I have the following system: x ˙ = − y y ˙ = x Given that ( x i n , y i n ) are the points obtained for i = 1 , 2 … 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 n → ∞ ( 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.

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I have the following system:                                          x            ˙                                              =        −        y                                                  y            ˙                                              =        x            Given that   (      x    i    n    ,      y    i    n    ) are the points obtained for   i  =  1  ,  2  …      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          n      →      ∞        (      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.

Rowan Dyer · Accepted Answer

Transform       x    1    ,      y    1   to polar coordinates   r  ,  θ. Show then using trigonometric identities that      x    n    =      r    n    cos  ⁡  (  n  θ  )  ,            y    n    =      r    n    sin  ⁡  (  n  θ  )  .Show that   r  =  1  +  O  (      h    2    ) and   θ  =  h  +  O  (      h    2    ) and draw conclusions for the case   n  h  =  1.

I have the following system: dot x=-y, dot y=x Given that (x^n_i,y^n_i) are the points obtained for i=1,2…n^2 using a time-step h=1/n starting at the initial point (x_0,y_0)=(1,0).

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