f$, then we have $$F'(x_1) = f'(f(x_1))f'(x_1) = f'(x_2)f'(x_1) = F'(x_2).$$ Thus, if $f'(x_2)$ and $f'(x_1)$ are both larger than one in absolute value, then the same will be true for $F$ and the orbit will be repelling. Finally, as you seem to know, make sure that the graph of $f$ intersects the graph of $y=x$ with small slope.
If $x_1 = 0$ and $x_2 = 1$, then one such graph might look like so:
Let $f$ is the function in question, i.e., your dynamics is described by $x_{t+1} = f(x_t)$.
Hint: A period-2 orbit is equivalent to a fixed point of $f ∘ f$, i.e. of the dynamics described by $x_{t+1} = f(f(x_t))$. The criteria for stability translate.