Chapter 5.8, Problem 10E

A software package that supports the construction of Poincaré maps is required for the problems in this section.

To show that the limit set of the Poincaré map given in (3) depends on the initial values, do the following:

a. Show that when $\omega =2$ or $3$, the Poincaré map consists of the single point

$\left(x,v\right)=\left(A\sin \varphi +\frac{F}{{\omega }^2-1},\omega A\cos \varphi \right)$.

b. Show that when $\omega =1/2$, the Poincaré map alternates between the two points

$\left(\frac{F}{{\omega }^2-1}\pm A\sin \varphi ,\pm \omega A\cos \varphi \right)$.

c. Use the results of the parts (a) and (b) to show that when $\omega =2$, $3$, or $1/2$, the Poincaré map (3) depends on the initial values $\left(x_0,v_0\right)$.

$\begin{array}{l}{x}_n:=x\left(2\pi n\right)=A\sin \left(2\pi \omega n+\varphi \right)+F/{\omega }^2-1,\\ {\upsilon }_n:=\upsilon \left(2\pi n\right)=\omega A\cos \left(2\pi \omega n+\varphi \right),\end{array}$ $\left(3\right)$

for $n=0,1,2,\dots$.