Euler's method for system of linear ODE's

Question

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.

Answer 1

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 $nh=1$.

Answer 2

$$ \begin{bmatrix} \dot x \\ \dot y\end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix}$$

Using SymPy to compute the matrix exponential,

>>> from sympy import *
>>> t = Symbol('t', real=True)
>>> A = Matrix([[ 0,-1],
                [ 1, 0]])
>>> exp(t * A)
Matrix([[cos(t), -sin(t)],
        [sin(t),  cos(t)]])

Can you take it from here?