I have the following system:
$$ \begin{aligned} \dot x &= -y \\ \dot y &= x \end{aligned} $$
Given that $(x_i^n,y_i^n)$ are the points obtained for $i=1,2\dots n^2$ using a time-step $h=1/n$ starting at the initial point $(x_0,y_0)=(1,0)$.
I am struggling to find the following limit:
$$\lim\limits_{n \to \infty} (x_n^n,y_n^n)$$
I can intutively think of the above limit ending up somewhere on the unit circle but I am unable to obtain the value of the exact limit. Pardon if its too simple, but I am surely missing out something. Please help out, thanks in advance.
Best Answer
Transform $x_1,y_1$ to polar coordinates $r,\theta$. Show then using trigonometric identities that $$ x_n=r^n\cos(n\theta),~~ y_n=r^n\sin(n\theta). $$ Show that $r=1+O(h^2)$ and $\theta=h+O(h^2)$ and draw conclusions for the case $nh=1$.