Since the correct answer depends on what your professor thinks, and the fact that there are plenty of relations, I'll put my favorite:
EDIT: You might have noticed, that $2$-,$4$-,$3$-,$5$-,$6$- and $7$-point cycles start at increasing values of $r$. See here for a list. So the relation, that your professor bears in mind, could be: Can you show that there is an upper bound for $r$ using the golden ratio. You might have heard already that chaos starts at 4.00 (pm), so you have to choose $r>4$ (and better lock your door at $t<4$).
The logistic map relates to the Golden Ratio, such that a beautiful proof exists, which shows that $x_{n+1}=F(x_n)=rx_n(1-x_n)$ is chaotic, if $r>\phi^3$:
Since $r>4$, the interval $I=[0,1]$ is split into three parts. If $x_0$ lies inside the middle part, it will leave $I$ after one iteration. It is left to show that this will happen for the left ($I_0$) and right ($I_1$) interval too.
Therefore, we require that |F(x)'|>\lambda>1 for all $x\in I_0 \bigcup I_1 $. This union set is further called $A$. We have |F(0)'|=|F(1)'|=r, so $r>1$. The roots of $F(x)=1$, giving the right/left edges of $I_{0,1}$, are $x_{-,+}=(1\pm \sqrt{1-4/r})/2.$ Substituting $x_{-,+}$ in F(x)' gives $r^2-4r=1, \; \text{ which has roots at } \;\; r_{+,-}=2\pm \sqrt{5}.$ Since $r_-<0$, you choose $r_+=2+\sqrt{5}=2\phi +1=\phi^2+\phi=\phi^3$.
From here on, I copied what I needed from here, since I'm not at all experienced on this field:
By the chain rule, it follows that |F_n(x)'|>\lambda^n as well. Indeed, if this were so, we could choose two distinct point $x$ and $y$ in $A$ with the closed interval $[x,y]\subset A$. Choose $n$ so that $\lambda^n |y-x|>1$. By the Mean Value Theorem, it then follows that $|F^n(y)-F^n(x)|>\lambda^n |y-x|>1,$ which implies that at least one of $F^n(y)$ or $F^n(x)$ lies outside of $I$.
On the very nice linked page, you'll also find that the set $A$ is Cantor Set. (This distracted me quite a time, because I always tried to find the no-middle-third set.)
As WNY points out in his comment, the logistic map becomes chaotic for all $r>4$, but the proofs don't seem share the same kind of beauty.
In total, I think there might be plenty of relations between the logistic map and the golden ratio.