I want to plot the phase portrait and also the direction field of the following system of equations:
diff(x(t), t) = y(t)*(1+x(t)^2+y(t)^2),
diff(y(t), t) = x(t)*(1+x(t)^2+y(t)^2).
Taking the critical point (0,0) and its eignvalues (-1,+1), it is clear that it is a saddle point.
Now, is there some trick to provide 'nice' initial conditions, such that the phase portrait shows clearly the solution curves?
I considered the following Ics:
[x(0)=-2, y(0)=-2], [x(0)=3, y(0)=0]
But it is not a good choice, because the solution curves are not nicely shown. How does one specify the initial conditions such that this happens? Is it just by trial and error?