Exercise 28

Gary

In your graph your rotated spiral formula doesn't seem right.
It should be $e^{i\alpha t}(e^{at}e^{ibt})=e^{at}e^{i(\alpha+b)t}$ - the same spiral.

Vasco



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Gary

In part (1) we show that the spiral is an invariant curve of $\mathcal{F}_{\tau}$

In part (iii) we show that if we rotate the original spiral about the origin through an arbitrary angle then the new spiral is also an invariant curve of $\mathcal{F}_{\tau}$.

In part (iv) we show that the spirals in part (iii) are the only invariant curves of $\mathcal{F}_{\tau}$.

If we wanted to find a counter example we wouldn't change the nature of $\mathcal{F}_{\tau}$ but rather the nature of the curve we applied $\mathcal{F}_{\tau}$ to, in order to show that these curves were also invariant curves of $\mathcal{F}_{\tau}$.

So we would have to apply $\mathcal{F}_{\tau}$ to a $W(t)$ of the form $Z(t)+v$ where $v$ is the translation, and show that

$\mathcal{F}_{\tau}[W(t)]=\mathcal{F}_{\tau}[Z(t)+v]\rightarrow W(t+\tau)$.

If you work this through it becomes

$\mathcal{F}_{\tau}[W(t)]=W(t+\tau)+v(e^{a\tau}e^{ib\tau}-1)$

which shows that this $W$ is not an invariant curve of $\mathcal{F}_{\tau}$

Similarly, if $W(t)=\overline{Z(t)}$ then we can show that this $W$ is also not an invariant curve of $\mathcal{F}_{\tau}$.

In your answer to part (iii) you say that certain things in the exercise are not entirely clear. When Needham says 'the spiral' I think he must mean as defined in the intro to the exercise.

Vasco