\(\(\( (*\ vector\ field*) \)\(
\)\(\(f[x_, y_] := y;\)

    \(g[x_, y_] := \(-x\);\)

    \)\)\)

\(\(\( (*initial\ values\ *) \)\(
\)\(x[0] = 1; \ 
    y[0] = 0;
 (*time\ step*) 

    \(e = 0.1;\)
 (*number\ of\ steps*) 

    \(nmax = 500;\)\)\)\)

\(\(\( (*\ Euler\ integration\ *) \)\(
\)\(Do[
      x[n + 1] = x[n] + e*f[x[n], y[n]]; 
      y[n + 1] = 
		y[n] + e*g[x[n], y[n]], {n, 0, nmax}]\)\)\)

ListPlot[Table[{x[n], y[n]}, {n, 0, nmax}]]

ListPlot[Table[{n, x[n]^2 + y[n]^2}, {n, 0, 500}]]

Fit[Table[{n, Log[x[n]^2 + y[n]^2]}, {n, 0, 500}], {1, u}, 
      u]

Log[1 + e^2]